Abramowitz & Stegun
Abramowitz & Stegun
Abramowitz & Stegun
Washington, D.C.
Preface to the Ninth Printing
The enthusiastic reception accorded the “Handbook of Mathematical
Functions” is little short of unprecedented in the long history of mathe-
matical tables that began when John Napier published his tables of loga-
rithms in 1614. Only four and one-half years after the first copy came
from the press in 1964, Myron Tribus, the Assistant Secretary of Com-
merce for Science and Technology, presented the 100,OOOth copy of the
Handbook to Lee A. DuBridge, then Science Advisor to the President.
Today, total distribution is approaching the 150,000 mark at a scarcely
diminished rate.
The successof the Handbook has not ended our interest in the subject.
On the contrary, we continue our close watch over the growing and chang-
ing world of computation and to discuss with outside experts and among
ourselves the various proposals for possible extension or supplementation
of the formulas, methods and tables that make up the Handbook.
In keeping with previous policy, a number of errors discovered since
the last printing have been corrected. Aside from this, the mathematical
tables and accompanying text are unaltered. However, some noteworthy
changes have been made in Chapter 2: Physical Constants and Conversion
Factors, pp. 6-8. The table on page 7 has been revised to give the values
of physical constants obtained in a recent reevaluation; and pages 6 and 8
have been modified to reflect changes in definition and nomenclature of
physical units and in the values adopted for the acceleration due to gravity
in the revised Potsdam system.
The record of continuing acceptance of the Handbook, the praise that
has come from all quarters, and the fact that it is one of the most-quoted
scientific publications in recent years are evidence that the hope expressed
by Dr. Astin in his Preface is being amply fulfilled.
LEWIS M. BRANSCOMB, Director
National Bureau of Standards
November 1970
Foreword
This volume is the result of the cooperative effort of many persons and a number
of organizations. The National Bureau of Standards has long been turning out
mathematical tables and has had under consideration, for at least IO years, the
production of a compendium like the present one. During a Conference on Tables,
called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramo-
witz of t,hat Division mentioned preliminary plans for such an undertaking, but
indicated the need for technical advice and financial support.
The Mathematics Division of the National Research Council has also had an
active interest in tables; since 1943 it has published the quarterly journal, “Mathe-
matical Tables and Aids to Computation” (MTAC),, editorial supervision being
exercised by a Committee of the Division.
Subsequent to the NBS Conference on Tables in 1952 the attention of the
National Science Foundation was drawn to the desirability of financing activity in
table production. With its support a z-day Conference on Tables was called at the
Massachusetts Institute of Technology on September 15-16, 1954, to discuss the
needs for tables of various kinds. Twenty-eight persons attended, representing
scientists and engineers using tables as well as table producers. This conference
reached consensus on several cpnclusions and recomlmendations, which were set
forth in tbe published Report of the Conference. There was general agreement,
for example, “that the advent of high-speed cornputting equipment changed the
task of table making but definitely did not remove the need for tables”. It was
also agreed that “an outstanding need is for a Handbook of Tables for the Occasional
Computer, with tables of usually encountered functions and a set of formulas and
tables for interpolation and other techniques useful to the occasional computer”.
The Report suggested that the NBS undertake the production of such a Handbook
and that the NSF contribute financial assistance. The Conference elected, from its
participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz,
J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to
help implement these and other recommendations.
The Bureau of Standards undertook to produce the recommended tables and the
National Science Foundation made funds available. To provide technical guidance
to the Mathematics Division of the Bureau, which carried out the work, and to pro-
vide the NSF with independent judgments on grants ffor the work, the Conference
Committee was reconstituted as the Committee on Revision of Mathematical
Tables of the Mathematics Division of the National Research Council. This, after
some changes of membership, became the Committee which is signing this Foreword.
The present volume is evidence that Conferences can sometimes reach conclusions
and that their recommendations sometimes get acted on.
V
,/”
VI FOREWORD
Active work was started at the Bureau in 1956. The overall plan, the selection
of authors for the various chapters, and the enthusiasm required to begin the task
were contributions of Dr. Abramowitz. Since his untimely death, the effort has
continued under the general direction of Irene A. Stegun. The workers at the
Bureau and the members of the Committee have had many discussions about
content, style and layout. Though many details have had t’o be argued out as they
came up, the basic specifications of the volume have remained the same as were
outlined by the Massachusetts Institute of Technology Conference of 1954.
The Committee wishes here to register its commendation of the magnitude and
quality of the task carried out by the staff of the NBS Computing Section and their
expert collaborators in planning, collecting and editing these Tables, and its appre-
ciation of the willingness with which its various suggestions were incorporated into
the plans. We hope this resulting volume will be judged by its users to be a worthy
memorial to the vision and industry of its chief architect, Milton Abramowitz.
We regret he did not live to see its publication.
P. M. MORSE, Chairman.
A. ERD~LYI
M. C. GRAY
N. C. METROPOLIS
J. B. ROSSER
H. C. THACHER. Jr.
JOHN TODD
‘C. B. TOMPKINS
J. W. TUKEY.
Handbook of Mathematical Functions
with
1. Introduction
The present Handbook has been designed to tional importance. Many numerical examples
provide scientific investigators with a compre- are given to illustrate the use of the tables and
hensive and self-contained summary of the mathe- also the computation of function values which lie
matical functions that arise in physical and engi- outside their range. At the end of the text in
neering problems. The well-known Tables of each chapter there is a short bibliography giving
Funct.ions by E. Jahnke and F. Emde has been books and papers in which proofs of the mathe-
invaluable to workers in these fields in its many matical properties stated in the chapter may be
editions’ during the past half-century. The found. Also listed in the bibliographies are the
present volume ext,ends the work of these authors more important numerical tables. Comprehen-
by giving more extensive and more accurate sive lists of tables are given in the Index men-
numerical tables, and by giving larger collections tioned above, and current information on new
of mathematical properties of the tabulated tables is to be found in the National Research
functions. The number of functions covered has Council quarterly Mathematics of Computation
also been increased. (formerly Mathematical Tables and Other Aids
The classification of functions and organization to Computation).
of the chapters in this Handbook is similar to The ma.thematical notations used in this Hand-
that of An Index of Mathematical Tables by book are those commonly adopted in standard
texts, particularly Higher Transcendental Func-
A. Fletcher, J. C. P. Miller, and L. Rosenhead. tions, Volumes 1-3, by A. ErdBlyi, W. Magnus,
In general, the chapters contain numerical tables, F. Oberhettinger and F. G. Tricomi (McGraw-
graphs, polynomial or rational approximations Hill, 1953-55). Some alternative notations have
for automatic computers, and statements of the also been listed. The introduction of new symbols
principal mathematical properties of the tabu- has been kept to a minimum, and an effort has
lated functions, particularly those of computa- been made to avoid the use of conflicting notation.
IX /-
.
X INTRODUCTION
4. Interpolation
The tables in this Handbook are not provided Let us suppose that we wish to compute the
with differences or other aids to interpolation, be- value of xeZ&(x) for x=7.9527 from this table.
cause it was felt that the space they require could We describe in turn the application of the methods
be better employed by the tabulation of additional of linear interpolation, Lagrange and Aitken, and
functions. Admittedly aids could have been given of alternative methods based on differences and
without consuming extra space by increasing the Taylor’s series.
intervals of tabulation, but this would have con- (1) Linear interpolation. The formula for this
flicted with the requirement that linear interpola- process is given by
tion is accurate to four or five figures.
For applications in which linear interpolation jp= (1 -P)joSPfi
is insufficiently accurate it is intended that
Lagrange’s formula or Aitken’s method of itera- where jO, ji are consecutive tabular values of the
tive linear interpolation3 be used. To help the function, corresponding to arguments x0, x1, re-
user, there is a statement at the foot of most tables spectively; p is the given fraction of the argument
of the maximum error in a linear interpolate, interval
and the number of function values needed in p= (x--x0>/(x1-~0>
Lagrange’s formula or Aitken’s method to inter-
polate to full tabular accuracy. and jP the required interpolate. In the present
As an example, consider the following extract instance, we have
from Table 5.1.
jo=.89717 4302 ji=.89823 7113 p=.527
zez El (2) ze*El (z)
775 . 89268 7854 d0 . 89823 7113 The most convenient way to evaluate the formula
;:; : 89384
89497 6312
9666 g. I .89927
90029 7306
7888
on a desk calculating machine is.to set o and ji
in turn on the keyboard, and carry out t d e multi-
E : 89608
89717 4302
8737 8: 4
ix :.90227
90129 4695
60”3 plications by l-p and p cumulatively; a partial
check is then provided by the multiplier dial
[ 1‘453 reading unity. We obtain
The numbers in the square brackets mean that j.6z,E.‘;9;72;&39717 4302)+.527(.89823 7113)
the maximum error in a linear interpolate is
3X10m6, and that to interpolate to the full tabular
accuracy five points must be used in Lagrange’s Since it is known that there is a possible error
and Aitken’s methods. of 3 X 10 -6 in the linear formula, we round off this
8 A. C. Aitken On inte elation b iteration of roportional
result to .89773. The maximum possible error in
parts, with.
out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676 (1932). this answer is composed of the error committed
INTRODUCTION XI
by the last roundingJ that is, .4403X 10m5, plus The numbers in the third and fourth columns are
3 X lo-‘, and so certainly cannot exceed .8X lo-‘. the first and second differences of the values of
(2) Lagrange’s formula. In this example, the xezEl(x) (see below) ; the smallness of the second
relevant formula is the 5-point one, given by difference provides a check on the three interpola-
tions. The required value is now obtained by
f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi linear interpolation :
+A&)fa
Tables of the coefficients An(p) are given in chapter fn=.3(.89772 9757)+.7(.89774 0379)
25 for the range p=O(.Ol)l. We evaluate the
formula for p=.52, .53 and .54 in turn. Again, = 239773 7192.
in each evaluation we accumulate the An(p) in the
multiplier register since their sum is unity. We
now have the following subtable. In cases where the correct order of the Lagrange
polynomial is not known, one of the prelimina
x m=&(x) interpolations may have to be performed witT
7.952 .89772 9757 polynomials of two or more different orders as a
10622
7.953 .89774 0379 -2 check on their adequacy.
10620 (3) Aitken’s method of iterative linear interpola-
7.954 .89775 0999 tion. The scheme for carrying out this process
in the present example is as follows:
Here
1 Yo 20-x S2fl
yo,n=-
x.--20 Yn x,-x
safz
1 Yo.1 x,-x
Yo.1 ,n=- x,-x
G--z1 l/O.” wa
,,/
XII INTRODUCTION
10Qf.6,= .473(89717 4302) + .061196(2 2754) - .012(34) can be used. We first compute as many of the
+ .527(89823 7113) + .063439(2 2036) - .012(39) derivatives ftn) (~0) as are significant, and then
= 89773 7193. evaluate the series for the given value of 2.
An advisable check on the computed values of the
We may notice in passing that Everett’s derivatives is to reproduce the adjacent tabular
formula shows that the error in a linear interpolate values by evaluating the series for z=zl and x1.
is approximately
In the present example, we have
mPwfo+ F2(P)wl= m(P) + ~2(P)lk?f0+wJ
f(x) =xeZEt(x)
Since the maximum value of IEz(p)+Fz(p)I in the f’(z)=(l+Z-‘)f(Z)-1
range O<p<l is fd, the maximum error in a linear f”(2)=(1+2-‘)f’(Z)--Z-Qf(2)
interpolate is approximately f”‘(X) = (1 -i-z-y’(2) -22~Qf’(5) +22-y(2).
~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ ; - .00113
.01074 0669
7621 -.ooooo
.00056 6033
3159 53
+(~-x,)q$+ . . . 3 .00012 1987 .ooooo
.a9773 7194
0017 9
5. Inverse Interpolation
With linear interpolation there is no difference The desired z is therefore
in principle between direct and inverse interpola-
tion. In cases where the linear formula rovides z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083 57
an insufficiently accurate answer, two met fl ods are
available. We may interpolate directly, for To estimate the possible error in this answer,
example, by Lagrange’s formula to prepare a new we recall that the maximum error of direct linear
table at a fine interval in the neighborhood of the interpolation in this table is Aj=3X lOwe. An
approximate value, and then apply accurate approximate value for dj/dx is the ratio of the
inverse linear interpolation to the subtabulated first difference to the argument interval (chapter
values. Alternatively, we may use Aitken’s 25), in this case .OlO. Hence the maximum error
method or even possibly the Taylor’s series in x is approximately 3XlO-e/(.OlO), that is, .0003.
method, with the roles of function and argument (ii) Subtabulation method. To improve the
interchanged. ap roximate value of x just obtained, we inter-
It is important to realize that the accuracy of po Pate directly for p=.70, .7l and .72 with the aid
an inverse interpolate may be very different from of Lagrange’s 5-point formula,
that of a direct interpolate. This is particularly
X xe=El (x) 6 QQ
true in regions where the function is slowly
varying, for example, near a maximum or mini- 8. 170 . 89999 -.-_
1 0151
mum. The maximum precision attainable in an 8.171 . 90000 3834 -2
inverse interpolate can be estimated with the aid of 1 0149
the formula 8. 172 90001 3983
AxmAj/df
dx
Inverse
gives
linear interpolation in the new table
4 : 89927
90129 6033
7888 8. 31 8. 17083
17023 5712
1505 8. 1706,l 9521 -. . 00072
00129 2112
6033
3 . 89823 7113 8. 0 8. 17113 8043 2 5948 8. 17062 2244 -. 00176 2887
% : 90227
89717 4302
4695 7. 49
8. 8. 16992
17144 0382
9437 21 8142
7335 231
415 8. 17062 2318
265 -. .00227
00282 5G98
4695
The estimate of the maximum error in this discrepancy in the highest interpolates, in this
result is the same as in the subtabulation method. case xo .I ,2.3 A, and ZLI .2.8 .s.
An indication of the error is also provided by the I
6. Bivariate Interpolation
Bivariate interpolation is generally most simply interpolation is then carried out in the second
performed as a sequence of univariate interpola- direction.
tions. We carry out the interpolation in one An alternative procedure in the case of functions
direction, by one of the methods already described, of a complex variable is to use the Taylor’s series
for several tabular values of the second argument expansion, provided that successive derivatives
in the neighborhood of its given value. The of the function can be computed without much
interpolates are differenced as a check, and difficulty.
Stability-decreasing 7t
Jn+*-~Jn+J.-l=O
P”(X), P.,(z) @<l)
nE,+,+xE,,=e-=.
Qnh), Q:(x)
Particularly for automatic work, recurrence re- J&4, Z.@)
lations provide an important and powerful com- Jn+Hcd , Zn+&)
puting tool. If the values of P&r) or Jn(z) are Em(z) (n >r)
known for two consecutive values of n, or E',(z) F,,(t, p) (Coulomb wave function)
is known for one value of n, then the function may
be computed for other values of n by successive Illustrations of the generation of functions from
applications of the relation. Since generation is their recurrence relations are given in the pertinent
carried out perforce with rounded values, it is chapters. It is also shown that even in cases
vital to know how errors may be propagated in where the recurrence process is unstable, it may
the recurrence process. If the errors do not grow still be used when the starting values are known
relative to the size of the wanted function, the to sufficient accuracy.
process is said to be stable. If, however, the Mention must also be made here of a refinement,
relative errors grow and will eventually over- due to J. C. P. Miller, which enables a recurrence
whelm the wanted function, the process is unstable. process which is stable for decreasing n to be
It is important to realize that st,ability may applied without any knowledge of starting values
depend on (i) the particular solution of the differ- for large n. Miller’s algorithm, which is well-
ence equation being computed; (ii) the values of suited to automatic work, is described in 19.28,
x or other parameters in the difference equation; Example 1.
XIV INTRODUCTION
8. Acknowledgments
The production of this volume has been the bibliographic references and assisted in preparing
result of the unrelenting efforts of many persons, the introductory material.
all of whose contributions have been instrumental Valuable assistance in the preparation, checkin
in accomplishing the task. The Editor expresses and editing of the tabular material was receive IFi
his thanks to each and every one. from Ruth E. Capuano, Elizabeth F. Godefroy,
The Ad Hoc Advisory Committee individually David S. Liepman, Kermit Nelson, Bertha H.
and together were instrumental in establishing Walter and Ruth Zucker.
the basic tenets that served as a guide in the forma- Equally important has been the untiring
tion of the entire work. In particular, special cooperation, assistance, and patience of the
thanks are due to Professor Philip M. Morse for members of the NBS staff in handling the myriad
his continuous encouragement and support. of detail necessarily attending the publication
Professors J. Todd and A. Erdelyi, panel members of a volume of this magnitude. Especially
of the Conferences on Tables and members of the appreciated have been the helpful discussions and
Advisory Committee have maintained an un- services from the members of the Office of Techni-
diminished interest, offered many suggestions and cal Information in the areas of editorial format,
carefully read all the chapters. graphic art layout, printing detail, preprinting
Irene A. Stegun has served eff ectively as associate reproduction needs, as well as attention to pro-
editor, sharing in each stage of the planning of motional detail and financial support. In addition,
the volume. Without her untiring efforts, com- the clerical and typing stafI of the Applied Mathe-
pletion would never have been possible. matics Division merit commendation for their
Appreciation is expressed for the generous efficient and patient production of manuscript
cooperation of publishers and authors in granting copy involving complicated technical notation.
permission for the use of their source material. Finally, the continued support of Dr. E. W.
Acknowledgments for tabular material taken Cannon, chief of the Applied Mathematics
wholly or in part from published works are iven Division, and the advice of Dr. F. L. Alt, assistant
on the first page of each table. Myrtle R. Ke Yiling- chief, as well as of the many mathematicians in
ton corresponded with authors and publishers the Division, is gratefully acknowledged.
to obtain formal permission for including their
material, maintained uniformity throughout the M. ABRAMOWITZ.
1. Mathematical Constants
DAVID S. LIEPMAN ’
Contents
Page
Table 1.1. Mathematical Constants ............... 2
enr e--nr
1) 2.3140 69263 27792 69006 - 2) 4.3213 91826 37722 49774
2) 5.3549 16555 24764 73650 - 3) 1.8674 42731 70798 88144
4) 1.2391 64780 79166 97482 - 5) 8.0699 51757 03045 99239
5) 2.8675 13131 36653 29975 - 6) 3.4873 42356 20899 54918
6) 6.6356 23999 34113 42333 - 7) 1.5070 17275 39006 46107
8j 1.5355 29353 95446 69392 - 9i 6. 5124 12136 07990 07282
9) 3.5533 21280 84704 43597 -1oj 2.8142 68457 48555 27211
LO) 8.2226 31558 55949 95275 -11) 1.2161 55670 94093 08397
12) 1.9027 73895 29216 12917 -13) 5.2554 85176 00644 85552
13) 4.4031 50586 06320 29011 -14j 2.2711 01068 32409 38387
In n log10 12
0.6931 47180 55994 53094 172321 0102 99956 63981 19521 37389
1.0986 12288 66810 96913 952452 7712 12547 l!id62 43729 50279
1.3862 94361 11989 344642 0205 99913 27962 39042 74778
1.6094 37912 43410 xz::: 007593 9897 00043 36018 80478 62611
1.7917 59469 22805 50008 124774 7815 12503 83643 63250 87668
1.9459 10149 05531 33051 053527 4509 80400 14256 83071 22163
2.0794 41541 67983 59282 516964 0308 99869 91943 58564 12167
2.1972 24577 33621 93827 904905 5424 25094 39324 87459
2.3025 85092 99404 56840 179915 0000 00000 00000 00000 ~:~::
2.3978 95272 79837 05440 619436 0413 92685 15822 50407 50200
2.5649 49357 46153 67360 534874 1139 43352 30683 67692
2.8332 13344 05621 60802 495346 2304 48921 37827 39285 ::%i
2.9444 38979 16644 04600 090274 2787 53600 95282 89615
3.1354 94215 92914 96908 067528 3617 27836 01759 X%f
3.3672 95829 98647 40271 832720 4623 97997 89895 %Ei 32847
3.4339 87204 48514 62459 291643 4913 61693 83427 26796 66704
3.6109 17912 64422 44443 680957 1. 5682 01724 06699 49968 08451
72066 70430 78038 667634 1.6127 83856 71973 54945 09412
I: % 00115 69356 24234 728425 1.6334 68455 57958 65264 05088
*See page xx.
MATHEMATICAL CONSTANTS
In n log10 n
3.8501 47601 71005 85868 209507 1.6720 97857 93571 74644 14219
3. 9702 91913 55212 18341 444691 1.7242 75869 60078 90456 32992
4.0775 37443 90.571 94506 160.504 1.7708 52011 64214 41902 60656
4.1108 73864 17331 12487 513891 1.7853 29835 01076 70338 85749
4. 2046 92619 39096 60596 700720 1.8260 74802 70082 64341 49132
4.2626 79877 04131 54213 294545 1.. 8512 58348 71907 52860 92829
4. 2904 59441 14839 11290 921089 1. 8633 22860 12045 59010 74387
4.3694 47852 46702 14941 729455 I.. 8976 27091 29044 14279 94821
4.4188 40607 79659 79234 754722 1. 9190 78092 37607 39038 32760
4.4886 36369 73213 98383 178155 1. 9493 90006 64491 27847 23543
4. 5747 10978 50338 28221 167216 1. 9867 71734 f ?624 48517 84362
1. 1447 29885 84940 01741 43427 loglog (-1) 4.9714 98726 94133 85435 12683
(-1) 9. 1893 85332 04672 74178 03296 logl0e (-1) 4.3429 44819 03251 82765 11289
nln 10
2.3025 85092 99404 56840 17991 3. 1415 92653 5s”9”19 32384 62643
4.6051 70185 98809 13680 35983 6. 2831 85307 17958 64769 25287
6.9077 55278 98213 70520 53974 !a. 4247 77960 76937 97153 87930
9.2103 40371 97618 27360 71966 ( 1) 1. 2566 37061 43591 72953 85057
( 1) 1. 1512 92546 49702 28420 08996 ( 1) 1.5707 96326 79489 66192 31322
( 1) 1. 3815 51055 79642 74104 10795 ( 1) 1. 8849 55592 15387 59430 77586
( 1) 1. 6118 09565 09583 19788 12594 ( 1) 2. 1991 14857 51285 52669 23850
( 1) 1. 8420 68074 39523 65472 14393 ( 1)2.5132 74122 87183 45907 70115
( 1) 2. 0723 26583 69464 11156 16192 ( 1) 2.8274 33388 23081 39146 16379
7P n *-”
3.1415 92653 58979 32384 62643 -1) 3. 1830 98861 83790 67153 77675
9.8696 04401 08935 86188 34491 ; -1) 1.. 0132 11836 42337 77144 38795
( 1) 3. 1006 27668 02998 20175 47632 3 -2j 3.2251 53443 31994 89184 42205
( 1) 9. 7409 09103 40024 37236 44033 4 -2) 1. 0265 98225 46843 35189 15278
C 2) 3.0601 96847 85281 45326 27413 -3) 3.2677 63643 05338 54726 28250
( 2j 9.6138 91935 75304 43703 02194 x -31 1. 0401 61473 29585 22960 89838
( 3) 3.0202 93227 77679 20675 14206 7 -4j 3. 3109 36801 77566 76432 59528
( 3) 9.4885 31016 07057 40071 28576 -4) 1. 0539 03916 53493 66633 17287
( 4) 2.9809 09933 34462 11666 50940 i -5) 3.3546 80357 20886 91287 39854
( 4) 9.3648 04747 60830 20973 71669 10 -5) 1. 0678 27922 68615 33662 04078
1. 5707 96326 79489 66192 31322 4.7123 88980 38468 98576 93965
1. 0471 97551 19659 77461 54214 s-ii 4.1887 90204 78639 09846 16858
(-1) 7.8539 81633 97448 30961 56608 ;h; J2 4.4428 82938 15836 62470 15881 *
1. 7724 53850 90551 60272 98167 (-1) 5. 6418 95835 47756 28694 80795
1.4645 91887 56152 32630 20143 r-1 13 (- 1) 6. 8278 40632 55295 68146 70208
1.3313 35363 80038 97127 97535 ?r-l/4 (-1) 7. 5112 55444 64942 48285 87030
2. 1450 29397 11102 56000 77444 +f3 (-1) 4. 6619 40770 35411 61438 19885
2.3597 30492 41469 68875 78474 ,-%I4 (- 1) 4. 2377 72081 23757 59679 10077
5. 5683 27996 83170 78452 84818 *-3/Z (-1) 1. 7958 71221 25166 56168 90820
t: 1) 2. 2459 15771 83610 45473 42715 r--c ( -2) 4.. 4525 26726 69229 06151 35273
2. 5066 28274 63100 05024 15765 (2r)-'12 (-1) 3. 9894 22804 01432 67793 99461
1. 2533 14137 31550 02512 07883 (a/,)"2 (-1) 7. 9788 45608 02865 35587 98921
2. 2214 41469 07918 31235 07940 2'f2/?r (-1) 4.5015 81580 78533 03477 75996
57. 2957 79513 08232 08767 98155’ 1’ 0. 0002 90888 20866 57215 96154r
0. 0174 53292 51994 32957 69237r 1” 0. 0000 04848 13681 10953 59936r i
0. 5772 15664 90153 28606 06512 In Y -0. 5495 39312 98164 48223 37662
Contents
Page
Table 2.1. Common Units and Conversion Factors . . . . . . . . . 6
Table 2.2. Names and Conversion Factors for Electric and Magnetic
Units . . . . . . . . . . . . . . . . . . . . . . . 6
Table 2.3. Adjusted Values of Constants . . . . . . . . . . . . . 7
Table 2.4. Miscellaneous Conversion Factors. . . . . . . . . . . . 8
Table 2.5. Conversion Factors for Customary U.S. Units to Metric
Units . . . . . . . . . . . . . . . . . . . . . . . 8
Table 2.6. Geodetic Constants . . . . . . . . . , . . . . . . . . 8
The tables in this chapter supply some of rise to the CGS system, often used in physics
the more commonly needed physical con- and chemistry.
stants and conversion factors. Table 2.1. Common Units and Conversion
All scientific measurements in the fields of Factors
mechanics and heat are based upon four in-
ternational arbitrarily adopted units, the
magnitudes of which are fixed by four agreed ~
on standards:
Length- the meter -fixed by the vacuum The SI unit of electric current is the ampere
wavelength of radiation corresponding to the defined by the equation 2r,,,Z1ZJ4~= F giving
transition 2Plu-5Da of krypton 86 the force in vacua per unit length between
(1 meter - 1650763.73h). two infinitely long parallel conductors of in-
finitesimal cross-section. If F is in newtons,
Mass-the kilogram -fixed by the interna- and rrn has the numerical value 477 X lo-‘,
tional kilogram at S&vres, France. then I1 and Zr are in amperes. The custom-
Time-the second- fixed as l/31,556,925.9747 ary equations define the other electric and
of the tropical year 1900 at 12” ephemeris magnetic units of SI such as the volt, ohm,
time, or the duration of 9,19‘2,631,770 cycles farad, henry, etc. The force between elec-
of the hyperfine transition frequency of cesi- tric charges in a vacuum in this system is
urn 133. given by Q, Qn/4nrerg= F, re having the nu-
Temperature-the degree-fixed on a ther- merical value 10r/4nc2 where c is the speed
modynamic basis by taking the temperature of light in meters per second (r,= 8.854
for the triple point of natural water as 273.16 x 10-12).
“K. (The Celsius scale is obtained by adding The CGS unrationalized system is obtained
-273.15 to the Kelvin scale.) by deleting 4n in the denominators in these
Other units are defined in terms of them by equations and expressing F in dynes, and r
assigning the value unity to the proportion- in centimeters. Setting r,,, equal to unity de-
ality constant in each defining equation. The fines the CGS unrationalized electromagnetic
entire system, including electricity units, is system (emu), re then taking the numerical
called the Systi.?me International d’unitds value of 1/c2. Setting re equal to unity de-
(SI). Taking the l/100 part of the meter as fines the CGS unrationalized electrostatic
the unit of length and the l/1000 part of the system (esu), r,,, then taking the numerical
kilogram as the unit of mass, similarly, gives value of l/cz.
Table 2.2. Names and Conversion Factors for Electric and Magnetic Units
= = = =
Quantity SI emu esu SI unit/ SI unit/
name name I name emu unit esu unit
- -
Current ampere I tbampere statampere 10-l -3x 100
Charge coulomb 1tbcoulomb statcoulomb LO-’ -3 x 109
Potential volt abvolt statvolt 108 -(1/3)X 10-Z
Resistance ohm abohm statohm 100 -(1/9)X 10-u
Inductance henry centimeter 100 %(1/9)X 10-l’
Capacitance farad centimeter 10-g -9x 10”
Magnetizing force amp. turns/ oersted 4*x IO-3* -3 x loo*
meter
Magnetomotive force amp. turns gil bert 4rX lo-I* -3/10**
Magnetic flu* weber maxwell __---___----_- 108 -(1/3)X 10-z
Magnetic flux density tesla gauss _-______-_____ 10’ -(1/3)X 10-B
Electric displacement --._-_-______ I_-..____..____ 10-J* -3x 105*
- - -
Example: If the value assigned to a current is 100 amperes its value in abamperes is 100X10-‘=lO.
*Divide this number by 4?r if unrationalized system is involved; other numbers are unchanged.
6
3. Elementary Analytical Methods
MILTON ABRAMOWITZ l
Contents
Page
Elementary Analytical Methods ................. 10
3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and
Geometric Progressions; Arithmetic, Geometric, Harmonic
and Generalized Means ............... 10
3.2. Inequalities ...................... 10
3.3. Rules for Differentiation and Integration ......... 11
3.4. Limits, Maxima and Minima .............. 13
3.5. Absolute and Relative Errors .............. 14
3.6. Infinite Series ..................... 14
3.7. Complex Numbers and Functions ............ 16
3.3. Algebraic Equations .................. 17
3.9. Successive Approximation Methods ........... 18
3.10. Theorems on Continued Fractions ............ 19
Numerical Methods ....................... 19
3.11. Use and Extension of the Tables ............ 19
3.12. Computing Techniques. ................ 19
References ............................ 23
Table 3.1. Powers and Roots . . . . . . . . . . . . . . . . . . 24
n’“, k=l(l)lO, 24, l/2, l/3, l/4, l/5
n=2(1)999, Exact or 10s
The author acknowledges the assistance of Peter J. U’Hara and Kermit C. Nelson in
the preparation and checking of the table of powers and roots.
3.1.18 liiM(t)=G
For a more extensive table see chapter 24. 3.2.2 min. a<M(t)<mnx. a
*See page II.
10
ELEMENTARY ANALYTICAL METHODS 11
3.2.3 min. a<G<max. a Minkowski’s Inequality for Sums
equality holds if all ak are equal, or t<O If p>l and &, bk>O for all x:,
and an an is zero 3.2.12
3.2.4 M(t)<&!(s) if t<s unless all ak are equal,
or s<O and an an is zero. (& (ak+bk)p~‘p<(& a;)“‘+($ bi$“j
Hiilder’s Inequality for Sums equality holds if and only if g(z) =cf(x) (c=con-
stant>O).
If ;++,p>1, fj>l
3.3. Rules for Differentiation and Integration
Derivatives
equality holds if and only if jgCs)I=clflr)Ip-’ Leibniz’s Theorem for Differentiation of an Integral
(c=constant>O).
If p=p=2 we get 3.3.7
b(c)
d
Schwa&s Inequality a(c) f (2, wx
3.2.11 & s
b(c) b
3 a(cj ,J(x,c)dx+f@, 4 $--f (a, 4 2
s
12 ELEMENTARY ANALYTICAL METHODS
Leibniz’s Theorem for Differentiation of a Product The following formulas are useful for evaluating
P(x)dx
3.3.8
S (ux”+ fJx+cy
where P(x) is a polynomial and
3.3.9
dX
-&=1g
(b2--4uc<O)
a
2az+b- (b2--4uc)t
3.3.17
I 2azfbf (P-4acy
3.3.10
#x -d2y
dy2=dr2
dy -3
0 zc
(b2-4m>O)
3.3.11 $= -B g-3 (g--j ($)-"
3.3.18
=2az+b -2
(P-4ac=O)
3.3.19
Integration by Parts
S S
3.3.12 jidu=w+dti 3.3.20
S (a+ bx$c+dxj=k
c+dx
bc In I-a+bx I (ad # bc)
S___
3.8.13 jkudx=(jidx) v-s(judx) 2 dx dx 1
3.3.21 =- arctan E!
u2+b2;C2 ub U
Integrals
(Integration
of Rational Algebraic
3.3.15
S $&In
S )ax+b) 3.3.25
S 1 W<O)
dx -d(a+ bx) 1’2
3.3.26 =h2 arctan
t(u+bx) (c+dx)11’2 C b(c+dx)
3.3.27 2bdx+ad+ bc
=+ arcsin (b>O, d<O)
bc-ad >
dx
3.3.29
S (a+bx)“P(c+dx)=[d(bc~ud)]1~2
arctan ~~~~-J” (d(u&-bc)<O)
3.3.30 d(u+bx)1’2-[d(ad-bc)]1’2
=[d(ad&]1/2 In d(a+bx)1’2+[d(ud-bc)]“2 @b-J--4>O)
I
ELEMENTARY ANALYTICAL METHODS 17
If zn=un+ivn, then ~~+l:=u,+,+iv~+~ where 3.8. Algebraic Equations
0-
3.7.24
21,2=- 2”, j-k gf, p= b2-4ac,
If --?r<~< ?r this is the principal root. The 3.8.2 Given Z3+a2z2+ulz+a0==0, let
other root has the opposite sign. The principa:
root is given by
3.7.27 d=[+(r+x)]+&-i[$(r--x)]*=ufiv where
2uv=y and where the ambiguous sign is taken tc If $+P>O, one real root and a pair of complex
be the same as the sign of y. c.onjugate roots,
3 . 7 . 28 Zl/n,Tl/nefe/n , (principal root if - ?r<0 5 7r) $+9=0, all roots real and at least two are
Other roots are Pet(B+2rn’ln (k=l,2,3, a . ., n-1) equal, .
p3+r2<0, all roots real (irreducible case).
Inequalities
Let
3.7.29 l&l-I221 _<1z1~~2111211+I~21 sl=[r-+(q3+r2)q+, sz=[T’-((p3+?3*]*
I I then
Complex Functions, Catwhy-Riemann Equations
f(z)=f(x+iy)=u(x,yy)+iv(x,y)whereu(x,y),v(z,y
aA real, is unaly& at those points z=z+$ a
which
au
-=-, av au
-=-- av
3.7.30
a~ by by ax
If z=Tefff,
If zl, z2, z3 are the roots of the cubic equation
3.7.31 g=; ;, ; f$=-$
Z~+Z~+Z~=-CIC~
Laplace’s Equation
~~~2+~$,+&~,:=~~
3.9.5
z2 j= -a3, z2 j2,2t= -&,
Newton’s Rule
3.9.1 Let z=zl be an approximation to x=[ will converge quadratically to x=5: (if instead of
where f(t) =0 and both x1 and [ are in the interval the condition (2) above),
a$r<b. We define
(1) Monotonic convergence, f(zO)r’(zo) >0
GI+1=G+C&n) (n=l, 2, . . .). and f’(s), j”(z) do not change sign in the
interval (Q, t), or
Then, if f’(z)>0 and the constants cn are
(2) Osdato y conwgence, f(xJf” (x0)<0
negative and bounded, the sequence x,, converges
monotonically to the root [. and f’(s), f”(z) do not change sign in the
If c,,=c=constant<O and f’(z)>O, then the interval (x0, x1), xo<E<xl.
process converges but not necessarily monotoni-
Newton’s Method Applied to Real nth Roots
cally.
Degree of Convergence of an Approximution Process 3.9.6 Given x”=N, if zk is an approximation
x=N’l” then the sequence
3.9.2 Let zl, z2, x3, . . . be an infinite sequence
of approximations to a number f. Then, if xk+l=- ; [$i+(n-l)xk]
1%n+~-~I<&n-tlk, (n=l, 2, . . .)
will converge quadratically to a.
where A and k are independent of n, the sequence
is said to have convergence of at most the kth
degree (or order or index) to [. If k=l and
A<1 the convergence is linear; if k=2 the con-
vergence is quadratic.
Regula Falsi (False Position)
Aitken’e G-Process for Acceleration of Sequences
3.9.3 Given y=f(z) to find 5 such that f(.$)=o,
choose ~0 and x1 such that f(rO) and f(zl) have 3.9.7 If 2k, &+I, zri+2 are three successive iterates
opposite signs and compute in p, sequence converging with an error which is
approximately in geometric progression, then
f 1~o-JoX,
x*=x, -Hi f,= jlVfO * (5k--k+1)*=;tk~k+2-2:+1.
&=xk-
A*& A*Xk ’
Then continue with x2 and either of x0 or x1 for
which f(;ro) or j(zl) is of opposite sign to f(zl).
Regula falsi is equivalent to inverse linear inter- is an improved estimate of x. In fact, if zk”x+*
polation, OGtk) then Z=s+O(P), Ix\<~.
ELEMENTARY ANALYTICAL METHODS
*
3.10. Theorems on Continued Fractions (4) A,B,_l-A,-lB,=(-l)n-’ kiI al;
Definitions (5) For every n>O,
j,=b, 1 claI ClC& c2c3a3 &I-lW%
c,bl+ czbz+ caba+ ’ * * c,b,’
(6) l+b,+b,b,+ . . . +bzb3. . . b,
=-- 1 bz _- b3 b,
l- b,+l- b,+l- * ’ ‘--b,+l
=b,,+&e&. ..
d+$+ . . . +;=-& --& . . . $yu
I 1 2 n1 n
If the number of terms is finite, j is called a
ternlinating continued fraction. If the number
1
---
a0
x+A ...l
aof aoGa2
t(-1,n----5
_ . . . a,
of ternls is infinite, j is called an infinite cont’inued
1 aox _- a12 %-1X
fraction and the terminating fraction =- ___
uo+ al-x+ I12-xf * . . +un-2
(2) If j.=+
n
A,=b,A,-~+a,A,-2
Bn=bnBn-l+anBn-2
where A-1=l, A,,=bo, B-1=0, B,=l. 0 .2 .4 .6 .8
FIGURE 3.1
1 i y:=xn*
*n=0,,5t 29 1, 2, 5.
Numerical Methods
3.11. Use and Extension of the Tables I Linear interpolation in Table 3.1 gives
(919.826)“4-5.507144.
Example 1. Computti xl9 and x4’ for x=29 By Newton’s method for fourth roots with
using Table 3.1. N=919.826,
3p=x9. x10
1
4 ~7~3+3(5.507144)-]=5.50714 3845
= (1.45071 4598. 1013)(4.207072333. 1014) [ .
=6.10326 1248. 102’ Thus,
Repetition yields the same result.
x4’= (x*4)2/x
~“~=5.50714 3845/10$=1.74151. 1796,
= (1.25184 9008. 1036)2/29 ~-~“=zt/x=.18983 05683.
=5.40388 2547. lO6*
3.12. Computing Techniques
Example 2. Compute x-3’4 for x=9.19826.
Example 3. Solve the quadratic equation
(9.19826)“‘= (919.826/100)1’4= (919.826)1’4/10t x2- 18.2x+.056 given the coeflicients as 18.2 f .l,
*see page II.
..
<
20 ELEMENTARY ANALYTICAL METHODS
.056f .OOI. From 3.8.1 the solution is Example 5. Solve the cubic equation x3- 18.12
z=4(18.2f-[(18.2)2-4(.05B)]:) -34.8=0.
To use Newton’s method we first form the
=3(18.2~[:J31]t)=3(18.2~18.~) table of f(z)=23-1S.1r-34.8
= 18.1969, .OOJ
4” -43.2
f(x) .
The smaller root may be obtained more accurately
from 5 - .3
* .05fi/18.1969= .0031& .OOOl. 6 72.6
7 181.5
Example 4. Compute (-3 + .0076i)i.
From 3.7.26, (-3+.0976i)~=u+iv where We obtain by linear inverse interpolation:
O-(-.3)
Y r!y *, j”= (t”+y’)t x,=5+ 72.6-(-.3)=5’oo4’
u=2G? I,-= ( >
Thus Using Newton’s method, f’(x) =3x2- 18.1 we get
r=[(-3)2+(.0076)2]~=(9.00005776)~=3.00000 9627
21 =zo-f&J/f’ (d
Ij= 3.00000 9627- (-3)
2 1 f=
.73205 2196 =5.004- C--.07215 9936jz5
57.020048 '
00526
'
.0076 Repetition yields x1=5.00526 5097. Dividing
u=&=2(1.73205 21g6)=.00219 392926
f(x) by x-5.00526 5097 gives x2+5.00526 5097x
We note that the principal square root has been i-6.95267 869 the zeros of which are -2.50263 2549
computed. f.83036 8OOi.
-
Example 6. Solve the quartic equation I
~‘-2.37752 4922x3+6.07350 5741.x’
-11.17938 023s+9.05265 5259=0. We seek that value of y, for which y(nJ =O.
Resolution Into Quadratic Factors Inverse interpolation in ~(a,) gives ~(a,) =O for
(22 + p12 + qd w + p2x + 92) pl -2.003. Then,
by Inverse Interpolation
QI Qz PI P2 Y h)
2. 0041 4. 51706 7640 -2. 55259 257 17506 765 .00078 552
2. 0042 4.,51684 2260 -2. 55282 851 . 17530 358 . 00001 655
2. 0043 4. 51661 6903 -2. 55306 447 . 17553 955 -. 00075 263
Qz PI P2 Y (Ql)
_.
2. 00420 2152 4. 51683 7410 -2. 55283 358 17530 8659 -. 00000 0011
-
4 ELEMENTARY ANALYTICAL METHODS 21
Double Precision Multiplication and Division on a Method @)--If N and d are numbers each not
Desk Calculator
more than 19 digits let N=N1+NolOQ, d=dI+
Example 7. MultiplyM=20243 97459 71664 32102 dolO where No and do contain 10 digits and N,
by m=69732 82428 43662 95023 on a 10X10X20 and dl not more than 9 digits. Then
desk calculating machine. N NolOQ+N, zs- 1 [.N-y]
Let MO=20243 97459, Ml=71664 32102, mO= d=,lOQ+d, dolO
69732 82428, ml=43662 95023. Then Mm= Here
M0m0102’+ (Mom,+Mlmo) 101o+M~ml.
(1) Multiply ,W1m1=31290 75681 96300 28346 N= 14116 69523 40138 1761,
and record the digits 96300 28346 appearing in d=20243# 97459 71664 3210
positions 1 to 10 of the product dial. No= 14116 69523, do=20243 97459,
(2) Transfer the digits 31290 75681 from posi- d,=71664 3210
tions 11 to 20 of the product dial to positions 1 to
10 of the product dial. (1) NodI= 10116 63378 42188 8830 (productdial).
(3) Multiply cumulatively M,mo+Mom,+31290 (2) (Nod,)/do=49973 55504 (quotient dial).
75681=58812 67160 12663 25894 and record the (3) N- (N&/d,= 14116 69!;22 90164 62106
digits 12663 25894 in positions 1 to 10. (product dial).
(4) Transfer the digits 58812 67160 from posi- (4) [N- (NodI)/do]/dolOQ= .69732 82428=first 10
tions 11 to 20 to positions 1 t,0 10. digits of quotient in quotient dial. Remainder
(5) Multiply cumulatively Mom,+58812 67160 =r=O8839 11654, in positions 1 to 10 of product
=14116 69523 40138 17612. The results as ob- dial.
tained are shown below, (5) r/(d010Q)=.43662 9502.10-“O=next 9 digits of
9630028346 quotient. N/d=.69732 82428 43662 9502. This
1266325894 method may be modified to give the quotient of
14116695234013817612 20 digit numbers. Method (1) may be extended
141166952340138176121266325894963~?28346 to quotients of numbers containing more than 20
digits by employing higher order interpolation.
If the product Mm is wanted to 20 digits, only
the result obtained in step 5 need be recorded. Example 9. Sum the series S= l-&+*-i
Further, if the allowable error in the 20th place is + to 5D using the Euler transform.
a unit’, the operation MImI may be omitted. The sum of the first 8 terms is .634524 to 6D.
When either of the factors M or m contains less If u,=ljn we get
than 20 digits it is convenient to position the
numbers as if they both had 20 digits. This n %7 Au, A*u, A3u, A%,,
multiplication process may be extended to any 9 . 111111
higher accuracy desired. -11111
10 . 100000 2020
Example 8. Divide N=14116 69523 40138 17612 -9091 -505
by d=20243 97459 71664 32102. 11 . 090909 1515 156
Method (1)--linear interpolation. -7576 -349
12 . 083333 1166
N/20243 97459.101’= .69732 82430 90519 39054 -6410
N/20243 97460.10”= .69732 82427 46057 26941 13 . 076923
Difference=3 44462 12113.
From 3.6.27 we then obtain
Difference X.71664 32102=24685 64402&10-*O (-.011111)+.002020
SC 634524+.111111
-_
(note this is an 11 X 10 multiplication). 2 22 23
m sin 2
Example 10. Evaluate the integral -
s Cl J:
dx $?jk-‘=glI k-‘+l& (k+10)-2
=- G to 4D using the Euler transform.
(Icfl)= y dx
- F dx=g s,. 1
s0 +jY&p- ...
=& s,’ sin;;;;t) dt+% (-l)f g dt. where f(k) = (k+10)-2. Thus,
. 18260 A A2 A3 A4
to 5D for x= .2. Here al=x, an=(n-l)2x2 for
n>l, &,=O, b,=2n-1, A-l=l, Bdl=O, A,,=O,
. 14180
-2587
.11593 799
- 1788 -321
.09805 478 153 A0
-=
-1310 - 168 Bo ’
.08495 310
- 1000
.07495 A
-r,*g
Bl
The sum to k=3 is 1.49216. Applying the
Euler transform to the remainder we obtain
A=.197368
B2
f (.14180)-h (-.02587)+& (.00799)
A3
B=.197396
-; (-.00321)+$ (.00153) 3
[II
A4
+ .00005
= .07862 Bq = 15.36
We obtain the value of the integral as 1.57018 as Note that in carrying out the recurrence method
compared with 1.57080. for computing continued fractions the numerators
A, and the denominators B, must be used as
Example 11.
Sum the series $I kep==f using
P originally computed. The numerators and de-
the Euler-Maclaurin summation formula. nominators obtained by reducing An/B, to lower
From 3.6.28 we have for n= a, terms must not be used.
ELEMENTARY ANALYTICAL METHODS 23
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[3.27] L. J. Comrie, Barlow’s tables of squares, cubes,
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[3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc.,
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[3.14] I. M. Longman, Note on a method for computing [3.28] H. B. Dwight, Tables of integrals and other mathe-
infinite integrals of oscillatory functions, Proc. matical data, 3d ed. (The Macmillan Co., New
Cambridge Philos. Sot. 52, 764 (1956). York, N.Y., 1957).
[3.15] S. E. Mikeladze, Numerical methods of mathe- [3.29] Gt. Britain H.M. Nautical Almanac Office, Inter-
matical analysis (Russian) (Gos. Izdat. Tehn- polation and allied tables (Her Majesty’s Sta-
Teor. Lit., Moscow, U.S.S.R., 1953). tionery Office, London, England, 1956).
[3.16] W. E. Milne, Numerical calculus (Princeton Univ. [3.30] B. 0. Peirce, A short table of integrals, 4th ed.
Press, Princeton, N.J., 1949). (Ginn and Co., Boston, Mass., 1956).
[3.17] L. M. Milne-Thomson, The calculus of finite differ- [3.31] G. Schulz, Formelsammlung zur praktischen Mathe-
ences (Macmillan and Co., Ltd., London, England, matik (de Gruyter and Co., Berlin, Germany,
1951). 1945).
24 ELEMENTARY ANALYTICAL METHODS
6 7 8 9
:
3
2:
125
625
2:: 3:;
2401
516;
4096
7%
4 1296 6561
3125 7716 16807 32168 59049
2 15625 46656 1 17649 2 62144 5 31441
78125 2 79936 8 23543 20 97152 47 82969
i 3 90625 16 79616 57 64801 167 77216 430 46721
19 53125 100 77696 403 53607 1342 17728 3874 20489
190 97 65625 604 66176 2824 75249 ( 9)1.0737 41824 ( 9)3.4867 84401
24 (16)5.9604 64478 (18)4.7383 81338 (20) 1.9158 12314 (21)4.7223 66483 (22)7.9766 44308
l/2 2.2360 61977 2.4494 89743 2.6457 51311 2.8284 27125 3.0000 00000
l/3 1.7099 75947 1.8171 20593 1.9129 31183 2.0000 00000 2.0800 83823
l/4 1.4953 48781 1.5650 84580 1.6265 76562 1.6017 92831 1.7320 50808
l/5 1.3797 29662 1.4309 69081 1.4757 73162 1.5157 16567 1.5518 45574
13
1’0:
1000
1::
1331
12
1728
--_
169
2197
1’946
2744
10000 14641 20736 28561 38416
1
10
100
00000
00000
00000
17
194
1 61051
71561
87171 358
-. __._
79
2 48832
85984
31808
3 71293
48 26809
627 48517 1054
5 37824
75 29536
13504
1000 00000 2143 58881 4299 81696 8157 30721
( 9 11.0000 00000 ( 9)2.3579 47691 9)5.1597 80352 10)1.0604 49937
(10 1.0000 00000 (10)2.5937 42460 10)6.1917 36422 11)1.3785 84918
24 (24) 1.0000 00000 (24)9.8497 32676 (25)7.9496 84720 (26)5.4280 07704 (27)3.2141 99700
w 3.1622 77660 3.3166 24790 3.4641 01615 3.6055 51275 3.7416 57381
v3 2.1544 34690 2.2239 80091 2.2894 28485 2.3513 34688 2.4101 42264
l/4 1.7782 79410 1.8211 60287 1.8612 09718 1.8988 28922 1.9343 36420
v5 1.5848 93192 1.6153 94266 1.6437 51830 1.6702 71652 1.6952 18203
16
2:: 256 2% 3::
3375 4096 4913
50625
7 59375
65536
10 48576 14
83521
19857
1
18
5832
04976
89568 24
1 6859
30321
76099
113 90625 167 77216 -.-
241 _37569-- 340 12224 470 45881
1708 59375 2684 35456 41U3 38673 6122 20032 8938 71739
I 914.2949 67296 I 9j6.9757 57441 10)1.1019 96058
(lOj6;8719 47674 (llj1.1858 78765 11 1.9835 92904
11)5.7665 03906 (12)1.0995 11628 (12)2.0159 93900 12 1 3.5704 67227
24 (28)1.6834 11220 (28)7.9228 16251 (29)3.3944 86713 (30)1.3382 58845 (30)4.8987 62931
l/2 3.8729 83346 4.0000 00000 4.1231 05626 4.2426 40687 4.3588 98944
l/3 2.4662 12074 2.5198 42100 2.5712 81591 2.6207 41394 2.6684 01649
l/4 1.9679 89671 2.0000 00000 2.0305 43185 2.0597 67144 2.0877 97630
l/5 1.7187 71928 1.7411 01127 1.7623 40348 1.7826 02458 1.8019 83127
23
42000 441 4:: 5::
8000 9261 10648 13824
1 60000 1 94481 2 34256 3 31776
32 00000 40 84101 51 53632 79 62624
640 00000 857 66121 1133 79904 1911 02976
9) 1.2800 00000 ( 9)1.8010 88541
10)2.5600 00000 (10)3.7822 85936
(11)5.1200 00000 (11)7.9428 00466
(13)1.0240 00000 (13)1.6679 88098
24 (31)1.6777 21600 (31)5.4108 19838 (32)1.6525 10926 (32)4.8025 07640 (33)1.3337 35777
w 4.4721 35955 4.5825 75695 4.6904 15760 4.7958 31523 4.8989 79486
l/3 2.7144 17617 2.7589 24176 2.8020 39331 2.8438 66980 2.0844 99141
l/4 2.1147 42527 2.1406 95143 2.1657 36771 2.1899 38703 2.2133 63839
l/5 1.8205 64203 1.8384 16287 1.8556 00736 1.8721 71231 1.8881 75023
ELEMENTARY ANALYTICAL METHODS 25
POWERS AND ROOT!3 nk Table 3.1
6::
15625
26
676
21
729
19683
7:: 8::
17576 21952 24389
3 90625 4 56976 5 31441 6 14656 7 07281
97 65625 118 81376 143 48907 172 10368 2105 11149
2441 40625 3089 15776 3674 20489 4818 90304 5948 23321
9 8.0318 10176 10 1.0460 35320 10 1.3492 92851
11 2.0882 70646 11 2.8242 95365 11 3.7780 19983
12 5.4295 03679 12 7.6255 97405 13 1.0578 45595
14 I 1.4116 70957 14 I 2.0589 11321 II 14 2.9619 61661
(33)3.5527 13679 33)9.1066 85770 34)2.2528 39954 (34)5.3925 32264 (35)1.2518 49008
5.0000 00000 5.0990 19514 5.1961 52423 5.2915 02622 5.3851 64807
2.9240 17738 2.9624 96068 3.0000 00000 3.0365 88972 3.0723 16826
2.2360 67977 2.2581 00864 2.2795 07057 2.3003 26634 2.32!05 95787
1.9036 53939 1.9186 45192 1.9331 82045 1.9472 94361 1.9610 09057
31 34
9:: 961 10;; 1156
27000 29791 32768 35937 39304
6 10000 9 23521 10 48576 11 85921 13 36336
243 00000 286 29151 335 54432 391 35393 454 35424
7290 00000 0675 03681 9 1.2914'67969 9)1.5448 04416
10 2.1870 00000 10 2.7512 61411 '10 4.2618 44298
11 6.5610 00000
13 1.9683 00000
Ii 14 5.9049 00000
(35)2.8242 95365
5.4772 25575
II
11 8.5289
13 2.6439
14 8.1962
(35)6.2041
10374
62216
82870
26610 (36)1.3292 27996
'12 1.4064 08618
113I 4.6411 48440
15)1.5315 78985
(36)2.7818 55434 (36)5. 6950 03680
5.5677 64363 5.6560 54249 5.7445 62647 5.8309 51895
3.1072 32506 3.1413 80652 3~1748 02104 3.2075 34330 3.2396 11801
2.3403‘47319 2.3596 11062 2.3704 14230 2.3967 El727 2.41147 36403
1.9743 50486 1.9873 40755 2.0000 00000 2.0123 46617 2.0;!43 97459
35 37 30 19
1225 1369 1444 15%
42875 46656 50653 54672 59319
15 00625 16 79616 18 74161 20 85136 23 13441
525 21075 604 66176 693 43957 792 35168 902 24199
9 3.0109 36384
11 1.1441 55826
~12 4.3477 92138
14 1.6521 61013
15)2.7585 47354 15)3.6561 58440 15 I 6.2782 11848
24 (37)1.1419 13124 (37)2.2452 25771 (37)4.3335 25711 (37)8.2187 60383 (38)1.5330 29700
l/2 5.9160 79783 6.0000 00000 6.0827 62530 6.1644 14003 6.2449 97998
l/3 3.2710 66310 3.3019 27249 3.3322 21852 3.3619 75407 3. 3912 11443
l/4 2.4322 99279 2.4494 89743 2.4663 25715 2.4828 23796 2.4989 99399
l/5 2.0361 68005 2.0476 72511 2.0589 24137 2.0699 35054 2.0807 16549
40
1600 16G 17:: 18:; 19;:
64000 68921 74088 79507 85184
25 60000 28 25761 31 11696 34 18801 37 48096
1024 00000 1158 56201 1306 91232 1470 08443 1649 16224
9)4.7501 04241 9 6.3213 63049
11)1.9475 42739 11 I 2.7181 86111
12)7.9849 25229 13 1.1688 20028
14 1 5.0259 26119
(16)1.0485 76000 16)2.1611 48231
24 (38)2.8147 49767 (38)5.0911 10945 (38)9.0778 49315 39)1.5967 72093 (39)2.7'724 53276
l/2 6.3245 55320 6.4031 24237 6.4807 40698 6.5574 38524 6.6332 49581
l/3 3.4199 51893 3.4482 17240 3.4760 26645 3.5033 98060 3.5303 48335
l/4 2.5148 66859 2.5304 39534 2.5457 29895 2.5607 49602 2.5'755 09577
l/5 2.0912 79105 2.1016 32478 2.1117 85765 2.1217 47461 2.1'315 25513
45 47
2025 21;: 22d9 23::
91125
41 00625
97336 1 03823 1 10592 _ _._..
44 77456 48 79681 53 08416 57 64801
1845 28125 2059 62976 2293 45007 2548 03968
6 ( 9)8.3037 65625 (10 1.0779 21533 10)1.2230 59046
7 (11)3.7366 94531 11 5.0662 31205 11 5.8706 83423
(13 1.6815 12539 13 2.3811 28666 13 12.8179 28043
9” (14 7.5668 06426 15 1.1191 30473 1.3526 05461
10 (I6 I 3.4050 62892 ii16 5.2599 13224
24 (39)4.7544 50505 (39)8.0572 70802 (40)1.3500 46075 40)2.2376 37322 (40)3.6‘703 36822
l/2 6.7082 03932 6.7823 29983 6.8556 54600 6.9282 03230 7.0000 00000
l/3 3.5568 93304 3.5830 47871 3.6088 26080 3.6342 41186 3.6!593 05710
l/4 2.5900 20064 2.6042 90687 2.6183 30499 2.6321 48026 2.6457 51311
l/5 2.1411 27368 2.1505 60013 2.1598 30012 2.1689 43542 2.1'779 06425
The numbers in square brackets at the bottom of the page mean that the maximum
error in a linear interpolate is a X 10-P (p in parentheses), and that to interpolate to
the full tabular accuracy 11~ poJnts must be used in Lagrange's and Aitkens methods for
the respective functions W'.
k
52 53 54
: 25:: 26;: 2704 2809 2916
1 25600 1 32651 1 40608 1 48877 1 51464
: 62 50000 67 65201 73 11616 78 90481 85 03056
3125 00000 3450 25251 3802 04032 4181 95493 4591 65024
2 10 1.5625 00000 10 I 1.7596 28780 10 2.2164 36113
7 11 7.8125 00000 11 8.9741 06779 12 1.1747 11140
13 3.9062 50000 13 4.5167 94451 13 6.2259 69041
t 15 1.9531 25000 15 3.2997 63592
10 16 I 9.7656 25000 1.1904 24238 il17 1.7488 74104
24 (40)5.9604 64478 40)9.5870 33090 (41)1.5278 48342 (41i2.4133 53110 (41)3.7796 38253
l/2 1.0710
3.6840
67812
31499
7.1414 28429 7.2111 02551 7.2801
3.1562
09889
85154
7.3484
3.7197
69228
63150
l/3 3.1084 29769 3.7325 11157
2.6591 47948 2.6723 45118 2.6853 49614 2.6981 67876 2.7108 06011
::: 2.1867 24148 2.1954 01897 2.2039 44515 2.2123 56822 2.2206 43035
56 57
30:: 3136 3244 34::
1 66375 1 75616 1 85193 2 05379
91 50625 98 34496 105 56001 121 17361
5032 84375 5507 31776 6016 92057 7149 24299
10 2.7680 64063
12 1.5224 35234
'13 8.3733 93789
'15 I 4.6053 66584
:17)2.5329 51621
24 (41)5.8708 98173 (41)9.0471 67858 (42)1.3835 55344 (42)2.1002 54121 (42)3.1655 43453
l/2 7.4161 98487 1.4833 14774 7.5498 34435 7.6157 73106 7.6811 45748
l/3 3.8029 52461 3.8258 62366 3.8485 01131 3.8708 76641 3.8929 96416
l/4 2.7232 69815 2.7355 64800 2.1476 96205 2.7596 69021 2.7714 88002
l/5 2.2288 07384 2.2368 53829 2.2447 86134 2.2526 07878 2.2603 22470
60 64
3600 37;: 38:: 39% 40%
2 16000 2 26981 2 38328 2 50047 2 62144
129 60000 138 45841 147 76336 157 52961 167 77216
7176 00000 8445 96301 9161 32832 .._. ___
9974 36547._
10)4.6656 00000
24 (42)4.7383 81338 (42) 7.0455 68477 (43) 1.0408 79722 ; 43; 1.5281 75339 (43)2.2300 74520
l/2 7.7459 66692 7.8102 49676 7.8740 07874 7.9372 53933 8.0000 00000
l/3 3.9148 67641 3.9364 97183 3.9578 91610 3.9790 57208 4.0000 00000
l/4 2.7831 57684 2.7946 82393 2.8060 66263 2.8173 13241 2.8284 27125
l/5 2.2679 33155 2.2754 43032 2.2828 55056 2.2901 72049 2.2973 96710
66
4226: 4356 44:; 462648 4766;
2 74625 2 81496 3 00763 3 14432 3 28509
178 50625 189 74736 201 51121 213 81376 226 67121
( 9)1.1602 90625
10)7.5418 89063
12)4.9022 27891
3.1864 48129
I
('l8)1.3462 74334
24 (43)3.2353 44710 (43)4.6671 78950 (43)6.6956 88867 (43)9.5546 30685 (44)1.3563 70007
l/2 8.0622 51148 8.1240 38405 8.1853 52172 8.2462 11251 8.3066 23863
l/3 4.0207 25159 4.0412 40021 4.0615 48100 4.081'6 55102 4.1015 65930
l/4 2.8394 11514 2.8502 69883 2.8610 05553 2.8716 21711 2.8821 21417
l/5 2.3045 31620 2.3115 79249 2.3185 41963 2.3254 22030 2.3322 21626
72 73
49:: 50:: 5184 5329 54::
3 43000 3 57911 3 73248 3 89017 4 05224
240 10000 254 11681 268 13856 283 98241 299 86576
( 9 I 1.8042 29351 ( 9I 1.9349 17632 9 2.0730 -71593
(11 1.2810 02839 (11 1.3931 40695 11 1.5133 42263
13 1.0030 61300 13 1.1047 39852
I 14
12 9.0951
6.4515 20158
35312 14 1.2220 41363 14 I 8.0646 00919
(16 I 4.5848 50072
(18 3.2552 43551 [ :86{ :: :I;: i:::: 16 5 2976
1814: 25830
8871 58671
24 (44)1.9158 12314 (44)2.6927 76876 (44)3.7668 63772 ;44;5.2450 38047 (44)7.2704 49690
l/2 8.3666 00265 8.4261 49773 8.4852 81374 8.5440 03745 8.6023 25261
l/3 4.1212 85300 4.1408 17749 4.1601 67646 4.1793 39196 4.1983 36454
l/4 2.8925 07608 2.9027 83108 2.9129 50630 2.9230 12786 2.9329 72088
l/5 2.3389 42837 2.3455 81669 2.3521 58045 2.3586 55818 2.3650 82169
.1[(-,,9] ,i[(-;)4] ,i[(-55131 “:[(-55)2]
ELEMENTARY ANALYTICAL METHO:DS 27
POWERS AND ROOTS nk Table 3.1
75 76
5625 5776 59:; 6078: 62:;
4 21875 4 38976 4 56533 4'74552 4 93039
316 40625 333 62176 351 53041 370 15056 389 50081
81
6480: b56i 67;: 611:; 70::
12000 5 31441 5 51368 5 71787 5 92704
40: 60000 430 46721 452 12176 474 58321 497 87136
00000
40000
52000
21600
77280
41824 (19)1.2157 66546
(45)4.7223 66483 (45)6.X626 85441 (45) a.5414 66801 (46)1.1425 47375 (46)1.5230 10388
l/2 8.9442 71910 9.0000 00000 9.0553 85138 9. 1104 33579 9.1651 51390
l/3 4.3088 69380 4.3267 48711 4.3444 81486 4. 3620 70671 4.3795 19140
l/4 2.9906 97562 3.0000 00000 3.0092 16698 3. ~0183 49479 3.0274 00104
l/5 2.4022 48868 2.4082 24685 2.4141 41771 2A200 01407 2.4258 04834
86 87 88 89
72;: 7396 7569 .,
-I"44 7921
6 14125 6 36056 6 58503 6 81472 7 04969
522 00625 547 00816 572 89761 599 69536 627 42241
9)4.9842 09207
24 (46)2.0232 71747 (46)2.6789 39031 (46)3.5355 91351 (46)4.6514 04745 (46)6.1004 25945
l/2 9.2195 44457 9.:!736 18495 9.3273 79053 9.:3808 31520 9.4339 81132
l/3 4.3968 29672 4.41140 04962 4.4310 47622 4.4479 60181 4.4647 45096
l/4 3.0363 70277 3.0452 61646 3.0540 75810 3.13628 14314 3.0714 78656
l/5 2.4315 53252 2.4372 47818 2.4428 89656 2.4484 79851 2.4540 19455
92 94
8190:
29000
82::
53571
8464
7 78688
86:;
a 04357
8836
a 30584
65: 10000 74961 716 39296 748 05201 780 74896
9 5.9049 00000 21451 9)6.9568 83693
11 I 5.3144 10000 92520 11 6.4699 01834
13 4.7829 69000 10194 13 6.0170 08706
15 4.3046 72100 25276 15 5.!5958 18097
04890 98001 17 5.2041 10830
84401 6118i 19 4.8398 23072
24 46)7.9766 44308 (47) 1.0399 04400 (47)1.3517 a5726 47)1:7522 28603 (47)2.2650 01461
l/2 9.4868 32981 9.!;393 92014 9.5916 63047 9.6436 5Oi'bl 9.6953 59715
l/3 4.4814 04747 4.4979 41445 4.5143 57435 4. 5306 54896 4.5468 35944
l/4 3.0800 70288 3.0885 90619 3.0970 41015 3.1054 22799 3.1137 37258
l/5 2.4595 09486 2.4649 50932 2.4703 44749 2.4756 91866 2.4809 93182
96 97
90;: 9216 .-,
9P"9 9d48 98:;
8 57375 8 04736 9 12673 9 41192 9 70299
814 50625 849 34656 885 29281 922 36816 960 59601
9 7.7378 09375 9)9.0392 07868
11 I 7.3509 18906 11 8.8584 23E109 I 119 I 9.5099
9.4148 00499
01494
13 6.9633
15 I b.6342
17 6.3024
72961
04313
94097
I::{;:z: 2383%
(17)7.6023 10587
13 1 8.6812
15)8.5076
8.3374
55932
30226
77621
I 13
15 9.3206
9.2274 46944
53479
k
101 102 103 __.
1nr
: 10201 10404 10609 10816
10 30301 10 61200 10 92121 11 24864
: 1040 60401 1002 43216 1125 50001 1169 85856
5
7"
:
10
24 (48)1.0000 00000 48) 1.2691 34649 (48)1.6084 37249 (48)2.0327 94106 40)2.5633 04165
l/2 ( 1)1.0000 00000 1)1.0140 89157 1)1.0190 03903
l/3 4.6415 00034 4.6123 20720 4.6075 40148 4.7026 69375
l/4 3.1622 11660 3.1719 71028 3.1857 32501 3.1934 36868
l/5 2.5110 66432 2.5168 90229 2.5218 54546 2.5267 80003 2.5316 67508
24 (48)9.8497 32676 (49)1.2239 15658 (49)1.5170 b2893 (49)1.8708 09051 (49)2.3212 20685
w ( 1)1.0408 08048 ( 1) 1.0535 65375 ( 1):';;;; f3;;: ( 1)1.0630 14581 ( 1)1.0677 07825
l/3 4.7914 19857 4.0058 95534 4.8345 00127 4.0480 07586
l/4 3.2385 31040 3.2458 67180 3:2531 53123 3.2603 90439 3.2675 79071
l/5 2.5602 21376 2.5640 65499 2.5694 70314 2.5740 42354 2.5705 02140
20)4.4114 35079
24 (49)2.0625 17619 (49)3.5236 41704 (49)4.3297 20675 (49)5.3109 00621 (49)6.5031 99444
l/2 ( 1)1.0723 80529 ( 1)1.0770 32961 ( 1)1.0016 65303 ( l)l.O862 70049 ( 1)1.0900 71211
l/3 4.0629 44131 4.0769 90961 4.8909 73246 4.9040 60131 4.9106 64734
l/4 3.2147 22111 3.2810 18035 3.2008 68168 3.2950 73252 3.3028 33952
l/5 2.5830 90170 2.5075,66964 2.5920 12982 2.5964 20703 2.6000 14507
1: I1921 4.3310
6.5831
17 2.8493
40965
82267
69056 I
24 (52) 1.6834 11220 (52)1.9744 52704 ( 1)1.2328
52)2.3133 75387 (52)2.7076 61312 (52)3.lb59 00782
l/2
l/3
( 1)1.2247
5.3132
44871
92846
( 1)1.2288
5.3250
20573
74022
( 5.3368
82801
03297
( 1)1.2369
5.3484
31688
81241
( 1)1.2409
5.3601
67365
08411
l/4 3.4996 35512 3.5054 53712 3.5112 43086 3.5170 03963 3.5227 36670
l/5 2.7240 69927 2.7276 92374 2.7312 95679 2.7348 80069 2.7304 45765
1 3.6914
43278
51946
I 17)3.3316
15)2.1494
8.0041 05615
82490
22977 I
19 5.7955
21 9.0990
79555
59901
24 (52)3.6979 47627 (52)4.3150 94990 (52)5.0302 74186 (52)5.8582 79483 (52)b.8160 22003
l/2
l/3
( 1)1.2449
5.3716
89960
85355
( 1)1.2489
5.3832
99600
12612
( 1)1.2529
5.3946
96409
90712
( 1)1.2569
5.4061
80509
20176
( 1)1.2609
5.4175
52021
01515
l/4 3.5284 41525 3.5341 18843 3.5397 68931 3.5453 92093 3.5509 88625
l/5 2.7419 92987 2.7455 21947 2.7490 32856 2.7525 25920 2.7560 01343
I1311 11.1157
15
1.8075
2.9282
71008
49033
29434
I
17)4.2949 67296 17 4.7437 31683 I
19 7.6848 45327
I22 I 1.2449 44943
24 (52)7.9228 16251 (52)9.2007
( 1)1.2688
03274
57754
(53)1.0674
( 1)1.2727
81480 (53) 1.2373
( 1)1.2767
78329
14533
(53)1.4330
1)l. 2806
20335
l/2 ( 1)1.2649
5.4288
11064
35233 5.4401 21825 5.4513
92206
61778 5.4625 55571 5.4737
24847
03675
l/3
3.5565 58820 3.5621 0296b 3.5676 21345 3.5731 14235 3.5785 81908
::; 2.7594 59323 2.7629 00056 2.7663 23734 2.7697 30547 2.7731 20b81
1.1019 96058
24 (54)1.3382 58845 (54)1.5285 71637 (54)1.7446 70074 (54)1.9898 76639 (54)2.2679 20111
l/2 ( 1)1.3416 40786 ( l)l.3453 62405 ( 1)1.3490 73756 ( 1)1.3527 74926 ( l)l.3564 65997
l/3 5.6462 16173 5.6566 52026 5.6670 51108 5.6774 11371 5.6877 33960
l/4 3.6628 41501 3.6679 16217 3.6729 73940 3.6780 08871 3.6830 23210
l/5 2.8252 34501 2.8283 66697 2.8314 85080 2.8345 89786 2.8376 80950
3.0772 03640
5.8159 14881
24 (54)2.5829 82606 54)2.9397 51775 (54)3.3434 78670 (54)3.8000 41874 (54)4.3160 la526
l/2 ( 1)1.3601 47051 ( 1)1.3638
5.7082 18170
67473 ( 1)1.3674
5.7184 79065
79433 ( 1)l 5.7,a6
3711 54316
30920 ( 1)1.3747 72700
l/3 5.6980 19215 5.7307 93540
l/4 3.6880 17151 3.6929 90888 3.6979 44609 3:7il20 78502 3.7077 92751
l/5 2.8407 58702 2.8430 23174 2.0468 74493 2.8499 12786 2.8529 38178
_..
190 193 194
: 36100 37636
68 59000 E7 73 01384
: 63361 54496 88001
5 49020 92632 51842
22627 49854 54055
7" 84218 30326
97266 ::::2" 22953
: 86816 74059 a7299
10 04818 a6193 90487
24 (54)4.8987 62931 54)5.5564 93542 (54)6.2983 89130 (54)7.1346 95065 (54)8.0768 40718
w ( 1)1.3784 04875 l';.:;;; ff;;;; ( l)l.3856 40646 ( 1)1.3892 44399 (*)1.3928 38828
l/3 5.7408 97079 3:7175 63041 3.7224
5.7689 19436
98281 5.7789 96565 5.7869 60372
l/4 3.7126 07530 3.7212 56899 3.7320 75599
l/5 2.8559 50791 2.0589 50746 2.8619 38162 2.8649 13156 2.0670 75844
24 (54)9.1375 69069 55)1.0331 07971 (55)1.1673 18660 (55)1.3181 49187 (55)1.4875 57746
l/2 ( l)l.3964 24004 ( ";.;;;; it;;; ( 1)1.4035 66885 ( 1)1.4071 24728 ( 1)1.4106 73598
l/3 5.7988 89998 3: 1416 57367 3.7464
5.0186 20805
47867 5.8284 76683 5.8382 72461
3.7368 75706 3.7511 66123 3.7558 93499
2.8708 26340 2.8737 64756 2.8766 91203 2.8796 05790 2.8825 08624
: 20)9.4129 11168
10 (23)2.0143 62990
24 (55)5.4108 19838 (55)6.0642 75557 (55)6.7929 a5105 (55)7.6051 97251 (55)8.5100 19601
l/2 ( 1)1.4491
5.9439
37675
21953
( 1)1.4525
5.9533 a3905
41813 ( 1)1.4560 21978 ( 1)1.4594 51952 ( 1)1.4628 73884
l/3 5.9621 31958 5.9720 92620 5.9814 24030
3.8067 54096 3.8112 77876 3.8157 85604 3.8202 77414 3.8247 53435
$5" 2.9136 93459 2.9164 63134 2.9192 22328 2.9219 71130 2.9247 09627
10
24 (55)9.5175 03342 56)l.o63a 73589 (56)1.1885 94216 i56j1.3272 59512 (56)1.4813 53665
l/2 ( 1)1.4662 a7830 ";.;;;; ;;;;; ( 1)1.4730 91986 ( 1)1.4764 82306 ( 1)1.4798 64859
l/3 5.990f26415 3:8336 58625 6.0092 45007 6.0184 61655 6.0276 50160
l/4 3.8292 13796 3.8380 88048 3.8425 02187 3.8469 01167
l/5 2.9274 37906 2.9301 56052 2.9328 64149 2.9355 62280 2.9382 50529
1:
24 (56)2.0338 73334 (56)3.1521 18526 56) 3.5044 5568b 56)3.0943 62082 ( 56) 4.325'b 51908
l/2 ( 1)1.5000 00000 ( 1)1.5033 29638 1)l. 5066 51917 1)1.5099 bb887 ( 1’;. y; y;;
6.0822 01996 6.0911 99349 6.1001 70200 6.1091 14744
:s: 3.8729 83346 3.0772 79501 3.8815 61435 3.0050 29230 3:89OQ 83026
l/5 2.9541 76939 2.9567 90210 2.9594 10235 2.9620 130b2 2.964b 06713
24 (56)4.8025 07640 (56)5.3295 12896 (56)5.9116 89798 56) 6.5545 38267 (56) 1.2640 79321
l/2 ( 1)l. 5165 75089 ( 1)1.5198 68415 ( "lb.;::; ;;;$; ( l)l.5264 33152 ( 1'; :'6;; g;:;
6.1269 25615 b..1357 92440
if: 3.8943 22905 3.8985 48980 3:9027 61357 6.1534
3.9069 49494
60138 3:9111 45426
l/5 2.9bll 91438 2.9697 b7129 2.9723 33915 2.9148 91866 2.9774 41049
:x
._..
74649
49664
56201
24 (56)8.0469 01671 (56) 8.9102 12697 (56)9.861'3 93410 (57)1.0910 55818 (57)1.2065 61943
l/2 ( l)l.5329 70972 ( 1) 1.5362 29150 "k:;;; "b;;i; ( 1)1.5427 24862 ( 1)1.5459 62483
l/3 6.1710 05793 6.1791 46606 6.2058 21795
l/4 3.9153 17320 3.9194 15921 3: 9236 21327 6.1971
3.9277 54435
53635 3.9318 72942
l/5 2.9799 81531 2.9825 13380 2.9850 36660 2.9075 51438 2.9900 57776
%;
98010
24 57)9.1066 85770 57)9.9‘355 54265 (58)1.0945 38372 (58) 1.1993 27914 (58)1.3136 94086
l/2 ( 1)1.6124 51550 ( I)1 6155 49442 ( 1)1.6186 41406 ( 1)1.6217 27474 ( 1)1.6248 07681
l/3 6.3825 04299 6.3906
4:0193 76528
89807 6.3988
4.0232 34278
27910 6.4069
4.0270 58577
67760 6.4150 68660
l/4 4.0155 34273 4.0308 90325
l/5 3.0408 41703 3.0431 83226 3.0455 11602 3.0418 32879 3.0501 47105
1.9113 83974
24 (58) 1.4384 10548 (58) 1.5145 60235 (58) 1.1229 40472 58) 1.8846 68868 (58)2.0608 89564
( 1)1.6278 82060 ( 1)1.6309 50643 ( 1)1.6340 13464 1) 1.6310 10554 ( 1) 1.6401 21947
6.4231 58289 6.4312 27591 6.4392 16696 6.4413 05721 6.4553 14811
4.0341 02045 4.0385 02994 4.0422 93240 4.0460 72854 4.0498 41906
3.0524 54329 3.0547 54599 3.0570 47961 3.0593 34462 3.0616 14141
24 (58)2.2528 39954 (58)2.4618 51891 (5Q2.6893 89450 (58)2.9369 91176 (58) 3.2063 69049
m ( 1)1.6431 61673 ( 1)1.6462 01763 ( 1)1.6492 42250 ( 1) 1.6522 11164
l/3 6.4633 04070 6.4712 13627 6.4792 23603 6.4811 54111
l/4 4.0536 00464 4.0573 48596 4.0610 86370 4.0648 13851
l/5 3.0638 87063 3.0661 53254 3.0684 12165 3.0706 65640 3.0129 11923
ELEMENTARY ANALYTICAL METHODS 35
POWERS AND ROOTS nk Table 3.1
k
275 276 277 278 279
: 75625 16116 76719 77284 77841
201 96875 210 24576 212 53933 214 a4952 217 17639
: 9 5.8873 39441
12 1.6307 93025
2 14 4.5172 96680
17 1 1.2512 91180
i 19 3.4660 76569
21 9.6010 32097
1: 24 I 2.6594 85891
24 (58)3.4993 28001 (58)3.8178 42160 58)4.1640 35828 (58)4.5402 01230 (58)4.9488 11121
l/2 ( 1)1.6583 12395 ( 1)1.6613 24773 1)1.6643 31698 ( 1)1.6673 33200 ( 1) ;. ;;g ;g;
l/3 6.5029 57234 6.5108 30071 6.5186 a3915 6.5265 10079
l/4 4.0722 38199 4.0759 35196 4.0796 22161 4.0832 99156 4:0869 66245
l/5 3.0751 51657 3.0773 a4885 3.0796 11650 3.0818 31992 3.0840 45954
24 (58)5.3925 32264 (58)5.8742 39885 (58)6.3970 33126 58)6.9642 51599 (58)7.5794 93086
l/2 ( 1)1.6733 20053 ( 1)1.6763 05461 ( 1)1.6792 a5562 1)1.6822 60384 ( 1) ;. y; $94;;
6.5421 32620 63499 11620 6.3576 72186 6.5654 14427
:;: 4.0906 23489 4.0942 70950 4.0979 08689 4.1015 36766 4:1051 55240
l/5 3.0862 53577 3.0884 54901 3.0906 49967 3.0928 38815 3.0950 21484
III 12
149 5.4726
1.9135
6.6905 a5616
31410
07486
17 1.5651 72583
19 4.4163 935’39
22 1.2802 48566
24 3.6615 10900 24)4.0642 31407
24 (5a)a.2466 32480 (58)8.9690 42039 (58)9.7536 13040 (59)1.0602 77893 (59)1.1522 54005
l/2 ( 1)1.6aal 94302 ( 1)1.6911 53453 ( 1)1.6941 07435 ( 1)1.6970 56275 ( 1)1.7(100 00000
l/3 6.5808 44365 6.5085 32215 6.5962 02284 6.6038 54498 6.6114 89018
l/4 4.1087 64171 4.1123 63618 4.1159 53637 4.1195 34288 4.1231 05626
l/5 3.0971 98013 3.0993 68441 3.1015 32807 3.1036 91148 3.1058 43502
24)6.5226 83188
(59)2.8242 95365 59)3.0591 15639 (59)3.3125 81949 (59)3.5861 05682 (59)3.8811 99856
l/2 1)1.7349 35157 ( l)i.7378 14720 ( 1)1.7406 89519 ( 1)1.7435 59577
l/3 6.7017 59395 6.7091 72852 6.7165 69962 6.7239 50814
l/4 4.1652 55283 4.1687 10496 4.1721 57138 4.1155 95260
l/5 3.1291 34645 3.1312 17958 3.1332 95743 3.1353 68030 3.1314 34853
:
10 1.1156
12 I 3.6259
64063
08203
10 1.1574
12 3.7963
31706
15994
( 10)1.1716 11408
;
17 3.8298
20 1 1.2447
65540
06300
II 17 4.0842
20 1.3396
93150
48153
I
9
10
22 4.0452
25) 1.3147
95476
21030
22 4.3940
I 25 I 1.4412
45942
47069
I(
24 60)1.9284 15722 (60)2.0759 76350 (60)2.2343 23554 (60)2.4042 09169 (60)2.5864 34894
10
1/3
( l)l.8027
6.8753
75638
44335
( 1)1.8055
6.8823
47009
88750
( 1)1.8083
6.8894
14132
18774
( 1)1.8110
6.8964
77028
34481
( 1)1.813#3
6. 903,1
35715
35942
1/4 4.2459 10547 4.2491 72871 4.2524 27697 4.2556 75061 4.258') 15020
l/5 3.1796 30632 3.1815 84924 3.1835 34426 3.1854 79164 3.187,4 19165
:
340
39:
341
16281
51821
342
16964
_._
343
1 17649
144
1 18336
40; 01688 403 53607 407 07584
: 27096 57730
II
53398 57435
2 66909 35043 1.6284 13598
7 30158 61047 (1012
17 1.4003
15 4.8171
5.7004
1.6571 72660
40890
49439
07395
II
47684 81952
i 24602 10274
10 04689 77114 25 1.9609
22
20 2.3205
6.7456 15244
54607
83848
24 (60)5.6950 03680 (60)6.lloa 98859 (60)6.5558 12822 (60)7.0316 76479 (60)7.5405 43015
l/2 ( 1)1.8439 08891 ( 1);;;;; ;"8;;; ( 1)1.8493 24201 ( 1) +. i;'o; g;;;; ( 1)1.8547 23699
1/3 6.9795 32047 7.0067 96121
4.2940 76026 412972 29958 4.3003
6.9931 90657
76961 4:3035 17071 4.3066 50321
$2 3.2084 53751 3.2103 38860 3.2122 19552 3.2140 95850 3.2159 67776
I 30338 I 15
17 1.8069
6.3063 48816
76738
Ii
39497
76659
I 13924 25 2.2009
22
20 7.6811
2.6807 37377
95921
15737
24 (60)8.0845 95243 (60)8.6661 53376 (60)9.2876 83235 (60)9.9518 04932 (61) 1.0661 30203
l/2 ( 1)1.8574 17562 ( 1)1.8601 07524 ( 1)1.8627 93601 ( 1)1.8654 75811 ( 1)1.8681 54169
l/3 7.0135 79083 7.0203 48952 7.0271 05788 7.0338 49656 7.0405 80617
4.3097 76748 4.3128 96386 4.3160 09269 4.3191 15431 4.3222 14906
3.2178 35355 3.2196 98608 3.2215 57557 3.2234 1222,6 3.2252 62636
;61;2.2452 25771 (61)2.3997 87825 (61)2.5645 17652 51)2.7400 53237 i61)2.9270 70667
24
1)1.9052 55888 ( 1)1.9078 78403
l/2 7.1334 92490 7.1400 36982
l/3 4.3649 23697 4.3679 26743
l/4 3.2507 33187 3.2525 22254
l/5 3.2453 42223 3.2471 43191 3.2489 40172
g [ 41
(-(94
40 ELEMENTARY ANALYTICAL METHODS
24 (62)3.7924 56055 (62)4.0236 92707 (62)4.2684 06980 (62)4.5273 48373 (62)4.8013 06073
l/2 ( 1)2.0124 61180 ( 7.4047
1)2.0149 20630
44168 ( 1)2 7.4107
0174 24100
95055 ( 1)2.0199 00988 ( 1)2.0223 74842
l/3 7.3986 36223 7.4168 59539 7.4229 14120
l/J 4.4860 46344 4.4088 12948 414915 14446 4.4943 30860 4.4970 82211
l/5 3.3226 99030 3.3243 38251 3.3259 74245 3.3276 07026 3.3292 36609
il 10
18 2.9661
15
13
20 8.7980
5.1084
2.1200
1.2309 16532
43334
45063
03984
50201
3.7311 26518
26)1.5521 48631
24 (62)6.8101 13045 (62)7.2150 59801 (62)7.6430 25690 (62)8.0952 59269 (62)8.5730 73581
l/2 ( 1)2.0371 54879 ( 1)2.0396 07805 ( 1)2 0420 57786 ( 1)2.0445 04830 1)2.0469 48949
l/3 7.4590 35926 7.4650 22314 7.4709 99115 7.4769 66370 7.4829 24114
l/4 4.5134 85215 4.5162 01729 4:5u9 13349 4.5216 20097 4.5243 21992
l/5 3.3389 47722 3.3405 55305 3.3421 59799 3.3437 61218 3.3453 59575
k
425 427 428 429
: 1 80625 I 81476 1 82329 1 83184
767 65625 773 08776 778 54483 784 02752
: 10 3.2933 53858
1'13 11.4029 68743
2 115 I 5.9166 46847
t 18 2.5460 51557
i I 21 1.0846 17963
123 4.6204 12523
1'0 I 26 1.9683 21295
24 (63)1.2059 63938 (63)1.2759 40370 63)1.3497 98685 (63)1.4277 44370 63)1.5099 93273
w / 1)2.0639 76744 1)2.0663 97832 ( 1)2.0688
7.5361
16087
22043
1)2.0712 31518
7.5243 65204 7.5302 48212 7.5419 86732
it: 4.5431 01082 4.5457 64877 4.5484 23998 4.5510 78463
l/5 3.3548 86145 3.3564 63431 3.3580 37758 3.3596 09138 3.3611 77583
430 ._-
431 432 433
1 85761 1 86624 1 87489 1 88%6
800 62991 806 21568 811 82737 817 46504
24 (63)1.5967 72093 ,63)1.6883 18906 (63)1.7848 83700 63)1.8867 28946 (63)1.9941 30189
( 1)2.0736 44135 , 1)2.0760 53949 ( 1)2.0784 60969 1)2.0808 65205 ( 1)2.0832 66666
::: 7.5478 42314 7.5536 88825 7.5595 26299 7.5653 54712 7.5711 74278
l/4 4.5537 28292 4.5563 73502 4.5590 14114 4.5616 50145 4.5642 81614
l/5 3.3627 43107 3.3643 05720 3.3658 65436 3.3674 22267 3.3689 76223
[ 1 II3
3.3858 83431 3.3874 03811, 3.3889 21465, 3.3904 36406, 3.3919 48644
1
$ (--6)4 C-47)4
4 I
42 ELEMENTARY ANALYTICAL METHODS
491 .._
497 I_ 494
2 40100 41081 2 42064 2 43049 ;! 44036
1176 49000 70771 1190 95488 1198 23157 lZO!i 53784
10)5.7648 01000 04856 5.6594 98010 10 5.9072 81640
13 2.8247 52490 94384 2.8828 73021 13 2.9122 89849
lb 1.3841 28720 63943 1.4183 73526 lb 1.4357 58895
'18 6.1822 30728 14959 6.9783 97749 18 I 7.0782 91354
21 I 3.3232 93057 40045 3.4333 71692 21 3.4895 97638
68562 1.6892 18873 24 1.7203 71635
71639 8.3109 56854 26 I 8.4814 32162
24 (b4)3.6703 36822 (b4)3.8543 91376 (64)4.0472 72689 64)4.2493 84825 b4)4.4611 49467
l/2 ( 1)2.2135 94362 ( y. '8;;; y;; ( 1';. ii,": ;;;"7; 1)2.2203 60331 1)2.222B 11077
l/3 7.8837 35163 7.8997 91695 7.9051 29393
l/4 4.7048 85081 4:7072 83697 4: 7096 78653 4.7120 b99bO 4.7144 57633
l/5 3.4517 49066 3.4531 56794 3.4545 62231 3.4559 65384 3.457'3 66263
1.4735 91925
24 (65)1.5278 48342 (65)1.5999 46126 (65)1.6752 98008 (65)1.7540 44200 (65)1.8363 30669
l/2 ( 1)2.2803 50850 ( 1)2.2825 42442 ( 1)2.2847 31932 ( 1)2.2869 19325 ( 1)2.2891 04628
l/3 8.0414 51517 8.0466 02993 8.0517 47881 8.0568 86203 8.0620 17979
l/4 4.7753 01928 4.7775 96092 4.7790 86957 4.7821 74532 4.7844 58829
l/5 3.4930 16754 3.4943 59190 3.4956 99566 3.4970 37889 3.4983 74167
1
+;)a] .:[(-,P] ni[(-;)S] $[‘-;‘“I
ELEMENTARY ANALYTICAL METHODS 45
POWERS AND ROOTS nk Table 3.1
k
525 526 527 528 529
: 2 75625 2 76676 2 77729 2 78784 2 79841
3 1447 03125 1455 31576 1463 63183 1471 97952 1480 35889
4 10)7.5969 14063 110 7.6549 60898 10)7.7133 39744 10 7.7720 51866 (10)7.8310 98528
5 13)3.9883 79883 113 1 4.0265 09432 13 4.1036 43385
6 16 2.0938 99438 I16 2.1179 43961 16)2.1667 23707
7 19 11.0992 97205 I19 1.1140 385'24 19)1.1440 30117 19 1.1592 83632
8 21)5.7713 10327 I 21 5.8598 42634 21 6.0404 79020 21 6.1326 10416
9 124 I 3.0822 17226 24 I 3.1893 72923 I 24 I 3.2441 50910
10 127)1.6212 77821 27)1.6839 88903 (2711.7161 55831
24 (65)1.9223 09365 , 65)2.0121 38448 (65)2.1059 82534 (65)2.2040 12944 (65)2.3064 07963
l/2 ( 1)2.2912 87847 / 1)2.2934 68988 ( 1)2.2956 48057 ( 1)2.2978 25059 ( 1)2.3000 00000
l/3 8.0671 43230 8.0722 61977 8.0113 74241 8.0824 80041 8.0875 79399
l/4 4.7867 39859 4.7890 17632 4.7912 92160 4.7935 63454 4.7958 31523
l/5 3.4997 08406 3.5010 40614 3.5023 70797 3.5036 98962 3.5050 25117
(27)1.9572 57'189
24 (65)3.0233 66304 (65)3.1619 49669 (65)3.3066 09101 (65)3.4575 98937 (65)3.6151 83652
l/2 ( 1)2.3130 06701 ( 1)2.3151 67381 ( 1)2.3173 26045 1)2.3194 82701 ( 1)2.3216 37353
l/3 8.1180 41379 8.1230 96201 8.1281 44739 8.1331 87014 8.1382 23044
l/4 4.8093 72829 4.8116 18626 4.8138 61283 4.8161 00810 4.8183 31217
l/5 3.5129 40196 3.5142 52463 3.5155 62774 3.5168 71134 3.5181 77550
24 (65)5.8708 98173
19 1.5419
21 I 8.4959
24 1 4.6812
27 2.5793
(65)6.1325
17693
66491
17536
83922
11516
19 1.5616
21 I 8.6201
24
27
4.7582
2.6265
(65)6.4052
13462
'I6308
98682
80873
76258
1
19 1.5815
21 8.7458
24 4.8364
27)2.6745
(65)6,6896
24482
30384
44203
53644
46227 (65)6.9860 92851
l/2 ( 1)2.3452 07880 ( 1)2.3473 38919 ( 1)2.3494 68025 1)2.3515 95203 ( 1)2.3537 20459
l/3 8.1932 12706 8.1981 75283 8.2031 31859 ( 8.2080 82453 8.2130 27082
l/4 4.8427 34641 4.8449 34384 4.8471 31136 4.8493 24905 4.8515 15700
l/5 3.5324'21650 3.5337 05234 3.5349 86956 3.5362 66821 3.5375 44836
_. - .-
64.1 642 643 644
4 09600 4 10881 4 12164 4 13449 4 14736
2621 44000 2633 74721 2646 09288 2658 47707 2670 89984
24 (67)2.2300 74520 (67)2.3152 22362 67)2.4034 80891 (67)2.4949 58638 (67)2.5897 67740
l/2 ( 1)2.5298 22128 ( 1) 2 ‘6;;; ‘2:;;; 1)2.5337 71892 ( 1)2.5357 44467 ( 1)2.537'7 15508
l/3 8.6177 30760 8.6311 82992 8.6356 55108
l/4 5.0297 33719 5:0316 97308 8.6267
5.0336 06237
58602 5.0356 17605 5.0375 74325
l/5 3.6411 28406 3.6422 65548 3.6434 01272 3.6445 35581 3.6450 68481
24 (67)2.6880 24057 (67)2.7898 47292 (67)2.8953 61105 (67)3.0046 93247 (67)3.1179 75679
w ( 1)2.5396 85020 ( 1)2.5416 53005 ( 1); p9'; y; ( 1)2.5455 84412 ( 1)2.5475 47841
l/3 8.6401 22598 8.6445 85472 8.6534 97422 8.657') 46522
l/4 5.0395 28767 5.0414 80939 5:0434 30845 5.0453 78492 5.0473 23886
l/5 3.6467 99973 3. b479 30063 3.6490 58755 3.6501 86051 3.6513 11957
1 1
4--97 .;[(-;I”] “;[(-;)3] 4’-37
50 ELEMENTARY ANALYTICAL METHODS
656 657
4 29025 4 30336 4 31649 4 34281
2810 11375 2823 0041b 2835 93393 2861 91179
(67)3.8885 81447 67) 4.0335 93654 (67)4.1837 80288 (b7) 4.3393 17689 (67)4.5003 87920
( 1)2.5592 96778 1)2.5612 49695 ( 1)2.5670 99531
8. b845 45603 8.6889 62971 8.7021 88202
5.0589 49271 5.0608 79069 5.0666 55239
3.6580 38399 3.6591 54676 3.6602 69592 3.6613 83152 3.6624 95358
: lQ1.8974
2874
4 35600
96000
73600 11
2888
1.9089
4 36921
04781
99602
2901
4 38244
17528 2914
4 39569
34241
09058
_-.4 54944
2927 - 40896
682 --a
4 62400 4 63761 4 65124 4 66489
3144 32000 4 67856
3158 21241 3172 14568 3186 11987 3200 13504
(11 2.1507 42651 11 2.1634 03354 11 2.1761 19871 11)2.1888 92367
I 14
16
11 I 2.1381
9.8867
1.4539 33568
37600
48262 (14 1.4646 55745 14 1.4754 41087 14 1.4862 89872
116 9.9743 05627 17J1.0062 50822 17 I 1.0151 35983
119 6.7925 02132 19 6.8626 30603 19 6.9333 78761
I22 4.6256 93952 22 1 4.6803 14071 22 4.7354 97694
125 3.1500 97581 3.1919 14196 25 I 3.2343 44925
2.1139 22820 128 2.1452 16453 28)2.2090 57584
24 (67)9.5546 30685 167)9.8976 17949 68) 1.0252 38701 (6aJ1.0619 32441 6a)l.o998 a2878
( 1)2.6076 a0962 I 1)2.6095 97670 lJ2.6115 12971 ( 112.6134 26869 1)2.6153 39366
8.7936 59344 8.7979 67850 8.8022 72141 a.8065 12225 a.8108 68115
5.1065 45762 5.1084 22134 5.1102 96441 5.1121 68688 5.1140 38880
3.6855 45546 3.6866 28893 3.6871 10968 3.6887 91774 3.6898 71315
24 68) 1.3563 70007 '68)1.4043 42816 (68) 1.4539 39271 (68) 1.5052 11857 (6a)1.5582 14678
l/2 1)2.6267 a5107 : 1)2.6286 a7886 ( lj2.6305 a9288 ( 1)2.6324 a9316 ( 1)2.6343 a7974
l/3 8.8365 55922 8.8408 22729 a.8450 a5422 a.8493 44010 a.8535 98503
l/4 5.1252 17173 5.1270 73128 5.1289 27069 5.1307 79001 5.1326 28931
1/5 3.6963 22179 3.6973 92956 3.6984 62494 3.6995 30796 3.7005 97866
24 68)1.6130 03502 (68) 1.6696 35809 (6a)1.7281 70846 (68)1.7886 69670 (68) 1.8511 95210
l/2 1)2.6362 a5265 ( 1)2.63al al192 ( 1)2.6400 75756 ( lJ2.6419 68963 ( 1)2.6438 60813
10 a.8578 48911 8.8620 95243 8.8663 37511 8.8705 15722 a.8748 09888
5.1344 76863 5.1363 22801 5.1381 66751 5.1400 08719 5.1418 48708
3.7016 63101 3.1027 28321 3.7037 91713 3.7048 53884 3.7059 14839
52 ELEMENTARY ANALYTICAL METHODS
i
9"
10 2q2.9481 74939
24 (6a)1.9158 12314 (68)1.9825 a7808 (68)2.0515 90555 (68)2.1228 91511 (68)2.1965 63787
l/2 ( 1)2.6457 51311 ( 1)2.6476 40459 ( 1’;;;;: gg; ( 1) 2 "8:;; '0;;;; ( 1)2.6532 99832
a.8790 40017 8.8832 66120 8.8959 20362
;g 5.1436 86124 5.1455 22771 511473 56056 511491 a8981 5.1510 19154
3.7069 74581 3.7080 33112 3.7090 90435 3.7101 46554 3.7112 01473
I 19
17)1.3249
22 I 9.4599
4.8226
6.7543 a3171
20408
18825
29584
24 (68)2.6927 76016 (bQ2.7852 89985 68)2.8808 44702 (68)2.9795 36544 (ba)3.0814 63889
v2 1)2.6645 a2519 ( 1)2.6664 58325 1)2.6683 32813 ( 1)2.6702 05985 ( 1)2.6720 77843
l/3 a.9211 21404 a.9253 07760 8.9294 90191 6.9336 68708 8.9376 43321
l/4 5.1619 59433 5.1637 76065 5.1655 90782 5.1674 03588 5.1692 14489
l/5 3.7115 05928 3.7185 52523 3.7195 97942 3.7206 42186 3.7216 85260
715
5 11225
716
5 12656
717 718 ._
719
5 14089 5 15524 5 16961
3655 25875 3670 61696 3686 61613 3701 46232 3716 94959
24 (68)3.1867 28051 (6CQ3.2954 33372 (68)3.4076 87302 (68)3.5236 00491 (68)3.6432 86875
( 1)2.6739 48391 ( 1)2.6758 17632 ( 1)2.6776 85568 ( 1)2.6795 52201.
a.9420 14037 8.9461 80866 8.9503 43817 8.9545 02899
5.1710 23488 5.1728 30591 5.1746 35801 5.1164 39125
3.7221 27165 3.1231 67905 3.7248 07483 3.7258 45902 3.7260 a3164
24 (68)6.1786 86185 (68)6.3836 27605 (68)6.5950 74542 (68)6.8132 24254 (68)7.0382 79698
l/2 ( 1) ;. ;;:; 2";;;; 1) $.-p; y7’; ( q.;;;; '0;;;; ( l'y& ( 1';. y; 2;;;;
l/3
l/4 5:2068 11253 512085 61314 5:2103 49693 5:2121 SW&
16213 5:2138 80938
l/5 3.7433 24423 3.7443 42461 3.7453 59393 3.7463 75222 3.7473 89950
28)5.4101 09038
24 (68)8.5457 57129 (68)8.8253 48404 (68)9.1136 94019 (68)9.4110 55807 (68)9.7177 03069
l/2 ( 1)2.7294 68813 ( 1';. ;;;; 2";;;; ( u;.;;;; y;; ( l)$m~ ;9"6";; ( 1)2.7367 a6437
9.0653 67701 9.081!5 63122
:5: 5.2244 31847 5:2261 84131 512279 34653 5:2296 a3419 5.2314 30432
l/5 3.7534 55355 3.7544 62453 3.7554 68472 3.7564 7341; 3.7574 77202
1
54 ELEMENTARY ANALYTICAL METHODS
.,-
795 796 797 ,-
7911
: 6 32025 6 33616 6 35209 6 36804 6' 38401
5024 59875 5043 58336 5062 61573 5081 69592 5100 82399
I
: 11 4.0755 58368
5 14 1 3.2563 71136
I 11
17 4.0349
14 2.5630
3.2158 04737
07803
19075 17 2.6018 40538
7” II 20
26 2.0427
23
29 1.2975
1.0341
1.6280 17219
49232
52362
45624 20 2.0788 70590
8 23 1.66101 17601
26 I 1.3271 53063
1: 29)1.0603 95298
24 (69)4.0626 65702 :69)4.1871 02820 (69)4.3151 87922 (69)4.4470 23172 (69)4.5827 13463
l/2 ( 1’;. y; ;‘i:;; ( 1’;. ;‘6;; y& ( 1)2.8231 18843 ( 1) $ ;;;i "3;;;; ( 1)2.8266# 58805
l/3 9.2715 59160 9.279? 08064
l/4 5:3099 66512 5:3116 35526 5.3133 02968 5:3149 68841 5.316b 33150
l/5 3.8025 36800 3.8034 92932 3.8044 48104 3.8054 02317 3.8063 55574
56 ELEMENTARY ANALYTICAL METHODS
%:f
80748
22686
00899
I17 I 4.3473
14
11 2.8663
3.5300
45103
44224
95910
13479
00190
34554
41493
52533
11)4.3903 34592
40945 41510s
06129 32047 25547
59670 21222 52870
24 (6936.3626 85441 (69)6.5539 10420 (69)6.7506 36166 (69)6.9530 13847 (69)7.1611 98588
l/2 ( 1)2.8460 49694 ( 1)2.8478
9.3255 06173
32030 ( 1)2,8495 61370 ( 1)2.8513 15486 ( 1)2.8530 68524
l/3 9.3216 97518 9.3293 63391 9.3331 9160H 9.3370 16687
l/4 5.3348 38230 5.3364 84023 5.3381 28295 5.3397 71049 5.3414 12288
l/5 3.8167 78910 3.8177 20859 3.8186 61880 3.8196 01974 3.9205 41144
9"
10 (29)1.3913 34555
24 (69)8.5414 66801 ( 69) 8.7949 98523 (69) 9.0557 33244 ( 69) 9.3238 66467 (69) 9.5995 98755
w ( lj2.8635 64213 ( "29';;;; ;i;;; ( 1)2.8670 54237 ( 1)2.8687 97658 ( 1)2.8705 40019
l/3 9.3599 01623 5:3528 58822 9.3675 05121 9.3713 02245 9.3750 96295
l/4 5.3512 28095 5.3544 88059 5.3561 15810 5.3577 42079
l/5 3.8261 56858 3.8270 89612 3.8280 21458 3.8289 5239; 3.8298 02432
1 1
ELEMENTARY ANALYTICAL METHODS 57
POWERS AND ROOTS nk Table 3.1
k
1
2
825 826 827 _-.
828 829
6 80625 b 82276 b 83929 b a5564 b 87241
3 5615 15625 5635 59976 5656 09283 5676 63552 5697 22789
4 11 4.6325 03906
14 3.8218 15723 3.8683 b5913
2 17 3.1529 97971
20 I 2.6012 23326
i 23 2.1460 09244 23 2.1879 83671
26 I 1.7704 57626 26 1.8094 62496
1: 29)1.4bOb 27542 i 29 I 1.4964 25484
24 69)9.8831 35853 (70)1.0174 68882 (70)1.0474 47415 (70)1.0782 71392 (70)1.1099 63591
l/2 1)2.8722 81323 ( 1)2.8740 21573 ( 1)2.8757 b0769 ( 1)2.8774 98914 ( 1)2.8792 36010
l/3 9.3788 87277 9.3826 75196 9.3864 60060 9.3902 41873 9.3940 20643
l/4 5.3593 bb869 5.3609 90182 5.3626 12021 5.3642 32391 5.3658 51293
l/5 3.8308 11564 3.8317 39795 3.8326 67128 3.8335 93565 3.8345 19107
III 11
17 4.8612
14 3.3893
4.0591 24550
b8999
27006 I
20 2.8301 23115
23 2.3631 52801
26 1.9732 32589 I
29 1.6476 49211 (29)1.6'375 41959
(70)1.3197 00592 (70)1.3581 59133 (70)1.3976 90431 70)1.4383 23072 (70) 1.4800' 86372
( 1y;; y; ( 1y;;; ;;;;; ( 1’;. i;:f ;:q2:; 1)2.8948 22965 1) 2.8965 49672
9.4278 93606 9.4316, 42272
513755 34071 5:3?71 42790 5:3787 50067 5.3803 55904 5.3819 60304
3.8400 53677 3.8409 73010 3.8418 91464 3.8428 09040 3.8437 25741
2
7
i
10 29)1;812i 90531
24 (70)1.5230 10388 (70)1.5671 25939 ( 70) 1. b124 b4626 70)1.6590 58848 70)1.7069 41821
l/2 ( 1)2,8982 75349 ( 1)2.9000 00000 ( 1)2.9017 23626 1)2.9034 46228 1)2.9051 b7809
l/3 9.4353 87961 9.4391 30677 9.4428 70428 9.4466 07220 9.4503 41057
l/4 5.3835 b3271 5.3851 64807 5.3867 64916 5.3883 63600 5.3899 60862
l/5 3.8446 41568 3.0455 56523 3.8464 70609 3.8473 83826 3.8482 96177
:
3 6141
850
7 22500
25000 6162
_.___
7 7A?“l
95051 6184
7 852
25904
70208 6206
857
7 27609
50477 6228
854
7 29%6
35864
5” 11)5.2200
14)4.4370
62500
53125 I 11)5.2446
14)4.4632
70884
14922
11)5.2693
14 4.4094
66172
99979
(11)5.2941
(14)4.5159
48569
08729
11)5.3190
14 4.5424
18279
41610
F (17)3.7714
(20)3.2057
95156
70883
(17 3.7981
1
20 3.2322
95899
64710
17 3.8250
I
20 3.2589
53982
45993
(17)3.8520
(20 3.2858
70146
15835
17 1 3.8792
20)3.3128
45135
75345
10
i (23)2.7249
1.9687
05250
44043
23)2.7506
26)2.3408
I 29)1.9920
57268
09335
28744
23)2.7766 21986 (23 1 2.8028
(26)2.3907
(29)2.0393
00907
89174
43165
23)2.8291 95545
24 (70)2.0232 71747 (70)2.0811 79034 (70)2.1406 72719 (70)2.2017 94325 70)2.2645 86409
l/2 ( 1)2.9154 75947 ( 1)2.9171 90429 ( 1)2.9189 03904 ( 1)2.9206 16373 1)2.9223 27839
1/q
l/3
l/5 ,
9.4726
5.3995
3.8537
82372
14744
52195
9.4763
5.4011
3.8546
95693
02137
58534
9.4801
5.4026
3.8555
06107
88131
64021
9.4838
5.4042
3.8564
13619
72729
68659
9.4875
5.4058
3.8573
18234
55935
72440
858
6250
7 31025
26315 6272
_-._.
856
7 327%
22016 6294
857
7 34449
22793 6316
1 36164
28712 6338
859
7 37881
39779
(11 1 5.3439 75506 I 11)5.3690 20457 11)5.3941 53336 11)5.4193 74349
(14 4.5690 99058 14 4.5958 81511 14)4.6227 89409 I 14)4.6498 23191
I 20)3.3401
I 17)3.9065
26)2.4417
23)2.8558
79694
25639
15345
07421
17 1 3.9340
20
1
3.3675
(23 2.8826
14574
67035
38067
1
17 3.9617
20 3.3952
23 2.9096
30523
03059
89021
(17)3.9895
(20)3.4230
48298
32440
(17 4.0175
lZOj3.4510
23 2.9644
28654
57114
58061
24
(29)2.0876
(70)2.3290
66620
92589 (70)2.3953 57569
I
26 2.4936
29 2.1370
70)2.4634
03491
18192
27165 (70)2.5333 48329
I
(26 2.5464
(29 2.1874
(70)2.6051
69474
17279
69182
l/2 ( 1)2.9240 38303 ( 1)2.9257 47768 1)2.9274 56234 ( 1)2.9291 63703 ( 1)2.9308 70178
l/3 9.4912 19958 9.4949 18797 9.4986 14756 9.5023 07842 9.5059 98059
5.4074 31751 5.4090 18180 5.4105 97225 5.4121 74889 5.4137 51174
::z 3.8582 75391 3.8591 77490 3.8600 78746 3.8609 79161 3.8618 78737
24 (70)3.5355 91351 (70)3.6344 25075 70)3.7359 03403 (70)3.8400 93943 (70)3.9470 65953
l/2 ( 1)2.9495 76241 ( 1) ;. p; ;;g;: 1)2.9529 64612 ( 1) ;. ;;y; ;;;g ( 1)2.9563 49100
l/3 9.5464 02709 9.5537 12362 9.5610 10846
l/4 5.4310 00130 5:4325 60090 5.4341 18707 514356 75984 5.4372 31924
l/5 3.8117 19185 3.0726 08827 3.8734 97651 3.0743 85661 3.8752 72857
1 1
21,EMENTARY ANALYTICAL METHODS 59
POWERS AND ROOTS nk Tabile 3.1
k
a75 a76 877 a78 879
7 65625 7 67376 69129 7 70884 7 72641
6699 21875 6122 21376 6745 26133 676.3 36152 6791 51439
;11)5.8886 59254 11)5.9155 94186 (11)5.9426 21415 (11)5.6697 41149
14)5.1584 65506 14)5.1879 76101 (14 5.2176 21602
'17)4.5188 15784 (17 4.5498 55041 (17 4.5810 71767
'20 3.9584 82626 (20 13.9902 22871 (20 4.0221 81011
23 3.4676 30781 (23)3.4994 25458 (23 i 3.5314 74928
26 I 3.0376 44564 (26 3.1006 34987
29)2.6609 76638 [:~~::E E: (29 12.7223 57518 (29j2.7535 23268
(70)4.0568 90376 70)4.1696 39882 (70)4.2853 88904 (70)4.4042 13682 (70)4.5261 92303
( 1)2.9580 39892 1)2.9597 29717 ( l’p; 18579 ( 1)2.9631 06478 ( 1)2.9647 93416
9.5646 55914 9.5682 98205 37725 9.5755 74480 9.5192 08475
5.4387 86530 5.4403 39803 514418 91747 5.4434 42365 5.4449 91658
3.8761 59242 3.8770 44816 3.8779 29583 3.8788 13542 3.8796 96696
7169
895
8 01025
17375 7193
896
8 02816
23136 7717
a97
8 04609
34213 7241
898
8 06404
50792
-..
8 08201
7265 72699
111 1 6.4451 35299 56429 11)6.5028 74112
14 5.7748 41228 38917 14 5.8395 80953
17 I 5.1742 57740 03608 17 1 5.2439 43696
120 4.6361 34935 76231 20 4.7090 61439
23 4.1539 76902 23 4.2287 31112
26 3.7219 63304 E2” 26 I 3.1974 05980
29 3.3348 79120 86240 29)3.4100 70570
24 (70)6.9783 51604 70)7.1679 04854 (70)7.3623 86846 (70)7.5619 20026 (70)7.7666 29743
l/2 ( 1)2.9916 55060 1)2.9933 25909 ( 11;. ;;i; 95826 ( 1)2.9966 64813 ( 1)2.9983 32870
l/3 9.6369 81200 9.6405 69057 54244 9.6477 36769 9.6513 16634
5.4696 02417 5.4711 29599 5:4726 55504 5.4741 a0133 5.4151 03489
3.8937 19006 3.8945 88722 3.8954 57662 3.8963 25828 3.8971 93220
905 .__
9"h 907 908 909
8 19025 8 20836 8 22649 _ _.._.
R 744h4 8 26281
7412 17625 7436 77416 7461 42643 7486 13312 7510 89429
11)6.7080 19506 I 11
17 I 6.1720
14 5.6042
6.7974 08873
18909
47257 (11)6.8274 02910
14)6.0707 57653 (14)6.2061 09245
17)5.4940 35676 17)5.6413 53304
20 4.9721 02287 20)5.0886 30769 20)5.1279 90153
23 4.4997 52570 4.5799 24265 I 23)4.6613 43049
26 4.0722 76076 26)4.2371 60832
29 I 3.6854 09848 3.7676 70117 I 29)3.8515 79196
24 (70)9.1109 96943 (70)9.3557 09844 (70)9.6067 14616 (70)9.8641 65825 (71)1.0128 22166
l/2 ( 1)3.0083 21791 ( 1';. y; y; ( "6 ;'8;: y;;;; ( 1)3.0149 62686
v3 9.6727 40271 9.6869 70141
l/4 5.4848 17035 5:4863 31551 5:4893 56824 5.4908 67587
l/5 3.9023 81426 3.9032 43449 3.9041 04712 3.9049 65216 3.9058 24962
24 (71)1.0399 04400 (71)1.0676 79852 (71)1.0961 65476 (71)1.1253 78622 (71)1.1553 37042
l/2 ( 1)3.0166
9.6905
20626
21083
( 1)3.0182 77655 ( 1)3.0199 33774 ( 1)3.0215
9.7011
88986
58327
1)3.0232
9.7046
43292
98896
l/3 9.6940 69425 9.6976 15172
l/4 5.4923 77104 5.4938 85370 5.4953 92410 5.4968 98203 5.4984 02760
l/5 3.9066 83951 3.9075 42186 3.9083 99668 3.9092 56397 3.9101 12376
.__
97n 923 924
8 46400 8 51929 8 53776
7786 88000 7812 29961 7837 77448 7063 30467 7888 89024
(11)7.2893 34582
(14)6.7353 45154
23 5.2221 26266
26 4.8148 00417
29)4.3438 84542 29 I 4.4392 45985
24 (71)1.3517 85726 (71)1.3874 94035 (71)1.4241 05308 (71)1.4616 41363 (71)1.5001 24518
l/2 ( 1)3.0331
9.7258
50178
88262
( 1)3.0347 98181 ( 1)3.0364 45290 ( 1)3.0380 91506 ( 1)3.0397 36831
l/3 9.7294 10859 9.7329 30906 9.7364 48410 9.7399 63373
l/4 5.5074 04268 5.5089 00236 5.5103 94986 5.5118 88520 5.5133 80842
l/5 3.9152 32576 3.9160 83344 3.9169 33373 3.9177 82664 3.9186 31220
1 1 1
+-;P] q-37)3] q-37)2] “[‘-.;‘l]
:ELEMENTARY ANALYTICAL METHODS 61
POWERS AND ROOTS nk Table 3.1
019
936 937 938 I/ I
.__
WI” 983 984
: 9 60400 9 66289 9 68256
941192000 9498 62087 9527 63904
: (11)9.2236 81600
(14)9.0392 07968
2 (17 8.8584 23809
20 1 8.6812 55332
l3 23)8.5076 30226 i 11
23 I19.1784
20
17
14 9.0223
9.3371.
8.7182
8.8689 79843
12862
44315
26396
99386
lo' I
24 (71)6.1578 03365 71)6.3103 89657 (71)6.4665 95666 (71)6.6265 03443 (71)6.7901 96812
l/2 ( 1)3.1304 95168 1)3.1320 91953 ( 1)3.1336 87923 ( l';.;:;; ;;3;; ( 1)3.1368 77428
l/3 9.9328 83884 9.9362 61267 9.9396 36356 9.9463 19667
l/4 5.5950 82813 5.5965 09584 5.5919 35265 5: 5993 59857 5.6007 83363
l/5 3.9650 18474 3.9658 27331 3.9666 35529 3.9674 43069 3.9682 49952
I
17 9.5872 81759
! 20 9.3206 53479 20 9.5201 70787
8 I 23 I 9.2274 46944 23 9.4535 29591
9 26)9.1351 72475 26 9.3873 54884
10 I 29)9.0438 20750 29 9.3216 43400
24 (71)7.8567 81408 (71)8.2466 98779 (71)8.4485 45822 (71)8.6551 22630
( 1)3.1464 26545 ( 1)3.1480 15248 ( 1)3.1496 03150 ( 1)3.1511 90251 ( 1)3.1527 76554
://: 9.9665 54934 9.9699 09547 9.9732 61904 9.9766 12009 9.9199 59866
5.6093 01690 5.6107 17644 5.6121 32527 5.6135 46340 5.6149 59086
$2 3.9730 77521 3.9738 79839 3.9746 81509 3.9754 82534 3.9762 82913
Contents
Page
Mathematical Prcbperties ..................... 67
4.1. Logarithmic Function ................... 67
4.2. Exponential Function ................... 69
4.3. Circular lknctions .................... 71
4.4. Inverse C rcular Functions ................. 79
4.5. Hyperbol:.c Functions ................... 83
4.6. Inverse Hyperbolic Functions ............... 86
References . . . . . . . . . . . . . . . . . . . . . . , , . . . 93
Table 4.6. Circular Sines and Cosines for Radian Arguments (0 <z < 1.6)1 . 142
sin 2, co9 5, ~=0(.001)1.6, 23D
Table 4.8. Circular Sines and Cosines for Large Radian Arguments
(0~z~1000). . . . . . . . . . . . . . . . . . . . . . . . . 175
sin 2, cos 2, ;e=O(l)lOO, 23D, ~=100(1)1000, 8D
Table 4.12. Circular Functions for the Argument f z (05 z < 1) . . . . 200
The author acknowledges the assistance of Lois K. Cherwinski and Elizabeth F. Godefroy
io the preparation and checking of the tables.
4. Elementary Transcendental Functions
Logarithmic, Exponential, Circular and Hyperbolic IFunctions
Mathematical Properties
4.1. Logarithmic Function Logarithmic Identities
n 0
1
-I
x 4.1.9 In z=ln z,-ln ;z2
z=r+iy
(-r<arg zl-arg z2<n)
(see 4.2.21)
The general logarithmic function is the many-
valued function Ln z defined by Logarithms to General Base
k being an arbitrary integer. In z is said to be the 4.1.22 log,, z=ln z/in lO=log,, e In z
principal branch of Ln z. = (.43429 44819 . . .)ln 2
67
68 ELEMENTARY TRANSCENDENTAL FUNCTIONS
jl421, zffl)
log10 x=alt+u3t”+t(x), t=(x-1)/(x+1)
4.1.29
le(x)l56XlO-’
In (z+a)=ln a+2
[(&)+f (&> al= .86304 a,=.36415
+f (&J5+ * . .]
4.1.42
(a>O, 921--a#z)
4.1.34 z<-In (l--2)< & * The approximations 4.1.41 to 4.1.44 are from C. Hast-
ings, Jr., Approsimations for digital computers. Princeton
(x<l, ‘XZO) ,Univ. Press, Princeton, N.J., 1955 (with permission).
ELEiMENTARY TRANSCENDENTAL FUNCTIONS 69
4.1.44 O_<z<l
4.1.52
In (1+2)=a~z+u~22+~~23+uq~4S~u~25+ug~6
4.1.53
fw’ + fws+f (2)
Approximation
4.1.45
in Terms of Chebyohev
O<z<l
Polynomials 3 -- &
S --
&+f)+ dz (n#--1)
Definite Integrals
Z’,*(X)=COS n8, cos ~=!kc-l (see chapter 22)
4.1.55
In (l+d=n$o -4,T,,*@)
4.1.56
n AZ n A,
0 . 37645 2813 6 -.. 00000 8503
1 34314 5750 00000 1250 4.1.57 Trdt-ln t-Zz(x)
- (see 5.1.3)
0
2 -: 02943 7252 ; -: 00000 0188
3 .00336 7089 9 00000 0029 4.2. Exponential Function
4 -. 00043 3276 10 -: 00000 0004
5 . 00005 9471 11 . 00000 0001 Beries Expansion
4.2.1
Differentiation Formulas
z 22 z3
e’=exp ~.=l+~+~+~+ . . . (z=z+iy)
4.1.46 & In z=Jj
where e is the real number defined in 4.1.16
4.1.49
S In z dz=z In z---z 4.2.5
d
z exp z=exp z
4.1.50
If N=u*,
of General
then z=Log,
Powers
4.2.27 e*?= fi
4.2.28 e2*ki = 1 (k any integer)
Exponential Inequalities
Continued Fractions
4.2.45 O<x<ln 2 d
4.2.51 d=d In a:
dz
e-z=1+a~x+a2x2+~3x3+~4x4+~~~5
+u6x*+u,x’+E(x) 4.2.52 -d y&(&-l
dz
le(x)112X10-10
4.2.53 $ Y=(l+ln z)z’
a, = - .99999 99995 u5= -- .00830 13598
u2= .49999 99206 u6= .00132 98820 Integration Formulas
aa=-. a’=--.00014 13161
u4= .04165 73475
4.2.54 eazdz= earla
s
4.2.46 6 O<xil
4.2.55
10Z=(l+a,x+azx2+a3;c”+a4x4)2+r(x) f
JE(x)~~7X10-4 J z”e”‘dz=s [(uz)n-n(az)n-1+n(n-1)(uz)n-2
+ . . . +(-1>m-in!+%!] (n20)
a1=1.14991 96 u3= .20800 30
a2= .6774323 a4= .12680 89 4.2.56
5 &=- eaz
4.2.47 O_<x<l
S s5 dz@>I>
(n-1)2"-'+laY
10z= (1 +alx+u2x2+a3x3+u4x4t a& (See chapters 5, 7 and 29 for other integrals
involving exponential functions.)
+w6+~7~7)2+~(~)
csc 2=-r--
sin 2
4.3.19 (zl+z2,=“~~t~;“;i”,,“zz’
1
cot
SW z=- 2 1
4.3.5
cos 2
Half-Angle Formulas
1
4.3.6 cot z=- tan 2 4.3.20 sin $= &(qc)’
Periodic Properties
22 l-z2 du
4.3.23 sin u=--,
1+z2 cos u=l+zz
Multiple-Angle Formulas
1
70
4.3.24 sin 22=2 sin ,a cos 2=,“,“t”,“,~2
4.3.44
Functions of Angles in Any Quadrant in Terms of Angles in the First Quadrant. (0 2 0 15, k any integer)
2kufe
cos $(&i-l) 0
-Jz
4
-
(d3- 1) -l/2
4 . 3 . 58
2x--i sinh
cot z_sincash 2y- cos 2x
2y
tan w& co -(2f43) 4 Modulus and Phase (Argument) of Circular Functions
csc j5<&- 1) 1 JR&-1, 24313 4.3.59 /sin zj= (sin2 z+sinh2 y)+
‘cash 2y-cos 2x 4
4.3.63
sin lj5/2 112
4-$ (Jc1) 0
Itan ‘l=(,cosh 2yfcos 2x
cot -1
-
4 -@+&I co cos z=l -$+2-S+.
. . .
.. @4<4
ELEMENTARY TRANSCENDENTAL FUNCTIONS 75
4.3.67 Inerlualities
23 225 1727
tan z=z+~+~+~+ ... 4.3.79
22n-1$- . .
+(-l).-i22.(22n-l)Bzn
(2n)! (lH<; > 4.3.80 sin 21 x_< tan 2
4.36%
12 7 31 4.3.81
csc 2=;+,+,,, 23+15120 z5+ . . .
+(-l)n-12(22n-‘-l)BZn
(2n) !
*2n-l,+
(bl<d 4.3.82 (O<x<l)
4.3.69 4.3.33 lsinh y/ 5 lsin ZJ5 cash y
z2 5z4 61z6
set ~=l+~+~+m+. .. 4.3.84 lsinh y/ 5 Jcos 21 Icosh y
4.3.85 lcsc z[ Icschlyl
4.3.70
+(- l)n&
(2n) !
22n-- . .*
( ) lH<5
4.3.86 /cos zl Scosh(z)
4.3.87 /sin zI Isinhlzl
1 2 23 2z5
cot z=;-j-~-py- ...
4.3.88 lcos zl< 2, /sin 2[<$[2( (l2l<O
_ (-- w122n&n 22n-1,- .
(2n) ! (l2l<7r)
Infinite Products
4.3.71
4.3.89 sin 2=2 jiI (1-&)
In (I4<a>
4.3.90 cos 2=,i1 (l-
4.3.72 (2&$ >
l* cos 2=.& (-- w2n-1w- m, p (14<37r>
Expansion in Partial Fractions
W-1 n(2n) !
4.3.73 4.3.91
ln tan
T=gI 2 (- l)‘-‘y..;;~l- 1) Bzn 22n
1
w<t’d 4.3.92 csc224 -
k-i--m (z-kT)2
where B, and E, are the Bernoulli and Euler
numbers (see chapter 23).
Limiting Values
4.3.93
sin 2
4.3.74 lim ----xl (z#O,f?r,f%, . . .>
z+o x
Continued Fractions
4.3.75 lim tan=1
z-10 x 2 22 22 22
4.3.94 tan z=c ---
3- 5- 7- ... (2 #i*n*)
4.3.76 lim n sin E=x
n+- 4.3.95
4.3.77 lim n tan 2=x a tan 2 (1-a’) tan2 2 (4--a2) tan2 2
tan az=------
?a+- 1+ 3f 5+
Polynomial Approximations 1
4.3.101 o<x-$
4.3.96 o<x<;
tan x
-= 2 1 +a~x2+u4x4+agx6+agx8+~,ox10
la(s)] <2x10-5
4.3.102 05x<;
a2= - .49670 a4= .03705 a2= - .33333 33410 aa= - .00020 78504
a4= - .02222 20287 alo= - .00002 62619
4.3.99 O<X$ a6=-. 00211 77168
a2= - .49999 99963 aa= .00002 47609 T: (x) = cos ti, cos 8=2x- 1 (see chapter 22)
n A, n A,
4.3.100 o<x$ 0 1.27627 8962 0 .47200 12 16
1 -.28526 1569 1 - .49940 3258
tan 2
--l+a2x2+a4x4+t(x)
2 2 .00911 8016 2 .02799 2080
3 - .00013 6587 3 - .00059 6695
(a(x)(<lXlo-3
4 .OOOOO1185 4 .OOOOO6704
a*= .31755 a,=.20330 5 -.ooooo 0007 5 - .ooooo 0047
7 The a proximations 4.3.96 to 4.3.103 are from B. Carl- n The approximations 4.3.104 are from C. W. Clenshan,
son, M. e oldstein, Rational approximation of functions, Polynomial approxinlations to elementary functions,
Los Alamos Scientific Laboratory LA-1943, Los Alamos, Math. Tables Aids Camp. 8, 143-147 (1954) (with per-
N. Mex., 1955 (with permission). mission).
*see page Ix.
ELEMENTARY TRANSCENDENTAL FUNCTIONS
77
Differentiation Formulas
4.3.122
4.3.105
d
2; sin z=cos 2 S z dz
sin”=(n-
-2 cos 2
1) sinn-l z-(n-
1
1) (n-2) sinnm2 2
d
4.3.106 z cos z=-sin z
4.3.123
4.3.107 -$ tan z=sec2 z
.
zn cos zdz=zn sin z-n
S zn-‘sin zdz
d 4.3.124
4.3.108 & csc z=-csc 2 cot 2
4.3.109
d
z set z=sec z tan 2
S S‘ sdz (n>l)
4.3.110
d
-& cot 2=-csc2 2
4.3.125
SLdz=z
cos2 2
tan z+ln cos 2
4.3.126
2 dz sin
S 2 2
4.3.111 cosn=+1) cop-1 z-(%-l) (La) cosn-2Z
4.3.112
++-2)
(n-1) S5 cm>21
4.3.114
.I-
cos z dz=sin ;:
S sin”’ 2 cosnm2 z dz
4.3.115
4.3.116
S tan zdz=-ln cosz==ln secz =-
sinme 2 cosnfl
m+n
2
I-
J csc zdz=ln tan :=ln (csc z-cot z)=- 1 In- 1-cosz
2 lfcosz
S sin”‘-* 2 COS”
(m#-n)
2 dz
4.3.117 4.3.128
1
s
seczdz=ln(secz+tanz)=lntan
S sinm z”Z,oP z=(n--1) sinmS1 2 cosnel 2
=Inverse
(n>l)
4.3.118
4.3.119
S cot zdz=ln sin z=--1n csc z =(m-1) sin:: 2 cosnml 2
m-b-2 dz
(m>l>
4.3.120
S 4.3.129
S tannzdz=ts-fian”-2zda (n#l>
4.3.121
S Adz=--z
sin* 2
cot z-rln sin z 4.3.130
S cot”zdz=---
COtn-’
n-l
2
S Cotn-2ZdZ (n#l)
78 ELEMENTARY TRANSCENDENTAL FUNCTIONS
4.3.131
clz =------ 2
a tan 4.3.141
S 0
lsin2 nt dt== *cos2nt dt=%
S 0
=- 1
(bz-a2)+
In
a tan
0
z +b-(b2-a*))
1 4.3.142
S 0
msirltmt &=;
(m>O)
S dz
a+b cos z
=-----
2
(a2-b2)*
arctan (a”-b2)i W>W 4.3.144
S0
-sin t2 dt=
S 0
-cos t2 dt=i
J
i
4.3.145
=&h-l
(b-a)
(b-a)
tan z+(b2-a2)+
2
tan c- (b2-a2)+
1 S 0
r/2
In sin t dt=
f
.O
s/2
In cos t dt=-; In 2
2
m cos mt
(b2>a?) 4.3.146
S
o T+T dt=ae-”
4.3.134
S dz
-------=tan
1+cos 2
%
(See chapters 5 and. 7 for other integrals
ing circular functions.)
involv-
4.3.135
S dz=-cot
1-cos 2
-2
2
(See [5.3] for Fourier transforms.)
4.3.147
4.3.136
Formulas for Solution of Plane Right Triangles
4.3.137
,
4.3.138 C b A
eazsin“-I bz
S earsinn bz dz=
a2+n2b2
(a sin bz-nb cos bz)
If A, B and C are the vertices (C the right
+ n(n-l)b2
a2+n2b2 S eazsinne2 bz dz
angle), and a, b and c the sides opposite respectively,
sin A=:=- 1
4.3.139
c CSCA
S0
cos mt cos nt dt=O
exsecant A=exsec A=sec A-l
ELEMENTARY TRANSCENDENTAL FUNCTIONS 79
4.3.148 4.4. Inverse Circular Functions
Formulaa for Solution of Plane Triangles Definitions
4.4.1
s * dt
~rcsm z= s o (l-t*)* (2=x+$
4.4.2
0
C l dt
A
4.4.3
arccos z=
S ----=9-arcsin
z (l-t*)* 2
2
A/ ‘1
C
(I
A
m
0 -I 0 +I
C
b
I
-i
sin A
y=,=-- sin B sin C
sin a sm b sm c
4.4.12 (x real)
t=Arctan z=arctan z+ka (9#-1)
arcsin x and arctan x 0 < y 5 a/2 - ?r/2 < y<O 4.4.31 Arccot
=Arcsin{wdz[(l--zT) (l-$)1’}
4.4.15 arccos (- 2) = 7r- arccos 2
=Arccos [zz(l-2~)~~zz1(l-z~)*]
4.4.16 arctan (- 2) = - arctan 2 4.4.36
4.4.17 arccsc (- 2) = - arccsc 2 Arctan z1 &Arccot z2
arctan x=-
1+:8x2+‘(“)
/e(x)1 55x10-5
21 A,
88137 3587 i!t A,
00000 3821
4.4.59
S arccos z dz:=z arccos z-(3-z2)t
:: -: 01113
10589 2925
5843 7 -: . 00000 0570
0086 4.4.60 arctan 2 dz==z arctan 2--3ln (1-j-z”)
s
3 -: 00138 1195 ii -. 00000 0013
4 .00018 5743 10 . 00000 0002 4.4.61
5 -. 00002 6215
arccsc 2 d2=2 arccsc 2fln [~+(2~-1)*]
c
For 2 >l, use arctan s=$7r-arctan (l/z) s
O<arccsc 2<5
4.4.51 -$Ji<X<$JZ
4.4.62
.
o<x<@ arcsec z dz=.z arcsec zTln [2+(22-l)+]
n A?& n 4
;< arcsec 2<ir
0 1.05123 1959 : 00000 5881 [
; : 05494
00408 6487
0631 : 00000 0777
0107 4.4.63
i .00004 6985
.00040 7890 9 :. 00000 0015
0002 S arccot 2 dz:=z arccot 2+$ In (l+z2)
4.4.65
4.4.52 & arcsin z=(l-z2)-+ 9+1
4.4.53
S 2" arcsin 2 dz=--
n-j-1
arcsin
(n#-1)
$ arccos z=-(I-z2)-f
4.4.66
d 1
4.4.54
z
arctan z=-
1+22 S z arccos z dz.7 arccos z-f (l--z*)+
d -1 4.4.67
4.4.55 z arccot 2=1+22
4.4.56
d 1
S 2n arccos 2 dz=Ez
n,+l
arccos
S
z arccot z dz=i (14-S) nrccot z+:
h.5.12 coth z=i cot iz
Periodic Properties
4.4.71
,$a+1 1.5.13 sinh (2+2k7ri) =sinh z
S zn arccot z dz=-
n+l
arcco t z J,-
(k any integer)
(n#-1) cash (zf2ksi) =cosh z
a.5.14
4.5. Hyperbolic Functions 1.5.15 tanh (z+k&)=tanh z
Definitions
Relations Between Hyperbolic Functions
ez-e-2
4.5.1 sinh z=- (z=x+iy> 4.5.16 cosh2 z-sinh2 z=l
2
4.5.17 tanh2 z+sech2 z= 1
e”+e-”
4.5.2 cash z=- coth2 z-csch2 z= 1
2 4.5.18
+sinh z1 sinh zz
4.5.44
cash z,-cash z2=2 s:inh (p) sinh (v)
4.5.45
sinh (21+z2)
tad zl+tanh %=E;h L 2 cosh 2
1 2
4.5.46
sinh (2, +2,)
Multiple-Angle Formulas coth z,+coth zz=- smh 2, sinh zz
4.5.31 sinh 22=2 sinh z cash ~=~~‘a~~f, Relations Between Squares of Hyperbolic Sines and
COosines
4.5.47
4.5.32 cash 22=2 cosh2 z--l=2 sinh2 z-j-1 sinh2 zL-sinh2 zz=sinh (zl+zz) sinh (zl--- z2)
4.5.34 sinh 32=3 sinh 2+4 sinh3 z Hyperbolic Functions in z;;;s of Real and Imaginary
4.5.36 sinh 42=4 sinh3 z cash 2+4 cosh3 z sinh z 4.5.49 sinh z=sinh x cos y+i cash x sin y
4.5.37 cash 4z=cosh4 z-j-6 sinh2 z cosh2 z+sinh’ z 4.5.50 cash z=cosh x cos yfi sinh x sin y
Products of Hyperbolic Sines and Cosines 4 . 5 .51 tanh 2=sinh 2x-j-i sin 2y
cash 2x+cos 2y
4.5.38 2 sinh z1 sinh zz=cosh (zl+z2)
4 . 5 .52 coth 2=sinh 2x---i sin 2y
-cash (Q-.zz>
cash 2x-cos 2y
4.5.39 2 cash 21 cash zz=cosh (zl+zJ De Moivre’s Theorem
4.5.62 smh
’
Series
~=2+9+3+~+
Expansions
23 ZfJ 2
. . .
4.5.69 cash z=fi
k=l C
l+
(2Zzf)%2 1
(14-c -1 Continued Fraction
22 22 24
4.5.63 cash z=l+$+$+;+ ... 4.5.70 tanh 2=& 3+ 5f 7+ . . .
. . . (l4< m)
2 #i ifnni
( >
Differentiation Formulas
* +
... +22W2”-1)B,,
(h)!
22n-l+
’’* d
4.5.71
z sinh z=cosh 2
d
4.5.65 4.5.72 cash z=sinh 2
z
csch z=I ’ ’ -j&o25+...
z--T;+360 23
4.5.73 & tanh z=sech2 z
-2(22”-‘-l)B2, 2”-1
+...
zd csch z=-csch
(2n)! 2
4.5.74 z coth z
(l4<3d
d 4.5.87
4.5.75 z sech z=-sech z tanh z
Wl)
Integration Formulas 4.5.88
4.5.77
s
sinh z dz=cosh z s coth” z de= -%!@.?+SCoth”-2
n-l
z dz
Wl>
4.5.78 cash zdz=sinh z (See chapters 5 and 7 for other integrals in-
s volving hyperbolic functions.)
4.5.80
S csch zdz=ln tanh :
4.6.1 arcsinh z=
S Oz(lTt2)1 (z=x+iy)
4.5.81
S sech z dz=arctan (sinh z)
4.6.2 arccosh z=
S iz (t2fl)t
4.5.82
S coth zdz=ln sinh z 4.6.3 arctanh z=
S z dt
-
0 1-P
4.5.83 The paths of integration must not cross the fol-
P r
en sinh z dz=z” cash z-n 9-i cash z dz lowing cuts.
J J
4.6.1 imaginary axis from --im to --i and i
4.5.84 to iw
+- S
n-l
m+n
sinh” z coshnm2 z dz
4.6.4
4.6.5
arccsch z = arcsinh 1/z
q
iy
S
I-
m-l 9 iy
-m+- sinhm+ z cash” z dz
ti
X --CD 4-I X X
(m+n#O>
3 0 -I 0 +I
-i
4.5.86
S dz
sinh” z coshn z=s
-1
sinh”-’
1
z coshn-l z
arcsinh z
t-
arccosh .z arctanh z
- m+n-2
m--l S dz
sinhmm2 z cash” z (m#l>
1 1
=-
n-l sinh”-’ 5 cash”-’ z
arccsch z 1 arcsech z 1 arccothz
+
m+n-2
n-l Ssinh”
dz
z cosh”-2 z Wl>
FIGURE 4.7. Branch cuts for
junctions.
inverse hyperbolic
ELESMENTARY TRANSCENDENTAL FUNCTIONS 87
4.6.7 arctanh z=arccoth z& $ri x+1
4.6.25 arccoth x=i In - (x2>1)
x-l
(see 4.5.60) (according as J&SO)
Fundamental Property
‘see pageII.
88 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Series Expansions
4.6.31 4.6.42 -$ arccoth. 2=(1-S)+
arcsinh z=z --.A-
2 . 3 z3++-& 25 Integration Formulas
1.3.5
-2.4.6.7 “+ ’ ’ * 4.6.43 arcsinh z dz=z arcsinh z-(1+,9)+
s
(lzl<l) 4.6.44 arccosh zdz=z arccosh z-(z2-1)f
s
1.3
=In 2z+1 2 ’ 222 - 2 * 4 * 424 4.6.45
Sarctanhzdz=z arctanh z++ln (l-9)
n+l
arctanh z -&Ssdz
(n#- 1)
4.6.55
4.6.39 2 arctanhz=(l-,9)-I
Sz arccsch z de=< arccsh z&-f (l+zz)* *
4.6.40 d
z arccsch z=T ’ (according as &z~O)
z(l+S))
4.6.66
(according as 9?z=O)
.z”arccsch z dz- -
p+1arccsch 2f
4.6.41 1 S n+l
2 arcsech z= 7
dz z(l-.zy (n# -1)
*See pageII.
ELEMENTARY TRANSCENDENTAL FUNCTIONS 89
4.6.57 4.6.59
z arcsech z dz=$ arcsech tri (l-z*)1 z*- 1
*
s
(acc.ording as %ZO) S z arccoth z dz- -2 arccoth z+
1
4.6.58 I 4.6.60
p+1 1 *ll+1
sz” arcsech z dz =-
nfl
arcsech z + -
-n+1 S zdz
(l--z*)’
z” arcc0t.h z dz=-
n+l
arccoth z+-
1
n+l
(n#-1) I (hrE-1)
Numerical Methods
4.7. Use and Extension of the Tables Example 2.
Compute xw314for x=9.19826 to 10D usjng t,he
NOTE: In the examples given it is assumed t,hat Table of Common Logarithms.
the arguments are esact. From Table 4.1, four-point Lagrangian intierpola-
tion gives log,, (9.19826) = .96370 56812. Then,
-; log,, (x) = - .72277 92609=9.27722 073!$1- 10.
Example 1. Computation of Common Logarithms.
Linear inverse interpolation in Table 4.1 yields
antilog (‘i.27722) =. 18933. For 10 place ndcurncy
To comput,e common lognrithms, the number subtnbulation with 4- oint Lngrnnginn intcr-
must be expressed in the form x. log, (1 Iz<lO, polnnt’s produces the tn Ele
- Q) 5 q _<= ). The common lognrithm of x .lOP
consists of nn integral part which is called the N A A:’
characteristic and a deamnl part which is called log,, N
the mnntissa. Table 4.1 gives the common .18933 .27721 94350
logarithm of x. 2 29379
.18934 .2’7724 23729 -13
2 29366
.18935 .2’7726 53095
X x. 10’ log,, 5 * 10’
By linear inverse interpolation
so09836 9.836. 1O-3 5.99281:35=(--200718 15) x-314= .18933 05685.
.09836 9.836. lo-’ -i.99281,85=(-1.00718 15) Example 3.
.9836 9.836.10~’ i.99281;35=(-0.00718 15) Convert log,, x t,o In x for x=.009836.
9.836 9.836. loo 0.99281 85 Using 4.1.23 nnd Table 4.1, In (.009836)=
In 10 log,, (.009836)=2.30258 5093 (-2.00718 15)
98.36 9.836.10! 1.99281:35 = -4.62170 62:.
983.6 9,836.10* 2.99281:35 Example 4.
Compute In x for x=.00278 t*o 6D.
Interpolation in Table 4.1 between 983 nnd 984 Using 4.1.7, 4.1.11 nndTable 4.2, ln (.00278)=
gives -99281 85 as t,he mantissa of 9836. In (.278.10-*)==ln (.278)-2 ln lo=-5.886304.
Note t.hat 5.99281 85=-3+.!39281 85. When Linenr lntrrpolntion between x=.002 nnd
p is negative the common logarithm can be x= .003 would ivc ln(.00278)=-5.808. TO
expressed in the alternative forms obtain 5 decimn f plncc accurncy with linenr
interpolatio’n it, is neccssnry t’hnt x>.175.
Example 11.
In (2+3i)=i In (22+32)+i arctan i
Compute eelsto 7s.
=L282475+i(.982794)
Let n=&. and d=the decimal part of Go*
In (-2+3i)=i In 13+i 7-arctan i
> Then
=1.282475fi(2.158799) exp x=exp (& ln lO)=exp [(n+d) In lo]
References
Texts [4.11] British Association for the Advancement of Science,
Mathematical Tables, vol. I. Circular and hyper-
14.11 B. Carlson! M. Goldstein, R,ttional approximation bolic functions, exponential, sine and cosine
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LA-1943 (Los Alamos, N. Mex., 1955). Hermitian probabilit functions, 3d ed. (Cam-
[4.2] C. W. Clenshaw, Polynomial approximations to bridge Univ. Press, 8 ambridge, England, 1951).
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14.41 G. H. Hardy, A course of pure mathematics, 9th ed.
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[4.5] C. Hastings, Jr., Approximations for digital com-
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[4.6] C. Hastings, Jr., Note 143, Math. Tables Aids
Comp. 6, 68 (1953). Laboratory., Tables of the
[4.7] E. W. Hobson, A treatise on plane tri~metry, [4.16] Harvard. Computation
4th ed. (Cambridge Univ. Press, function arcsin z (Harvard Univ. Press, Cam-
ambndge, bridge, Mass., 1956). z=x+iy, O<x<475,
England, 1918). 0 5~ 1475, 6D, varying intervals.
[4.8] H. S. Wall, Analytic theory of continued fractions
(D. Van Nostrand Co., Iuc., New York, N.Y., [4.17] Harvard Computation Laboratory, Tables of inverse
1948). hyperbolic functions (Harvard Univ. Press,
Cambridge, Mass., 1949). arctanh x, O<z<l;
Tables arcsinh z, 0+<3.5; arccosh z, 1+<3.5;
[4.9] E. P. Adams, Smithsonian mathematical formulae arcsinhs, arccosh x, 3.5_<~<22980, 9D, varying
and tables of elliptic functions, 3d reprint (The intervals.
l$~$onian Institution, Washington, D.C., [4.18] National Bureau of Standards, Tables of lo”, Applied
Math. Series 27 (U.S. Government Printing
[4.10] H. Andoyer, Nouvelles tables tri onometrlques Office, Washington, D.C., 1953). z=O(.OOOO1)1,
fondamentales (Hermann et fils, !baris, France, 10 D. Radix- table of 10n’lo-P, n=1(1)999,
1916). p=3(3)15, 15D.
94 ELEMENTARY TRANSCENDENTAL FUNCTIONS
14.191 National Bureau of Standards, Table of natural [4.28] National Bureau of Standards, Table of arcsin z
logarithms for arguments between zero and five (Columbia Univ. Fress, New York, N.Y., 1945).
to sixteen decimal places, 2d ed., Applied Math. arcsin z, s=0(.0001).989(.00001) 1, 12D; auxil-
Series 31 (U.S. Government Printing office, iary table of f(u) =[$a-arcsin (l--)]/(2v) %,
Washington, D.C., 1953). 2=0(.0001)5, 16 D. v=O(.OOOOl).OOO5, 13D.
[4.20] National Bureau of Standards, Tables of the ex- [4.29] National Bureau of Standards, Tables of arctan z,
ponential function eZ, 3d ed., Applied Math. Series 2d ed., Applied Math. Series 26 (U.S. Govern-
14 (U.S. Government Printing Office, Washing- ment Printing Office, Washington D.C., 1953).
ton, D.C., 1951). Z= -2.4999(.0001) .9999, s=0(.001,7(.01)50(.1)300(1)2000(10)10000, 12D.
18D, s=l(.OOOl) 2.4999, 15D, z=2.5(.001)4.999, [4.30] National Bureau of 8tandards, Table of hyperbolic
15D, z=5(.01)9.99, 12D, z== -.000099(.000001) sines and cosines, x=2 to x=10, Applied Math.
.000099, 18D, z= - 100(1)100, lQS, z= -9X Series 45 (U.S. Government Printing Office,
10-n(10-n)9X, lo-*, n= 10, 9, 8, 7, 18D; values of Washington, D.C., 1955). 2=2(.001)10, 9s.
[4.31] B. 0. Peirce, A short table of integrals, 4th ed.
[4.21] N~t~?al%ur~~~2,5u56oD. Standards, Table of the de- (Ginn and Co., Boston, Mass., 1956).
scending exponential, 2=2.5 to z=lO, Applied [4.32] J. Peters, Ten-place logarithm table, ~01s. 1, 2
Math. Series 46 (U.S. Government Printing (together with an appendix of mathematical
Office, Washington, D.C., 1955). s=2.5(.001)10, tables) (Berlin, 1~922; rev. ed., Frederick Ungar
ZOU. Publ. Co., New York, N.Y., 1957).
[4.22] National Bureau of Standards, Tables of sines and [4.33] J. Peters, Seven-place values of trigonometric
cosines for radian arguments, 2d ed., Applied functions for every thousandth of a de ree
Math. Series 43 (U.S. Government Printing (Berlin-Friedenau., 1918; D. Van Nostrand 5 o.,
Office, Washingtor, D.C., 1955). sin 5, cos z, Inc., New York, N.Y., 1942).
z=O(.OO1)25.2, O(l)lOO, 8D, x=10-“(lo-“)9X [4.34] L. W. Pollak, R.echentafeln zur harmonischen
lo-n, n=5,4,3, 2, 1, 15D, ~=0(.00001) 91, 12D. Analyse (Johann Ambrosius Barth, Leipzig,
[4.23] xational Bureau of Standards, Tables of circular Germany, 1926).
and hyperbolic sines and cosines for radian [4.35] A. J. Thompson, Standard table of logarithms to
arguments, 2d ed., Applied Math. Series 36 (U.S. twenty decimal places, Tracts for Computers, No.
Government Printing O&e, Washington, D.C., 22 (Cambridge Univ. Press, Cambridge, England,
1953). sin x, cos z, sinh Z, cash 2, z=O(.OOOl) and New York, :N.Y., 1952).
1.9999, O(.l)lO, QD. [4.36] J. Todd, Table of arctangents of rational numbers,
[4,24] National Bureau of Standards, Table of circular NBS Applied Math. Series 11 (U.S. Government
and hyperbolic tangents and cot,angents for radian Printing Office, ‘Washington, D.C., I D51). arctan
arguments, 2d printing (Columbia Univ. Press, m/n and arccot m/n, O<m<n<lOO, 12D; re-
Fey;; prk, N.Y., 1947). tan 5, cot Z, tanh z, ductions of arctan m/n, O<m<n<lOO; reduc-
) 2=0(.0001)2, 8D or 8S, z=O(.l)lO, tions of arctan 1%for reducible n <2089.
10D. [4.37] U.S. Department of Commerce, Coast and Geodetic
[4.25] National Bureau of Standards, Table of sines and Survey, Natural sines and cosines to eight decimal
cosines to fifteen decimal places at hundredths of places, Special Publication No. 231 (U.S. GOV-
a degree, Applied Math. Series 5 (U.S. Government ernment Printing Office, Washington, D.C., 1942).
Printing Office, Washington, D.C., 1949). sin 5, [4.38] C. E. Van Ostrand, Tables of the exponential
cos z, z=O”(.O1o)QOo, 15D; supplementary table function and of the circular sine and cosine to
of sin Z, cos z, z=l”(lo)8Qo, 30 D. radian arguments, Memoirs of the National
[4.26] National Bureau of Standards, Table of secants and Academy of Sciences 14, 5th Memoir (U.S.
cosecants to nine significant figures at hundredths Government Printing Office, Washington, D.C.,
of a degree! Applied Math. Series 40 (U.S. Gov- 1921).
ernment Printing Office, Washington, D.C., 1954). [4.39] B. V. Vega, Logarithmic tables of numbers and
[4.27] National Bureau of Standards, Tables of functions trigonometrical functions (G. E. Stechert & Co.,
and of zeros of functions, Collected short tables
of the Computation Laboratory, Applied Math. Ncm York, N.Y., 1905); log10 2, z=1(1)100000;
Series 37 (U.S. Government Printing Office, logarithms of the trigonometrical functions for
Washington, D.C., 1954). every ten seconds.
ELEMENTARY TRANSCENDENTAL FUNCTIONS 95
COMMON LOGARITHMS Table 4.1
[ 1
(-;I6
[(-F2
1 C-l)6
[ 1
For use of common logarithms see Examples 1-3. For 100<~,<135 interpolate in the range
1000<:~~<1350.Compiled from A. J. Thompson, Standard table of logarithms to twenty decimal
places, Tracts for Computers, No. 22. Cambridge Univ. Press, Cambridge, England, 1952 (with
permission).
96 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[ 1
C-l)4
[93 1 [ 1
(73
[c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 97
COMMON LOGARITHMS Table 4.1
X log10 x X loglcl 2 X log10 2 2 log10 x X log10 2
600 77815 12504 650 81291 33566 700 84509 80400 750 87506 12634 800 90308 99870
601 77887 44720 651 81358 39886 701 84571 80180 751 87563 99370 801 90363 25161
602 77959 64913 652 al424 75957 702 84633 71121 752 87621 78406 802 90417 43683
603 78031 73121 653 81491 31813 703 84695 53250 753 87679 49762 803 90471 55453
604 78103 69386 654 81557 77483 704 84757 26591 754 a7737 13459 804 90525 60487
605 78175 53747 655 81624 13000 705 84818 91170 755 87794 69516 805 90579 58804
606 78247 26242 656 81690 38394 706 84880 47011 756 87852 17955 806 90633 50418
607 78318 86911 657 81756 53696 707 84941 94138 757 87909 58795 807 90687 35347
608 78390 35793 658 81822 58936 708 85003 32577 758 87966 92056 808 90741 13608
609 78461 72926 659 81888 54146 709 85064 62352 759 88024 17759 809 90794 85216
610 78532 98350 660 81954 39355 710 85125 83487 760 88081 35923 810 90848 50189
611 78604 12102 661 82020 14595 711 85186 96007 761 88138 46568 811 90902 085412
612 78675 14221 662 82085 79894 712 85247 99936 762 88195 49713 812 90955 6029:2
613 78746 04745 663 82151 35284 713 85308 95299 763 88252 45380 813 91009 05456
614 78816 83711 664 82216 80794 714 85369 82118 764 88309 33586 814 91062 44049
615 78887 51158 665 82282 16453 715 85430 60418 765 88366 14352 815 91115 76087
616 78958 07122 666 82347 42292 716 85491 30223 766 88422 87696 816 91169 01588
617 79028 51640 667 82412 58339 7L7 85551 91557 767 88479 53639 817 91222 20565
618 79098 a4751 668 82477 64625 718 85612 44442 768 88536 12200 818 91275 33037
619 79169 06490 669 82542 61178 719 85672 88904 769 88592 63398 819 91328 39018
620 79239 16895 670 82607 48027 720 85733 24964 770 88649 07252 820 91381 38524
621 79309 16002 671 82672 25202 721 a5793 52647 771 88705 43781 821 91434 31571
622 79379 03847 672 82736 92731 722 85853 71976 772 88761 73003 822 91487 18175
623 79448 80467 673 82801 50642 723 85913 82973 773 88817 94939 823 91539 98352
624 79518 45897 674 82865 98965 724 a5973 85662 774 88874 09607 824 91592 72117
625 79588 00173 675 82930 37728 725 86033 80066 775 88930 17025 825 91645 39485
626 79657 43332 676 82994 66959 726 86093 66207 776 88986 17213 826 91698 00473
627 79726 75408 677 83058 86687 727 86153 44109 777 89042 10188 827 91750 55096
628 79795 96437 678 83122 96939 728 86213 13793 778 89097 95970 828 91803 03368
629 79865 06454 679 83186 97743 729 86272 75283 779 a9153 74577 829 91855 45306
630 79934 05495 680 83250 89127 730 86332 28601 780 89209 46027 91907 80924
631 80002 93592 681 83314 71119 731 86391 73770 781 89265 10339 :zi 91960 10238
632 80071 70783 682 83378 43747 732 86451 10811 782 89320 67531 832 92012 33263
633 80140 37100 683 83442 07037 733 86510 39746 783 89376 17621 833 92064 50014
634 80208 92579 684 83505 61017 734 86569 60599 784 89431 60627 834 92116 60506
635 80277 37253 685 83569 05715 735 86628 73391 785 89486 96567 835 92168 64755
636 80345 711% 686 83632 41157 736 86687 78143 786 89542 25460 836 92220 62774
637 80413 94323 687 83695 67371 737 86746 74879 787 89597 47324 837 92272 54580
638 80482 06787 688 a3758 84382 738 86805 63618 788 89652 62175 838 92324 40186
639 80550 08582 689 83821 92219 739 86864 44384 789 89707 70032 839 92376 19608
640 80617 99740 690 83884 90907 740 86923 17197 790 89762 70913 840 92427 92861
641 80685 80295 691 83947 80474 741 86981 82080 791 89817 64835 841 92479 59958
642 80753 50281 692 84010 60945 742 87040 39053 792 89872 51816 842 92531 20915
643 80821 09729 693 84073 32346 743 87098 88138 793 89927 31873 843 92582 75746
644 80888 58674 694 84135 94705 744 87157 29355 794 89982 05024 844 92634 24466
645 80955 97146 695 84198 48046 745 87215 62727 795 90036 71287 845 92685 67089
646 81023 25180 696 84260 92396 746 a7273 88275 796 90091 30677 846 92737 036:30
647 81090 42807 697 84323 27781 747 87332 06018 797 90145 83214 847 92788 34103
648 81157 50059 698 84385 54226 748 87390 15979 90200 28914 848 92839 58523
649 81224 46968 699 84447 71757 749 87448 18177 T2 90254 67793 849 92890 76902
650 81291 33566 700 84509 80400 750 87506 12634 800 90308 99870 850 92941 89257
[ 1
c-y
[c-p11 C-47)
[ I
1 [ 1
C-t,8
98 ELEMENTARY TRANSCENDENTAL F’UNCTIONS
850 92941 89257 900 95424 25094 950 97772 36053 1000 00000 00000 1050 02118 92991
851 92992 95601 901 95472 47910 951 97818 05169 ii01 00043 40775 1051 02160 27160
852 93043 95948 902 95520 65375 952 97863 69484 1002 00086 77215 1052 02201 57398
853 93094 90312 903 95568 77503 953 97909 29006 1003 00130 09330 1053 02242 83712
854 93145 78707 904 95616 84305 954 97954 83747 1004 00173 37128 1054 02284 06109
855 93196 61147 905 95664 85792 955 98000 33716 1005 00216 60618 1055 02325 24596
856 93247 37647 906 95712 81977 956 98045 78923 1006 00259 79807 1056 02366 39182
857 93298 08219 907 95260 72871 957 98091 19378 1007 00302 94706 1057 02407 49873
858 93348 72878 908 95808 58485 958 98136 55091 1008 00346 05321 1058 02448 56677
859 93399 31638 909 95856 38832 959 98181 86072 1009 00389 11662 1059 02489 59601
860 93449 84512 910 95904 13923 960 98227 12330 1010 00432 13738 1060 02530 58653
861 93500 31515 911 95951 83770 961 98272 33877 1011 00475 11556 1061 02571 53839
562 93550 72658 912 95999 48383 962 98317 50720 1012 00518 05125 1062 02612 45167
363 93601 07957 913 96047 07775 963 98362 62871 1013 00560 94454 1063 02653 32645
164 93651 37425 914 96094 61957 964 98407 70339 1014 0060:3 79550 1064 02694 16280
865 93701 61075 915 96142 10941 965 98452 73133 1015 00646 60422 1065 02734 96078
866 93751 78920 916 96189 54737 966 98497 71264 1016 00689 37079 1066 02775 72047
867 93801 90975 917 96236 93357 967 98542 64741 1017 00732 09529 1067 02816 44194
868 93851 97252 918 96284 26812 968 98587 53573 1018 00774 77780 1068 02857 12527
869 93901 97764 919 96331 55114 969 98632 37771 1019 00817 41840 1069 02897 77052
870 93951 92526 920 96378 78273 970 98677 17343 1020 00860 01718 1070 02938 37777
871 94001 81550 921 96425 96302 971 98721 92299 1021 00902 57421 1071 02978 94708
872 94051 64849 922 96473 09211 972 98766 62649 1022 00945 08958 1072 03019 47854
873 94101 42437 923 96520 17010 973 98811 28403 1023 00987 56337 1073 03059 97220
874 94151 14326 924 96567 19712 974 98855 89569 1024 01029 99566 1074 03100 42814
875 94200 80530 925 96614 17327 975 98900 46157 1025 01072 38654 1075 03140 84643
876 94250 41062 926 96661 09867 976 98944 98177 1026 01114 73608 1076 03181 22713
877 94299 95934 927 96707 97341 977 98989 45637 1027 01157 04436 1077 03221 57033
878 94349 45159 928 96754 79762 978 99033 88548 1028 01199 31147 1078 03261 87609
879 94398 88751 929 96801 57140 979 99078 26918 1029 01241 53748 1079 03302 14447
880 94448 26722 930 96848 29486 980 99122 60757 1030 01283 72247 1080 03342 37555
881 94497 59084 931 96894 96810 981 99166 90074 1031 01325 86653 1081 03382 56940
882 94546 85851 932 96941 59124 982 99211 14878 1032 01367 96973 1082 03422 72608
a83 94596 07036 933 96988 16437 983 99255 35178 1033 014:LO 03215 1083 03462 84566
884 94645 22650 934 97034 68762 984 99299 50984 1034 01452 05388 1084 03502 92822
885 94694 32707 935 97081 16109 985 99343 62305 1035 01494 03498 1085 03542 97382
886 94743 37219 936 97127 58487 986 99387 69149 1036 01535 97554 1086 03582 98253
887 94792 36198 937 97173 95909 987 99431 71527 1037 01577 87564 1087 03622 95441
888 94841 29658 938 97220 28384 988 99475 69446 1038 01619 73535 1088 03662 88954
889 94890 17610 939 97266 55923 989 99519 62916 1039 01661 55476 1089 03702 78798
890 94939 00066 940 97312 78536 990 99563 51946 1040 01703 33393 1090 03742 64979
891 94987 77040 941 97358 96234 991 99607 36545 1041 01745 07295 1091 03782 47506
892 95036 48544 942 97405 09028 992 99651 16722 1042 01786 77190 1092 03822 26384
893 95085 14589 943 97451 16927 993 99694 92485 1043 01828 43084 1093 03862 01619
894 95133 75188 944 97497 19943 994 99738 63844 1044 01870 04987 1094 03901 73220
895 95182 30353 945 97543 18085 995 99782 30807 1045 01911 62904 1095 03941 41192
896 95230 80097 946 97589 11364 996 99825 93384 1046 01953 16845 1096 03981 05541
897 95279 24430 947 97634 99790 997 99869 51583 1047 01994 66817 1097 04020 66276
898 95327 63367 948 97680 83373 998 99913 05413 1048 02036 12826 1098 04060 23401
899 95375 96917 949 97726 62124 999 99956 54882 1049 02077 54882 1099 04099 76924
900 95424 25094 950 97772 36053 1000 00000 00000 1050 02X18 92991 1100 04139 26852
c-y
[ I [ 1
‘-iI 6
[ 1
c-3815
[1 I
C-38)5
ELEMENTARY TRANSCENDENTAL FUNCTIt INS 99
COMMON LOGARITHMS Table 4.1
1105 04336 22780 1155 06258 19842 1205 08098 70469 1255 09864 37258 1305 11561 05117
1106 04375 51270 1156 06295 78341 1206 08134 73078 1256 09898 96394 1306 11594 31769
1107 04414 76209 1157 06333 331590 1207 08170 72701 1257 09933 52777 1307 11627 55876
1108 04453 97604 1158 06370 851594 1208 08206 69343 1258 09968 06411 1308 11660 77440
1109 04493 15461 1159 06408 34360 1209 08242 63009 1259 10002 57301 1309 11693 96466
1110 04532 29788 1160 06445 791392 1210 08278 53703 1260 10037 05451 1310 11727 12957
1111 04571 40589 1161 06483 22:197 1211 08314 41431 1261. 10071 50866 1311 11760 26917
1112 04610 47872 1162 06520 61:!81 1212 08350 26198 1262 10105 93549 1312 11793 38350
1113 04649 51643 1163 06557 971.47 1213 08386 08009 1263 10140 33506 1313 11826 47261
1114 04688 51908 1164 06595 29803 1214 08421 86867 1264 10174 70739 1314 11859 53652
1115 04727 48674 1165 06632 59;!54 1215 08457 62779 1265 10209 05255 1315 11892 57528
1116 04766 41946 1166 06669 851jO4 1216 08493 35749 1266 10243 37057 1316 11925 58893
1117 04805 31731 1167 06707 08fj60 1217 08529 05782 1267 10277 66149 1317 11958 57750
1118 04844 18036 1168 06744 28428 1218 08564 72883 1268 10311 92535 1318 11991 54103
1119 04883 00865 1169 06781 45:112 1219 08600 37056 1269 10346 16221 1319 12024 47955
1120 04921 80227 1170 06818 58017 1220 08635 98307 1270 10380 37210 1320 12057 39312
1121 04960 56126 1171 06855 68951 1221 08671 56639 1271 10414 55506 1321 12090 28176
1122 04999 28569 1172 06892 76:117 1222 08707 12059 1272 10448 71113 1322 12123 14551
1123 05037 97563 1173 06929 80:121 1223 08742 64570 1273 10482 84037 1323 12155 98442
1124 05076 63112 1174 06966 80969 1224 08778 14178 1274 10516 94280 1324 12188 79851
1125 05115 25224 1175 07003 781i66 1225 08813 60887 1275 10551 01848 1325 12221 58783
1126 05153 83905 1176 07040 73?17 1226 08849 04702 1276 10585 06744 1326 12254 35241
1127 05192 39160 1177 07077 64h28 1227 08884 45627 1277 10619 08973 1327 12287 09229
1128 05230 90996 1178 07114 52905 1228 08919 83668 1278 10653 08538 1328 12319 80750
1129 05269 39419 1179 07151 38051 1229 08955 18829 1279 10687 05445 1329 12352 49809
1130 05307 84435 1180 07188 201173 1230 08990 51114 1280 10720 99696 1330 12385 16410
1131 05346 26049 1181 07224 98976 1231 09025 80529 1281 10754 91297 1331 12417 80555
1132 05384 64269 1182 07261 74'165 1232 09061 07078 1282 10788 80252 1332 12450 42248
1133 05422 99099 1183 07298 47446 1233 09096 30766 1283 10822 66564 1333 12483 01494
1134 05461 30546 1184 07335 17024 1234 09131 51597 1284 10856 50237 1334 12515 58296
1135 05499 58615 1185 07371 83503 1235 09166 69576 1285 10890 31277 1335 12548 12657
1136 05537 83314 1186 07408 46;390 1236 09201 84708 1286 10924 09686 1336 12580 64581
1137 05576 04647 1187 07445 07190 1237 09236 96996 1287 10957 85469 1337 12613 14073
1138 05614 22621 1188 07481 64,106 1238 09272 06447 1288 10991 58630 1338 12645 61134
1139 05652 37241 1189 07518 18546 1239 09307 13064 1289 11025 29174 1339 12678 05770
1140 05690 48513 1190 07554 69514 1240 09342 16852 1290 11058 97103 1340 12710 47984
1141 05728 56444 1191 07591 171515 1241 09377 17815 1291 11092 62423 1341 12742 87779
1142 05766 61039 1192 07627 62354 1242 09412 15958 1292 11126 25137 1342 12775 25158
1143 05804 62304 1193 07664 04137 1243 09447 11286 1293 11159 85249 1343 12807 60127
1144 05842 60245 1194 07700 43,268 1244 09482 03804 1294 11193 42763 1344 12839 92687
1145 05880 54867 1195 07736 79,353 1245 09516 93514 1295 11226 97684 1345 12872 22843
1146 05918 46176 1196 07773 11797 1246 09551 80423 1296 11260 50015 1346 12904 50599
1147 05956 34179 1197 07809 41504 1247 09586 64535 1297 11293 99761 1347 12936 75957
1148 05994 18881 1198 07845 68181 1248 09621 45853 1298 11327 46925 1348 12968 98922
1149 06032 00287 1199 07881 91331 1249 09656 24384 1299 11360 91511 1349 13001 19497
1150 06069 78404 1200 07918 12160 1250 09691 00130 1300 11394 33523 1350 13033 37685
c 1 c-3815
[ 1 c-3814
[ I
c-3814
[ I c-3813
[ I C-38)3
100 ELEMENTARY TRANSCENDENTAL FUNCTIONS
3 In x X In 2 X In x
0.000 -m 0.050 -2.99573 22735 539910 0.100 -2.30258 50929 940457
0: 001 -6.90775 52789 821371 0.051 -2.97592 96462 578113 0.101 -2.29263 47621 408776
0.002 -6.21460 80984 221917 0.052 -2.95651 15604 007097 0.102 -2.28278 24656 978660
0.003 -5.80914 29903 140274 0.053 -2.93746 33654 300152 0.103 -2.27302 62907 525013
0.004 -5.52146 09178 622464 0.054 -2.91877 12324 178627 0.104 -2.26336 43798 407644
0.005 -5.29831 73665 480367 0.055 -2.90042 20937 496661 0.105 -2.25379 49288 246137
3.006 -5.11599 58097 540821 0.056 -2.88240 35882 469878 0.106 -2.24431 61848 700699
0.007 -4.96184 51299 268237 0.057 -2.86470 40111 475869 0.107 -2.23492 64445 202309.
0.008 -4.82831 37373 023011 0.058 -2.84731 22684 357177 0.108 -2.22562 40518 579174
0.009 -4.71053 07016 459177 oIo59 -2.83021 78350 764176 0.109 -2.21640 73967 529934
0.010 -4.60517 01859 880914 0.060 -2.81341 07167 600364 0.110 -2.20727 49131 897208
0.011 -4.50986 00061 837665 0,061 -2.79688 14148 088258 0.111 -2.19822 50776 698029
0.012 -4.42284 86291 941367 0.062 -2.78062 08939 370455 0.112 -2.18925 64076 870425
0.013 -4.34280 59215 206003 0.063 -2.76462 05525 906044 0.113 -2.18036 74602 697965
0.014 -4.26869 79493 668784 0.064 -2.74887 21956 224652 0.114 -2.17155 68305 876416
0.015 -4.19970 50778 799270 0.065 -2.73336 80090 864999 0.115 -2.16282 31506 188870
0.016 -4.13516 65567 423558 0.066 -2.71810 05369 557115 0.116 -2.15416 50878 757724
OiO17 -4.07454 19349 259210 0.067 -2.70306 26595 911710 0.117 -2.14558 13441 843809
0.018 -4.01738 35210 859724 0.068 -2.68824 75738 060304 0.118 -2.13707 06545 164723
0.019 -3.96331 62998 156966 0.069 -2.67364 87743 848777 0.119 -2.12863 17858 706077
0.020 -3.91202 30054 281461 0.070 -2.65926 00369 327781 0.120 -2.12026 35362 000911
0.021 -3.86323 28412 587141 0.071 -2.64507 54019 408216 0.121 -2.11196 47333 853960
0.022 -3.81671 28256 238212 0.072 -2.63108 91599 660817 0.122 -2.10373 42342 488805
0.023 -3.77226 10630 529874 0.073 -2.61729 58378 337459 0.123 -2.09557 09236 097196
0.024 -3.72970 14486 341914 0.074 -2.60369 01857 779673 0.124 -2.08747 37133 771002
0.025 -3.68887 94541 139363 0.075 -2.59026 71654 458266 0.125 -2.07944 15416 798359
0.026 -3.64965 87409 606550 0.076 -2.57702 19386 958060 0.126 -2.07147 33720 306591
0.027 -3.61191 84129 778080 0.077 -2.56394 98571 284532 0.127 -2.06356 81925 235458
0.028 -3.57555 07688 069331 0.078 -2.55104 64522 925453 0.128 -2.05572 50150 625199
0.029 -3.54045 94489 956630 0.079 -2.53830 74265 151156 0.129 -2.04794 28746 204649
0.030 -3.50655 78973 199817 0.080 -2.52572 86443 082554 0.130 -2.04022 08285 265546
0.031 -3.47376 80744 969908 0.081 -2.51330 61243 096983 0.131 -2.03255 79557 809855
0.032 -3.44201 93761 824105 0.082 -2.50103 60317 178839 0.132 -2.02495 33563 957662
0.033 -3.41124 77175 156568 0.083 -2.48891 46711 855391 0.133 -2.01740 61507 603833
0.034 -3.38139 47543 659757 0.084 -2.47693 84801 388234 0.134 -2.00991 54790 312257
0.035 -3.35240 72174 927234 0.085 -2.46510 40224 918206 0.135 -2.00248 05005 437076
0.036 -3.32423 63405 260271 0.086 -2.45340 79827 286293 0.136 -1.99510 03932 460850
0.037 -3.29683 73663 379126 0.087 -2.44184 71603 275533 0.137 -1.98777 43531 540121
0.038 -3.27016 91192 557513 0.088 -2.43041 84645 039306 0.138 -1.98050 15938 249324
0.039 -3.24419 36328 524906 0.089 -2.41911 89092 499972 0.139 -1.97328 13458 514453
0.040 -3.21887 58248 682007 0.090 -2.40794 56086 518720 0.140 -1.96611 28563 728328
0.041 -3.19418 32122 778292 0.091 -2.39689 57724 652870 0.141 -1.95899 53886 039688
0.042 -3.17008 56606 987687 0.092 -2.38596 67019 330967 0.142 -1.95192 82213 808763
0.043 -3.14655 51632 885746 0.093 -2.37515 57858 288811 0.143 -1.94491 06487 222298
0.044 -3.12356 56450 638759 0.094 -2.36446 04967 121332 0.144 -1.93794 19794 061364
0.045 -3.10109 27892 118173 0.095 -2.35387 83873 815962 0.145 -1.93102 15365 615627
0.046 -3.07911 38824 930421 0.096 -2.34340 70875 143008 0.146 -1.92414 86572 738006
0.047 -3.05760 76772 720785 0.097 -2.33304 43004 787542 0.147 -1.91732 26922 034008
0.048 -3.03655 42680 742461 0.098 -2.32278 78003 115651 0.148 -1.91054 30052 180220
0.049 -3.01593 49808 715104 0.099 -2.31263 54288 475471 0.149 -1.90380 89730 366779
0.050 -2.99573 22735 539910 0.100 -2.30258 50929 940457 0.150 -1.89711 99848 858813
[ 1
C-7612
L ’ J [ I
In 10 = 2.30258 50929 940457
*see page n.
ELEMEiNTARY TRANSCENDENTAL FUNCTIONS 103
NATURAL LOGARITHMS Table 4.2
X In 2 X In x X In x
0.450 -0.79850 76962 177716 0.500 -0.69314 71805 599453 0.550 -0.59783 70007 556204
0.451 -0.79628 79394 794587 0.501 -0.69114 91778 972723 0.551 -0.59602 04698 292226
0.452 -0.79407 30991 499059 0.502 -0.68915 51592 904079 0.552 -0.59420 72327 050417
0.453 -0.79186 31534 991030 0.503 -0.68716 51088 823978 0.553 -0.59239 72774 598023
0.454 -0.78965 80809 407891 0.504 -0.68517 90109 107684 0.554 -0.59059 05922 348532
0.455 -0.78745 78600 311866 0.505 -0.68319 68497 067772 0.555 -0.58878 71652 357025
0.456 -0.78526 24694 677510 0.506 -0.68121 86096 946715 0.556 -0.58698 69847 315547
0.457 -0.78307 18880 879324 0.507 -0.67924 42753 909539 0.557 -0.58519 00390 548530
0.458 -0.78088 60948 679521 0.508 -0.67727 38314 036552 0.558 -0.58339 63166 008261
0.459 -0.77870 50689 215919 0.509 -0.67530 72624 316143 0.559 -0.58160 58058 270379
0.460 -0.77652 87894 989964 0.510 -0.67334 45532 637656 0.560 -0.57981 84952 529421
0.461 -0.77435 72359 854885 0.511 -0.67138 56887 784326 0.561 -0.57803 43734 594407
0.462 -0.77219 03879 003982 0.512 -0.66943 06539 426293 0.562 -0.57625 34290 884460
0.463 -0.77002 82248 959030 0.513 -0.66747 94338 113675 0.563 -0.57447 56508 424467
0.464 -0.76787 07267 558818 0.514 -0.66553 20135 269719 0.564 -0.57270 10274 840782
0.465 -0.76571 78733 947807 0.515 -0.66358 83783 184009 0.565 -0.57092 95478 356961
0.466 -0.76356 96448 564912 0.516 -0.66164 85135 005743 0.566 -0.56916 12007 789541
0.467 -0.76142 60213 132397 0.517 -0.65971 24044 737079 0.567 -0.56739 59752 543850
0.468 -0.75928 69830 644903 0.518 -0.65778 00367 226540 0.568 -0.56563 38602 609857
0.469 -0.75715 25105 358577 0.519 -0.65585 13958 162484 0.569 -0.56387 48448 558061
0.470 -0.75502 25842 780328 0.520 -0.65392 64674 066640 0.570 -0.56211 89181. 535412
0.471 -0.75289 71849 657193 0.521 -0.65200 52372 287701 0.571 -0.56036 60693 261268
0.472 -0.75077 62933 965817 0.522 -0.65008 76910 994983 0.572 -0.55861 62876 023392
0.473 -0.74865 98904 902041 0.523 -0.64817 38149 172142 0.573 -0.55686 95622 673975
0.474 -0.74654 79572 870606 0.524 -0.64626 35946 610949 0.574 -0.55512 58826 625706
0.475 -0.74444 04749 474958 0.525 -0.64435 70163 905133 0.575 -0.55338 52381 847866
0.476 -0.74233 74247 507170 0.526 -0.64245 40662 444272 0.576 -0.55164 76182 862458
0.477 -0.74023 87880 937958 0.527 -0.64055 47304 407747 0.577 -0.54991 30124 740375
0.478 -0.73814 45464 906811 0.528 -0.63865 89952 758756 0.578 -0.54818 14103 097596
0.479 -0.73605 46815 712218 0.529 -0.63676 68471 238377 0.579 -0.54645 28014 091418
0.480 -0.73396 91750 802004 0.530 -0.63487 82724 359695 0.580 -0.54472 71754 416720
0.481 -0.73188 80088 763759 0.531 -0.63299 32577 401982 O;iiSl -0i54300 45221 302258
0.482 -0.72981 11649 315367 0.532 -0.63111 17896 404927 0.582 -0.54128 48312 506992
0.483 -0.72773 86253 295644 0.533 -0.62923 38548 162925 0.583 -0.53956 80926 316447
0.484 -0.72567 03722 655053 0.534 -0.62735 94400 219422 0.584 -0.53785 42961 539100
0.485 -0.72360 63880 446539 0.535 -0.62548 85320 861305 0.585 -0.53614 34317 502806
0.486 -0.72154 66550 816433 0.536 -0.62362 11179 113351 0.586 -0.53443 54894 051244
0.487 -0.71949 11558 995473 0.537 -0.62175 71844 732724 0.587 -0.53273 04591 540406
0.488 -0.71743 98731 289899 0.538 -0.61989 67188 203526 0.588 -0.53102 83310 835101
0.489 -0.71539 27895 072650 0.539 -0.61803 97080 731399 0.589 -0.52932 90953 305503
0.490 -0.71334 98878 774648 0.540 -0.61618 61394 238170 0.590 -0.52763 27420 823719
0.491 -0.71131 11511 876165 0.541 -0.61433 60001 356555 ;A;; -0.52593 92615 760389
0.492 -0.70927 65624 898289 0.542 -0.61248 92775 424908 -0.52424 86440 981314
0.493 -0.70724 61049 394469 0.543 -0.61064 59590 482016 0:593 -0.52256 08799 844116
0.494 -0.70521 97617 942145 0.544 -0.60880 60321 261944 0.594 -0.52087 59596 194921
0.495 -0.70319 75164 134468 0.545 -0.60696 94843 188930 0.595 -0.51919 38734 365073
0.496 -0.70117 93522 572096 0.546 -0.60513 63032 372320 0.596 -0.51751 46119 167873
0.497 -0.69916 52528 855083 0.547 -0.60330 64765 601558 0.597 -0.51583 81655 895350
0.498 -0.69715 52019 574841 0.548 -0.60147 99920 341215 0.598 -0.51416 45250 315053
0.499 -0.69514 91832 306184 0.549 -0.59965 68374 726064 0.599 -0.51249 36808 666877
0.500 -0.69314 71805 599453 0.550 -0.59783 70007 556204 0.600 -0.51082 56237 659907
[ I
C-76
[ 1
95
2 In z X In 2 2 In x
0.600 -0.51082 56231 659907 0.650 -0.43078 29160 924543 0.700 -0.35667 49439 387324
OibOl -0.50916 03444 469295 0.651 -0.42924 56367 735678 0.701 -0.35524 73919 475470
0.602 -0.50749 78336 733160 0.652 -0.42771 07170 554841 0.702 -0.35382 18749 563259
0.603 -0.50583 80822 549516 0.653 -0.42617 81497 057060 0.703 -0.35239 83871 714721
0.604 -0.50418 10810 473221 0.654 -0.42464 79275 249384 0.704 -0.35097 69228 240947
0.605 -0.50252 68209 512956 0.655 -0.42312 00433 468851 0.705 -0.34955 74761 698684
0.606 -0.50087 52929 128226 0.656 -0.42159 44900 380480 0.706 -0.34814 00414 888950
0.607 -0.49922 64879 226388 0.657 -0.42007 12604 975265 0.707 -0.34672 46130 855643
0.608 -0.49758 03970 159700 0.658 -0.41855 03476 568199 0.708 -0.34531 11852 884173
0.609 -0.49593 70112 722400 0.659 -0.41703 17444 796298 0.709 -0.34389 97524 500096
0.610 -0.49429 63218 147801 0.660 -0.41551 54439 616658 0.710 -0.34249 03089 467759
0.611 -0.49265 83198 105417 0.661 -0.41400 14391 304508 0.711 -0.34108 28491 788962
0.612 -0.49102 29964 698110 0.662 -0.41248 97230 451288 0.712 -0.33967 73675 701613
0.613 -0.48939 03430 459257 0.663 -0.41098 02887 962745 0.713 -0.33827 38585 678411
0.614 -0.48776 03508 349946 0.664 -0.40947 31295 057032 0.714 -0.33687 23166 425527
0.615 -0.48613 30111 756192 0.665 -0.40796 82383 262829 0.71!i -0.33547 27362 881294
0.616 -0.48450 83154 486173 0.666 -0.40646 56084 417479 0.716 -0.33407 51120 214914
0.617 -0.48288 62550 767492 0.667 -0.40496 52330 665133
0.618 -0.48126 68215 244463 0.668 -0.40346 71054 454913 ao@
. 5:i -0.33128
-0.33267 94383
57099 825167
339129
0.619 -0.47965 00062 975409 0.669 -0.40197 12188 539086 0.719 -Or32989 39212 610904
0.620 -0.47803 58009 429998 0.670 -0.40047 75665 971253 0.720 -0.32850 40669 720361
0.621 -0.47642 41970 486583 0.671 -0.39898 61420 104553 0.721 -0.32711 61416 971880
0.622 -0.47481 51862 429576 0.672 -0.39749 69384 589875 0.722 -0.32573 01400 893108
0.623 -0.47320 87601 946839 0.673 -0.39600 99493 374092 0.723 -0.32434 60568 233724
0.624 -0.47160 49106 127094 0.674 -0.39452 51680 698300 0.724 -0.32296 38865 964207
0.625 -0.47000 36292 457356 0.675 -0.39304 25881 096072 0.7i!5 -0.32158 36241 274623
0.626 -0.46840 49078 820385 0.676 -0i39i56 22029 391730 0.726 -0.32020 52641 573410
-0.46680 87383 492164 0.677 -0.39008 40060 698621 0.727 -0.31882 88014 486177
:* 2 -0.46521 51125 139384 0.678 -0Ij8860 79910 417415 0.728 -0.31745 42307 854511
0: 629 -0.46362 40222 816965 0.679 -0.38713 41514 234409 0.729 -0.31608 15469 734789
0.630 -0.46203 54595 965587 0.680 -0.38566 24808 119847 0.7'30 -0.31471 07448 397002
-0.46044 94164 409239 0.681 -0.38419 29728 326247 a. 731 -0.31334 18192 323585
i* t;: -0.45886 58848 352796 0.682 -0.38272 56211 386750 0.732 -0.31197 47650 208255
0:633 -0.45728 48568 379609 0.683 -0.38126 04194 113470 0.733 -0.31060 95770 954856
0.634 -0.45570 63245 449111 0.684 -0.37979 73613 595866 0.134 -0.30924 62503 676215
0.635 -0.45413 02800 894454 0.685 -0.37833 64407 199118 0.735 -0.30788 47797 693004
0.636 -0.45255 67156 420149 0.686 -0.37687 76512 562518 0.736 -0.30652 51602 532608
0.637 -0.45098 56234 099737 0.681 -0.37542 09867 597877 0.137 -0.30516 73867 928004
0.638 -0.44941 69956 373472 0.688 -0.37396 64410 487934 0.738 -0.30381 14543 816646
0.639 -0.44785 08246 046022 0.689 -0.37251 40079 684785 0.739 -0: 30245 73580 339353
0.640 -0.44628 71026 284195 0.690 -0.37106 36813 908320 0.740 -0.30110 50927 839216
0.641 -0.44472 58220 614670 0.691 -0.36961 54552 144672 0.741 -0.29975 46536 860502
0.642 -0.44316 69752 921759 0.692 -0.36816 93233 644675 0.742 -0.29840 60358 147566
0.643 -0.44161 05547 445177 0.693 -0.36672 52797 922330 0.743 -0.29705 92342 643779
0.644 -0.44005 65528 777834 0.694 -0.36528 33184 753326 0.744 -0.29571 42441 490452
0.645 -0.43850 49621 863646 0.695 -0.36384 34334 173449 0.745 -0.29437 10606 025775
0.646 -0.43695 57751 995352 0.696 -0.36240 56186 477174 0.746 -0.29302 96787 783762
0.647 -0.43540 89844 812365 0.697 -0.36096 98682 216132 0.747 -0i29169 00938 493197
0.648 -0.43386 45826 298624 0.698 -0.35953 61762 197646 0.748 -0.29035 23010 076598
0.649 -0.43232 25622 780471 0.699 -0.35810 45367 483268 0.749 -0.28901 62954 649176
0.650 -0.43078 29160 924543 0.700 -0.35667 49439 381324 0.750 -0.28768 20724 517809
0.750 -0.28768 20724 517809 0.800 -0.22314 35513 142098 0.850 -0.16251 89294 977749
0.751 -0.28634 96272 180023 0.801 -0.22189 43319 137778 0.851 -0.16134 31504 087629
0.752 -0.28501 89550 322973 0.802 -0.22064 66711 156226 0.852 -0.16016 87521 528213
0.753 -0.28369 00511 822435 0.803 -0.21940 05650 353754 0.853 -0.15899 57314 904579
0.754 -0.28236 29109 741810 0.804 -0.21815 60098 031707 0.854 -0.15782 40851 935672
0.755 -0.28103 75297 331123 0.805 -0.21691 30015 635737 0.855 -0.15665 38100 453768
0.756 -0.27971 39028 026041 0.806 -0.21567 15364 755088 0.856 -0.15548 49028 403950
0.757 -0.27839 20255 446883 0.807 -0.21443 16107 121883 0.857 -0.15431 73603 843573
0.758 -0.27707 18933 397654 0.808 -0.21319 32204 610417 0.858 -0.15315 11794 941748
0.759 -0.27575 35015 865071 0.809 -0.21195 63619 236454 0.859 -0.15198 63569 978817
0.760 -0.27443 68457 017603 0.810 -0.21072 10313 156526 0.860 -0.15082 28897 345836
0.761 -0i27312 19211 204512 0.811 -0.20948 72248 667241 0.861 -0.14966 07745 544063
0.762 -0.27180 87232 954908 0.812 -0.20825 49388 204591 0.862 -0.14850 00083 184440
0.763 -0.27049 72476 976800 0.813 -0.20702 41694 343265 0.863 -0.14734 05878 987091
0.764 -0.26918 74898 156166 0.814 -0.20579 49129 795968 0.864 -0.14618 25101 780814
0.765 -0.26787 94451 556012 0.815 -0.20456 71657 412743 0.865 -0.14502 57720 502577
0.766 -0.26657 31092 415458 0.816 -0.20334 09240 180300 0.866 -0.14387 03704 197019
0.767 -0.26526 84776 148809 0.817 -0.20211 61841 221342 0.867 -0.14271 63022 015952
0.768 -0.26396 55458 344649 0.818 -0.20089 29423 793900 0.868 -0.14156 35643 217869
0.769 -0.26266 43094 764931 0.819 -0.19967 11951 290676 0.869 -0.14041 21537 167450
0.770 -0.26136 47641 344075 0.820 -0.19845 09387 238383 0.870 -0.13926 20673 335076
0.771 -0.26006 69054 188076 0.821 -0.19723 21695 297088 0.871 -0.13811 33021 296343
0.772 -0.25877 07289 573609 0.822 -0.19601 48839 259571 0.872 -0.13696 58550 731574
0.773 -0.25747 62303 947151 0.823 -0.19479 90783 050672 0.873 -0.13581 97231 425348
0.774 -0.25618 34053 924099 0.824 -0.19358 47490 726654 0.874 -0.13467 49033 266016
0.775 -0.25489 22496 287901 0.825 -0.19237 18926 474561 0.875 -0.13353 13926 245226
0.776 -0.25360 27587 989183 0.826 -0.19116 05054 611590 0.876 -0.13238 91880 457456
0.777 -0.25231 49286 144896 0.827 -0.18995 05839 584457 0.877 -0.13124 82866 099540
0.778 -0.25102 87548 037454 0.828 -0.18874 21245 968774 0.878 -0.13010 86853 470204
0.779 -0.24974 42331 113888 0.829 -0.18753 51238 468421 0.879 -0.12897 03812 969601
0.780 -0.24846 13592 984996 0.830 -0.18632 95781 914934 0.880 -0.12783 33715 098849,
0.781 -0.24718 01291 424511 0.831 -0.18512 54841 266889 0.881 -0.12669 76530 459575
0.782 -0.24590 05384 368260 0.832 -0.18392 28381 609285 0.882 -0.12556 32229 753457
0.783 -0.24462 25829 913340 0.833 -0.18272 16368 152944 0.883 -0.12443 00783 781770
0.784 -0.24334 62586 317292 0.834 -0.18152 18766 233903 0.884 -0.12329 82163 444936
0.785 -0.24207 15611 997286 0.835 -0.18032 35541 312816 0.885 -0.12216 76339 742075
0.786 -0.24079 84865 529305 0.836 -0.17912 66658 974354 0.886 -0.12103 83283 770561
0.787 -0.23952 70305 647338 0.837 -0.17793 12084 926617 0.887 -0.11991 02966 725576
0.788 -0.23825 71891 242579 0.838 -0.17673 71785 000540 0.888 -0.11878 35359 899670
0.789 -0.23698 89581 362628 0.839 -0.17554 45725 149309 0.889 -0.11765 80434 682325
0.790 -0.23572 23335 210699 0.840 -0.17435 33871 447778 0.890 -0.11653 38162 559515
Oi791 -0i23445 73112 144832 0.841 -0.17316 36190 091890 0.891 -0.11541 08515 113277
0.792 -0.23319 38871 677112 0.842 -0.17197 52647 398103 0.892 -0.11428 91464 021277
0.793 -0.23193 20573 472891 0.843 -0.17078 83209 802816 0.893 -0.11316 86981 056380
0.794 -0.23067 18177 350013 0.844 -0.16960 27843 861799 0.894 -0.11204 95038 086229
0.795 -0.22941 31643 278052 0.845 -0.16841 86516 249632 0.895 -0.11093 15607 072817
0.796 -0.22815 60931 377540 0.846 -0.16723 59193 759138 0.896 -0.10981 48660 072066
0.797 -0.22690 06001 919220 0.847 -0.16605 45843 300827 0.897 -0.10869 94169 233409
0.798 -0.22564 66815 323283 ;.;&I; -0.16487 46431 902340 ;.;N: -0.10758 52106 799374
0.799 -0.22439 43332 158624 . -0.16369 60926 707897 . -0.10647 22445 105168
0.800 -0.22314 35513 142098 0.850 -0.16251 89294 977749 0.900 -0.10536 05156 578263
[ 1
(72
[ 1
C-67)2
[ 1
(72
x In z X In 2 X In x
0.900 -0.10536 05156 578263 0.950 -0.05129 32943 875505 1.000 0.00000 00000 000000
0.901 -0.10425 00213 737991 0.951 -0.05024 12164 367467 1.0011 0.00099 95003 330835
0.902 -0.10314 07589 195134 0.952 -0.04919 02441 907717 1 OOZ! 0.00199 80026 626731
0.903 -0.10203 27255 651516 0.953 -0.04814 03753 279349 1:003 0.00299 55089 797985
0.904 -0.10092 59185 899606 0.954 -0.04709 16075 338505 1.004 0.00399 20212 695375
0.905 -0.09982 03352 822109 0.955 -0.04604 39385 014068 1.005 0.00498 75415 110391
0.906 -0.09871 59729 391577 0.956 -0.04499 73659 307358 1.006 0.00598 20716 775475
0.907 -0.09761 28288 670004 0.957 -0.04395 18875 291828 1.007 0.00697 56137 364252
0.908 -0.09651 09003 808438 0.958 -0.04290 75010 112765 1.008 0.00796 81696 491769
0.909 -0.09541 01848 046582 0.959 -0.04186 42040 986988 1.009 0.00895 97413 714719
0.910 -0.09431 06794 712413 0.960 -0.04082 19945 202551 1.010 0.00995 03308 531681
0.911 -0.09321 23817 221787 0.961 -0.03978 08700 118446 1.011. 0.01093 99400 383344
0.912 -0.09211 52889 078057 0.962 -0.03874 08283 164306 1 OK! 0.01192 85708 652738
0.913 -0.09101 93983 871686 0.963 -0.03770 18671 840115 1:013 0.01291 62252 665463
0.914 -0.08992 47075 279870 0.964 -0.03666 39843 715914 1.014 0.01390 29051 689914
0.915 -0.08883 12137 066157 0.965 -0.03562 71776 431511 1.015 0.01488 86124 937507
0.916 -0iO8773 89143 080068 Oi966 -0iO3459 14447 696191 1.016 OiOi587 33491 562901
0.917 -0.08664 78067 256722 0.967 -0.03355 67835 288427 1.017 0.01685 71170 664229
0.918 -0iO8555 78883 616466 0.968 -0.03252 31917 055600 l.OlEl 0.01783 99181 283310
0.919 -0.08446 91566 264500 0.969 -0.03149 06670 913708 1.019 0.01882 17542 405878
0.920 -0.08338 16089 390511 0.970 -0.03045 92074 847085 1.020 0.01980 26272 961797
Oi921 -0.08229 52427 268302 0.971 -0iO2942 88106 908121 1.021. OiO2078 25391 825285
0.922 -0.08121 00554 255432 0.972 -0.02839 94745 216980 l.O2i! 0.02176 14917 815127
Oi923 -0.08012 60444 792849 0.973 -0.02737 11967 961320 1.023 OiO2273 94869 694894
0.924 -0.07904 32073 404529 0.974 -0.02634 39753 396020 1.024 0.02371 65266 173160
0.925 -0.07796 15414 697119 0.975 -0.02531 78079 842899 0.02469 26125 903715
0.926 -0.07688 10443 359577 0.976 -0.02429 26925 690446 :- i:: 0.02566 77467 485778
0.927 -0.07580 17134 162819 0.977 -0.02326 86269 393543 1: 02i: 0.02664 19309 464212
0.928 -0.07472 35461 959365 0.978 -0.02224 56089 473197 1.028 0.02761 51670 329734
0.929 -0.07364 65401 682985 0.979 -0.02122 36364 516267 1.029 0.02858 74568 519126
'0.930 -II.07257 06928 348354 0.980 -0.02020 27073 175194 1.030 0.02955 88022 415444
0.931 -6.07149 60017 050700 0.981 -0.01918 28194 167740 1.031. 0.03052 92050 348229
0.932 -0.07042 24642 965459 0.982 -0.01816 39706 276712 1 03i! 0.03149 86670 593710
0.933 -0.06935 00781 347932 0.983 -0.01714 61588 349705 11033 0.03246 71901 375015
0.934 -0.06827 88407 532944 0.984 -0.01612 93819 298836 1.034 0.03343 47760 862374
0.935 -0.06720 87496 934501 0.985 -0.01511 36378 100482 1.035 0.03440 14267 173324
0.936 -0.06613 98025 045450 0.986 -0.01409 89243 795016 1.036 0.03536 71438 372913
0.937 -0.06507 19967 437149 0.987 -0.01308 52395 486555 1.037 0.03633 19292 473903
0.938 -0.06400 53299 759124 0.988 -0.01207 25812 342692 1.038 0.03729 57847 436969
0.939 -0.06293 97997 738741 0.989 -0.01106 09473 594249 1.039 0.03825 87121 170903
0.940 -0.06187 54037 180875 0.990 -0.01005 03358 535014 1.040 0.03922 07131 532813
0.941 -0.06081 21393 967574 o1991 -0~00904 07446 521491 1.041. OiO4018 17896 328318
0.942 -0.05975 00044 057740 0.992 -0.00803 21716 972643 1 042 0.04114 19433 311752
0.943 -0.05868 89963 486796 0.993 -0.00702 46149 369645 1:043 0.04210 11760 186354
0.944 -0.05762 91128 366364 0.994 -0.00601 80723 255630 1.044 0.04305 94894 604470
0.945 -0.05657 03514 883943 0.995 -0.00501 25418 235443 1.045 0.04401 68854 167743
0.946 -0.05551 27099 302588 0.996 -0.00400 80213 975388 1.046 0.04497 33656 427312
0.947 -0.05445 61857 960588 0.997 -0.00300 45090 202987 1.047 0.04592 89318 883998
0.948 -0iO5340 07767 271152 0.998 -0.00200 20026 706731 1.048 0.04688 35858 988504
0.949 -0.05234 64803 722092 0.999 -0.00100 05003 335835 1.049 0.04783 73294 141601
0.950 -0.05129 32943 875505 1.000 0.00000 00000 000000 1.050 0.04879 01641 694320
[ 1
C-67)2
[
(-iI 1
I [C-67)
11
In 10=2.30258 50929 940457
ELEMENTARY TRANSCENDENTAL FUNCTIONS 107
NATURAL LOGARITHMS Table 4.2
X In 2 X In x X In x
1.050 0.04879 01641 694320 1.100 0.09531 01798 043249 1.150 0.13976 19423 751587
1.051 OiO4974 20918 948141 1.101 0.09621 88577 405429 1.151 0.14063 11297 397456
1.052 0.05069 31143 155181 1.102 0.09712 67107 307227 1.152 0.14149 95622 736995
1.053 0.05164 32331 518384 1.103 0.09803 37402 713654 1.153 0.14236 72412 869220
1.054 0.05259 24501 191706 1.104 0.09893 99478 549036 1.154 0.14323 41680 859078
1.055 0.05354 07669 280298 1.105 0.09984 53349 697161 1.155 0.14410 03439 737569
1.056 0.05448 81852 840697 1.106 0.10074 99031 001431 1.156 0.14496 57702 501857
1.057 0.05543 47068 881006 1.107 0.10165 36537 264998 1.157 0.14583 04482 115395
1.058 0.05638 03334 361076 1.108 0.10255 65883 250921 1.158 0.14669 43791 508035
1.059 0.05732 50666 192694 1.109 0.10345 87083 682300 1.159 0.14755 75643 576147
1.060 0.05826 89081 239758 1.110 0.10436 00153 242428 1.160 0.14842 00051 182733
1.061 0.05921 18596 318461 1.111 0.10526 05106 574929 1.161 0.14928 17027 157544
1.062 0.06015 39228 197471 1.112 0.10616 01958 283906 1.162 0.15014 26584 297195
1.063 0.06109 50993 598109 1.113 0.10705 90722 934078 1.163 0.15100 28735 365274
1.064 0.06203 53909 194526 1.114 0.10795 71415 050923 1.164 0.15186 23493 092461
1.065 0.06297 47991 613884 1.115 0.10885 44049 120821 1.165 0.15272 10870 176639
1.066 0.06391 33257 436528 1.116 0.10975 08639 591192 1.166 0.15357 90879 283006
1.067 0.06485 09723 196163 1.117 0.11064 65200 870637 1.167 0.15443 63533 044189
1.068 0.06578 77405 380031 1.118 0.11154 13747 329074 1.168 0.15529 28844 060353
1.069 0.06672 36320 429082 1.119 0.11243 54293 297882 1.169 0.15614 86824 899314
1.070 0.06765 86484 738148 1.120 0.11332 86853 070032 1.170 0.15700 37488 096648
1.071 0.06859 27914 656117 1.121 0.11422 11440 900229 1.171 0.15785 80846 155803
1.072 0.06952 60626 486102 1.122 0.11511 28071 005046 1.172 0.15871 16911 548209
1.073 OiO7045 84636 485614 1.123 0.11600 36757 563061 1.173 0.15956 45696 713384
1.074 0.07138 99960 866729 1.124 0.11689 37514 714993 1.174 0.16041 67214 059047
1.075 0.07232 06615 796261 1.125 0.11778 30356 563835 1.175 0.16126 81475 961223
1.076 0.07325 04617 395927 1.126 0.11867 15297 174986 1.176 0.16211 88494 764352
1.077 0.07417 93981 742515 1.127 0.11955 92350 576392 1.177 0.16296 88282 781397
1.078 0.07510 74724 868054 1.128 0.12044 61530 758672 1.178 0.16381 80852 293950
1.079 0.07603 46862 759976 1.129 0.12133 22851 675250 1.179 0.16466 66215 552339
1.080 0.07696 10411 361283 1.130 0.12221 76327 242492 1.180 0.16551 44384 775734
1.081 0.07788 65386 570712 1.131 0.12310 21971 339834 1.181 0.16636 15372 152253
1.082 0.07881 11804 242898 1.132 0.12398 59797 809912 1.182 0.16720 79189 839065
1.083 0.07973 49680 188536 1.133 0.12486 89820 458693 1.183 0.16805 35849 962497
1.084 0.08065 79030 174545 1.134 0.12575 12053 055603 1.184 0.16889 85364 618139
1.085 0.08157 99869 924229 1.135 0.12663 26509 333660 1.185 0.16974 27745 870945
1.086 0.08250 12215 117437 1.136 0.12751 33202 989596 1.186 0.17058 63005 755337
1.087 0.08342 16081 390724 1.137 0.12839 32147 683990 1.187 0.17142 91156 275310
1.088 0.08434 11484 337509 1.138 0.12927 23357 041392 1.188 0.17227 12209 404532
1.089 0.08525 98439 508234 1.139 0.13015 06844 650451 1.189 0.17311 26177 086448
1.090 0.08617 76962 410523 1.140 0.13102 82624 064041 1.190 0.17395 33071 234380
1.091 0.08709 47068 509338 1.141 0.13190 50708 799386 1.191 0.17479 32903 731631
1.092 0.08801 08773 227133 1.142 0.13278 11112 338185 1.192 0.17563 25686 431580
1.093 0.08892 62091 944015 1.143 0.13365 63848 126736 1.193 0.17647 11431 157791
1.094 0.08984 07039 997895 1.144 0.13453 08929 576062 1.194 0.17730 90149 704103
1.095 0.09075 43632 684641 1.145 0.13540 46370 062030 1.195 0.17814 61853 834740
1.096 0.09166 71885 258238 1.146 0.13627 76182 925478 1.196 0.17898 26555 284400
1.097 0.09257 91812 930932 1.147 0.13714 98381 472336 1.197 0.17981 84265 758361
1.098 0.09349 03430 873389 1.148 0.13802 12978 973747 1.198 0.18065 34996 932576
1.099 0.09440 06754 214843 1.149 0.13889 19988 666186 1.199 Oil8148 78760 453772
1.100 0.09531 01798 043249 1.150 0.13976 19423 751587 1.200 0.18232 15567 939546
[
6;) 1
I [c-y1
In 10 = 2.30258 50929 940457
108 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X In 2 X In x X In x
1.200 0.18232 15567 939546 1.250 0.22314 35513 142098 1.300 0.26236 42644 674911
1.201 0.18315 45430 978465 1.251 0.22394 32314 847741 1.301 0.26313 31995 303682
1.202 0.18398 68361 130158 1.252 0.22474 22726 779068 1.302 0.26390 15437 863775
1.203 0.18481 84369 925418 1.253 0.22554 06759 139312 1.303 0.26466 92981 427081
1.204 0.18564 93468 866293 1.254 0.22633 84422 107290 1.304 0.26543 64635 044612
1.205 0.18647 95669 426183 1.255 0.22713 55725 837472 1.305 0.26620 30407 746567
1.206 0.18730 90983 049937 1.256 0.22793 20680 460069 1.306' 0.26696 90308 542393
1.207 0.18813 79421 153944 1.257 0.22872 79296 081104 1.307 0.26773 44346 420849
1.208 0.18896 60995 126232 1.258 0.22952 31582 782488 1.308 0.26849 92530 350070
1.209 0.18979 35716 326556 1.259 0.23031 77550 622101 1.309 0.26926 34869 277629
1.210 0.19062 03596 086497 1.260 0.23111 17209 633866 1.310 0.27002 71372 130602
1.211 0.19144 64645 709552 1.261 0.23190 50569 827825 1.311 0.27079 02047 815628
1.212 0.19227 18876 471227 1.262 0.23269 77641 190214 1.312 0.27155 26905 218973
1.213 0.19309 66299 619131 1.263 0.23348 98433 683541 1.313 0.27231 45953 206591
1.214 0.19392 06926 373065 1.264 0.23428 12957 246657 1. 314 0.27307 59200 624188
1.215 0.19474 40767 925118 1.265 0.23507 21221 794836 1. 31'5 0.27383 66656 297279
1.216 0.19556 67835 439753 1.266 0.23586 23237 219844 1.316 Oi27459 68329 031255
1.217 0.19638 88140 053901 1.267 0.23665 19013 390020 1.317 0.27535 64227 611440
1.218 0.19721 01692 877053 1.268 0.23744 08560 150342 1.318 0.27611 54360 803155
1.219 0.19803 08504 991345 1.269 0.23822 91887 322506 1.319 0.27687 38737 351775
1.220 0.19885 08587 451652 1.270 0.23901 69004 704999 0.27763 17365 982795
1.221
1.222
0.19967
0.20048
01951
88607
285676
494036
1.271
1.272
0.23980
0.24059
39922
04649
073170
179304
:- 3322:
1:322
0.27838
0.27914
90255
57414
401883
294945
1.223 0.20130 68567 050353 1.273 0.24137 63195 752695 1.32:3 0.27990 18851 328186
1.224 0.20212 41840 901343 1.274 0.24216 15571 499716 1.3214 0.28065 74575 148165
1.225 0.20294 08439 966903 1.275 0.24294 61786 103895 1.325 0.28141 24594 381855
1.226 0.20375 68375 140197 1.276 0.24373 01849 225981 1.326 0.28216 68917 636708
1.227 0.20457 21657 287744 1.277 0.24451 35770 504022 1.3i!7 0.28292 07553 500705
1.228 0.20538 68297 249507 1.278 0.24529 63559 553431 1.328 0.28367 40510 542421
1.229 0.20620 08305 838978 1.279 0.24607 85225 967056 1.329 0.28442 67797 311083
1.230 0.20701 41693 843261 1.280 0.24686 00779 315258 1.330 0.28517 89422 336624
1.231 0.20782 68472 023165 1.281 0.24764 10229 145972 1.331 0.28593 05394 129746
1.232 0.20863 88651 113280 1.282 0.24842 13584 984783 1.332 0.28668 15721 181974
1.233 0.20945 02241 822072 1.283 0.24920 10856 334994 1.333 0.28743 20411 965716
1.234 0.21026 09254 831961 1.284 0.24998 02052 677694 1.334 0.28818 19474 934320
1.235 0.21107 09700 799405 1.285 0.25075 87183 471831 1.335 0.28893 12918 522129
1.236 0.21188 03590 354990 1.286 0.25153 66258 154276 1.336 0.28968 00751 144540
1.237 0.21268 90934 103508 1.287 0.25231 39286 139896 1. 337 0.29042 82981 198061
1.238 0.21349 71742 624044 1.288 0.25309 06276 821619 1.338 0.29117 59617 060367
1.239 0.21430 46026 470054 1.289 0.25386 67239 570503 1.339 Oi29192 30667 090355
1.240 0.21511 13796 169455 1.290 0.25464 22183 735807 1.340 0.29266 96139 628200
1.241 0.21591 75062 224702 1.291 0.25541 71118 645054 1.341 0.29341 56042 995415
1.242 0.21672 29835 112870 1.292 0.25619 14053 604101 1.342 0.29416 10385 494901
1.243 0.21752 78125 285741 1.293 0.25696 50997 897204 1.343 0.29490 59175 411005
1.244 0.21833 19943 169877 1.294 0.25773 81960 787088 1.344 0.29565 02421 009578
1.245 6.21913 55299 166709 1.295 0.25851 06951 515011 1.1345 0.29639 40130 538024
1.246 0.21993 84203 652614 1.296 0.25928 25979 300830 1.346 0.29713 72312 225361
1.247 0.22074 06666 978994 1.297 0.26005 39053 343068 1.347 0.29787 98974 282269
1.248 0.22154 22699 472359 1.298 0.26082 46182 818983 1.348 0.29862 20124 901153
1.249 0.22234 32311 434406 1.299 0.26159 47376 884625 1.349 0.29936 35772 256188
cc-pI
1.250 0.22314 35513 142098 1.300 0.26236 42644 674911 1.350 0.30010 45924 503381
c-y
[ I
In 10 = 2.30258 50929 940457
ELEMENTARY TRANSCENDENTAL FUNCTIONS 109
NATURAL LOGARITHMS Table 4.2
x In x X In x X In x
1.350 0.30010 45924 503381 1.400 0.33647 22366 212129 1.450 0.37156 35564 324830
1.351 0.30084 50589 780618 1.401 0.33718 62673 548700 1.451 0.37225 29739 020508
1.352 0.30158 49776 207723 1.402 0.33789 97886 123983 1.452 0.37294 19164 026043
1.353 0.30232 43491 886510 1.403 0.33861 28011 203239 1.453 P.37363 03845 881459
1.354 0.30306 31744 900833 1.404 0.33932 53056 036194 1.454 0.37431 83791 113276
1.355 0.30380 14543 316642 1.405 0.34003 73027 857091 1.455 0.37500 59006 234558
1.356 0.30453 91895 182038 1.406 0.34074 87933 884732 1.456 0.37569 29497 744942
1.357 Oi30527 63808 527321 1.407 0.34145 97781 322520 1.457 0.37637 95272 130678
1.358 0.30601 30291 365044 1.408 0.34217 02577 358507 1.458 0.37706 56335 864664
1.359 0.30674 91351 690067 1.409 0.34288 02329 165432 1.459 0.37775 12695 406486
1.360 0.30748 46997 479606 1.410 0.34358 97043 900769 1.460 0.37843 64357 202451
1.361 0.30821 97236 693290 1.411 0.34429 86728 706770 1.461 0.37912 11327 685624
1.362 0.30895 42077 273206 1.412 0.34500 71390 710503 1.462 0.37980 53613 275868
1.363 0.30968 81527 143956 1.413 0.34571 51037 023904 1.463 0.38048 91220 379873
1.364 0.31042 15594 212704 1.414 0.34642 25674 743810 1.464 0.38117 24155 391198
1.365 0.31115 44286 369231 1.415 0.34712 95310 952009 1.465 0.38185 52424 690306
1.366 0.31188 67611 485983 1.416 0.34783 59952 715280 1.466 Or38253 76034 644597
1.367 0.31261 85577 418125 1.417 0.34854 19607 085434 1.467 0.38321 94991 608447
1.368 0.31334 98192 003587 1.418 0.34924 74281 099358 1.468 0.38390 09301 923238
1.369 0.31408 05463 063118 1.419 0.34995 23981 779056 1.469 0.38458 18971 917403
1.370 0.31481 07398 400335 1.420 0.35065 68716 131694 1.470 0.38526 24007 906449
0.31554 04005 801773 1.421 0.35136 08491 149636 1.471 0.38594 24416 193005
:* z: 0.31626 95293 036935 1.422 0.35206 43313 810491 1.472 0.38662 20203 066845
1:373 0.31699 81267 858340 1.423 0.35276 73191 077153 1.473 0.38730 11374 804932
1.374 0.31772 61938 001576 1.424 0.35346 98129 897840 1.474 0.38797 97937 671449
1.375 0.31845 37311 185346 1.425 0.35417 18137 206138 1.475 0.38865 79897 917831
1.376 0.31918 07395 111519 1.426 0.35487 33219 921042 1.476 0.38933 57261 782808
1.377 0.31990 72197 465178 1.427 0.35557 43384 946994 1.477 0.39001 30035 492427
1.378 0.32063 31725 914668 1.428 0.35627 48639 173926 1.478 0.39068 98225 260100
1.379 0.32135 85988 111648 1.429 0.35697 48989 477304 1.479 0.39136 61837 286627
1.380 0.32208 34991 691133 1.430 0.35767 44442 718159 1.480 0.39204 20877 760237
1.381 0.32280 78744 271551 1.431 0.35837 35005 743139 1.481 0.39271 75352 856617
1.382 0.32353 17253 454782 1.432 0.35907 20685 384539 1.482 0.39339 25268 738951
1.383 0.32425 50526 826212 1.433 0.35977 01488 460348 1.483 0.39406 70631 557950
1.384 0.32497 78571 954778 1.434 0.36046 77421 774286 1.484 0.39474 11447 451887
1.385 0.32570 01396 393018 1.435 0.36116 48492 115844 1.485 0.39541 47722 546629
1.386 0.32642 19007 677115 1.436 0.36186 14706 260324 1.486 0.39608 79462 955674
1.387 0.32714 31413 326945 1.437 0.36255 76070 968879 1.487 0.39676 06674 780180
1.388 0.32786 38620 846128 1.438 0.36325 32592 988549 1.488 0.39743 29364 109001
1.389 0.32858 40637 722067 1.439 0.36394 84279 052308 1.489 0.39810 47537 018719
1.390 0.32930 37471 426004 1.440 0.36464 31135 879093 1.490 0.39877 61199 573678
1.391 0.33002 29129 413059 1.441 0.36533 73170 173850 1.491 0.39944 70357 826014
1.392 0.33074 15619 122279 1.442 0.36603 10388 627573 1.492 0.40011 75017 815691
1.393 0.33145 96947 976686 1.443 0.36672 42797 917338 1.493 0.40078 75185 570533
1.394 0.33217 73123 383321 1.444 0.36741 70404 706345 1.494 0.40145 70867 106256
1.395 0.33289 44152 733290 1.445 0.36810 93215 643955 1.495 0.40212 62068 426497
0.33361 10043 401807 1.446 0.36880 11237 365729 1.496 0.40279 48795 522855
11.z: 0.33432 70802 748248 1.447 0.36949 24476 493468 1.497 0.40346 31054 374913
1:398 0.33504 26438 116185 1.448 0.37018 32939 635246 1.498 0.40413 08850 950277
1.399 0.33575 76956 833441 1.449 0.37087 36633 385453 1.499 0.40479 82191 204607
1.400 0.33647 22366 212129 1.450 0.37156 35564 324830 1.500 0.40546 51081 081644
c-y
[ I [ 1
(-fV
X In x 5 In z X In x
1.500 0.40546 51081 081644 1.550 0.43825 49309 311553 1.600 0.47000 36292 457356
1.501 0.40613 15526 513249 1.551 0.43889 98841 944018 0.47062 84340 145776
1.502 0.40679 75533 419430 1.552 0.43954 44217 610270 :- El: 0.47125 28486 461675
1.503 0.40746 31107 708374 1.553 0.44018 85441 665500 1:603 0.47187 68736 274159
1.504 0.40812 82255 276481 1.554 0.44083 22519 454557 1.604 0.47250 05094 443228
1.505 0.40879 28982 008391 1.555 0.44147 55456 311975 1.605 0.47312 37565 819792
1.506 0.40945 71293 777018 1.556 0.44211 84257 561999 1.606 0.47374 66155 245699
1.507 0.41012 09196 443584 1.557 0.44276 08928 518613 1.607 0.47436 90867 553755
1.508 0.41078 42695 857643 1.558 0.44340 29474 485565 1.608 0.47499 11707 567746
1.509 0.41144 71797 857118 1.559 0.44404 45900 756395 1.609 0.47561 28680 102462
1.510 0.41210 96508 268330 1.560 0.44468 58212 614457 1.610 0.47623 41789 963716
1: 511 0141277 16832 906025 1.561 0.44532 66415 332950 1.611 Oi47685 51041 948373
1.512 0.41343 32777 573413 1.562 0.44596 70514 174942 1.612 0.47747 56440 844365
1.513 Oi41409 44348 062189 1.563 0.44660 70514 393396 1.613 Oi47809 57991 430718
1.514 0.41475 51550 152570 1.564 0.44724 66421 231193 1.614 0.47871 55698 477571
1.515 0.41541 54389 613325 1.565 0.44788 58239 921165 1.615 0.47933 49566 746199
1.516 0.41607 52872 201799 1.566 0.44852 45975 686114 1.616 0.47995 39600 989036
1.517 0.41673 47003 663952 1.567 0.44916 29633 738838 1.617 0.48057 25805 949698
1.518 0.41739 36789 734382 1.568 0.44980 09219 282161 1.618 0.48119 08186 362999
1.519 0.41805 22236 136358 1.569 0.45043 84737 508955 1.619 0.48180 86746 954981
1.520 0.41871 03348 581850 1.570 0.45107 56193 602167 1.620 0.48242 61492 442927
1.521 0.41936 80132 771558 1.571 0.45171 23592 734841 1.621 0.48304 32427 535391
1.522 0.42002 52594 394941 1.572 0.45234 86940 070148 1.622 0.48365 99556 932212
1.523 0.42068 20739 130248 1.573 0.45298 46240 761408 1.623 0.48427 62885 324542
1.524 0.42133 84572 644545 1.574 0.45362 01499 952115 1.624 0.48489 22417 394862
1.525 0.42199 44100 593749 1.575 0.45425 52722 775964 1.625 0.48550 78157 817008
1.526 0.42264 99328 622653 1.576 0.45488 99914 356874 1.626 0.48612 30111 256188
1.527 0.42330 50262 364954 1.577 0.45552 43079 809013 1.627 0.48673 78282 369007
1.528 0.42395 96907 443287 1.578 0.45615 82224 236825 1.628 0.48735 22675 803486
1.529 0.42461 39269 469252 1.579 0.45679 17352 735050 1.629 0.48796 63296 199081
1.530 0.42526 77354 043441 1.580 0.45742 48470 388754 1.630 0.48858 00148 186710
1.531 0.42592 11166 755467 1.581 0.45805 75582 273350 1.631 0.48919 33236 388768
1.532 0.42657 40713 183996 1.582 0.45868 98693 454621 1.632 0.48980 62565 419153
1.533 0.42722 65998 896771 1.583 0.45932 17808 988751 1.633 0.49041 88139 883281
1.534 0.42787 87029 450644 1.584 0.45995 32933 922341 1.634 0.49103 09964 378111
1.535 0.42853 03810 391605 1.585 0.46058 44073 292439 1.635 0.49164 28043 492167
1.536 0.42918 16347 254804 1.586 0.46121 51232 126562 1.636 0.49225 42381 805553
1.537 0.42983 24645 564588 1.587 0.46184 54415 442720 1.637 0.49286 52983 889979
1.538 0.43048 28710 834522 1.588 0.46247 53628 249440 1.638 0.49347 59854 308777
1.539 0.43113 28548 567422 1.589 0.46310 48875 545789 1.639 0.49408 62997 616926
1.540 0.43178 24164 255378 1.590 0.46373 40162 321402 1.640 0.49469 62418 361071
1.541 0.43243 15563 379787 1.591 0.46436 27493 556498 1.641 0.49530 58121 079538
1.542 0.43308 02751 411377 1.592 0.46499 10874 221913 1.642 0.49591 50110 302365
1.543 0.43372 85733 810238 1.593 0.46561 90309 279115 1.643 0.49652 38390 551310
1.544 0.43437 64516 025844 1.594 0.46624 65803 680233 1.644 0.49713 22966 339882
1.545 0.43502 39103 497088 1.595 0.46687 37362 368079 1.645 0.49774 03842 173352
1.546 0.43567 09501 652302 1.596 0.46750 04990 276170 1.646 0.49834 81022 548781
1.547 0.43631 75715 909291 1.597 0.46812 68692 328754 1.647 0.49895 54511 955033
1.548 0.43696 37751 675354 1.598 0.46875 28473 440829 1.648 0.49956 24314 872800
1.549 0.43760 95614 347316 1.599 0.46937 84338 518172 1.649 0.50016 90435 774619
1.550 0.43825 49309 311553 1.600 0.47000 36292 457356 1.650 0.50077 52879 124892
E 1
C-58)6
[(-;I51 c-p
I: 3
In 10 = 2.30258 50929 940457
ELEMXNTARY TRANSCENDENTAL FUNCTIONS 111
NATURAL LOGARITHMS Table 4.2
X In x 2 In 2 X In x
1.650 0.50077 52879 124892 1.700 0.53062 82510 621704 1.750 0.55961 57879 354227
1.651 0.50138 11649 379910 1.701 0.53121 63134 137247 1.751 0.56018 70533 037148
1.652 0.50198 66750 987863 1.702 0.53180 40301 511824 1.752 0.56075 79925 141997
1.653 0.50259 18188 388871 1.703 0.53239 14016 805512 1.753 {Or56132 86059 390974
1.654 0.50319 65966 014996 1.704 0.53297 84284 071240 1.754 0.56189 88939 499913
1.655 0.50380 10088 290262 1.705 0.53356 51107 354801 1.755 0.56246 88569 178291
1.656 0.50440 50559 630679 1.706 0.53415 14490 694874 1.756 0.56303 84952 129249
1.657 0.50500 87384 444259 1.707 0.53473 74438 123036 1.757 0.56360 78092 049601
1.658 0.50561 20567 131032 1.708 0.53532 30953 663781 1.758 0.56417 67992 629853
1.659 0.50621 50112 083074 1.709 0.53590 84041 334538 1.759 0.56474 54657 554211
1.660 0.50681 76023 684519 1.710 0.53649 33705 145685 1.760 0.56531 38090 500604
1.661 0.50741 98306 311578 1.711 0.53707 79949 100564 1.761 0.56588 18295 140691
1.662 0.50802 16964 332564 1.712 0.53766 22777 195504 1.762 0.56644 95275 139878
1.663 0.50862 32002 107906 1.713 0.53824 62193 419829 1.763 0.56701 69034 157332
1.664 0.50922 43423 990168 1.714 0.53882 98201 755880 1.764 0.56758 39575 845996
1.665 0.50982 51234 324071 1.715 0.53941 30806 179032 1.765 0.56815 06903 852601
1.666 0.51042 55437 446509 1.716 0.53999 60010 657705 1.766 0.56871 71021 817683
1.667 0.51102 56037 686569 1.717 0.54057 85819 153385 1.767 0.56928 31933 375593
1.668 0.51162 53039 365550 1.718 0.54116 08235 620636 1.768 0.56984 89642 154517
1.669 0.51222 46446 796980 1.719 0.54174 27264 007122 1.769 0.57041 44151 776482
1.670 0.51282 36264 286637 1.720 0.54232 42908 253617 1.770 0.57097 95465 857378
1.671 0.51342 22496 132567 1.721 0.54290 55172 294024 1.771 0.57154 43588 006965
1.672 0.51402 05146 625099 1.722 0.54348 64060 055391 1.772 0.57210 88521 828892
1.673 0.51461 84220 046869 1.723 0.54406 69575 457926 1.773 0.57267 30270 920708
1.674 0.51521 59720 672836 1.724 0.54464 71722 415014 1.774 0.57323 68838 873877
1.675 0.51581 31652 770298 1.725 0.54522 70504 833231 1.775 0.57380 04229 273791
1.676 0.51641 00020 598913 1.726 0.54580 65926 612362 1.776 0.57436 36445 699783
1.677 0.51700 64828 410718 1.727 0.54638 57991 645415 1.777 0.57492 65491 725143
1.678 0.51760 26080 450144 1.728 0.54696 46703 818639 1.778 0.57548 91370 917128
1.679 0.51819 83780 954038 1.729 0.54754 32067 011534 1.779 0.57605 14086 836981
1.680 0.51879 37934 151676 1.730 0.54812 14085 096876 1.780 0.57661 33643 039938
1.681 0.51938 88544 264786 1.731 0.54869 92761 940722 1.781 0.57717 50043 075246
1.682 0.51998 35615 507563 1.732 0.54927 68101 402434 1.782 0.57773 63290 486176
1.683 0.52057 79152 086690 1.733 0.54985 40107 334690 1.783 0.57829 73388 810034
1.684 0.52117 19158 201350 1.734 0.55043 08783 583501 1.784 0.57885 80341 578176
1.685 0.52176 55638 043250 1.735 0.55100 74133 988225 1.785 0.57941 84152 316024
1.686 0.52235 88595 796637 1.736 0.55158 36162 381584 1.786 0.57997 84824 543073
1.687 0.52295 18035 638312 1.737 0.55215 94872 589679 1.787 0.58053 82361 772910
1.688 0.52354 43961 737654 1.738 0.55273 50268 432003 1.788 0.58109 76767 513224
1.689 0.52413 66378 256630 1.739 0.55331 02353 721460 1.789 0.58165 68045 265821
1.690 0.52472 85289 349821 1.740 0.55388 51132 264377 1.790 0.58221 56198 526636
1.691 0.52532 00699 164432 1.741 0.55445 96607 860520 1.791 0.58277 41230 785747
1.692 0.52591 12611 840315 1.742 0.55503 38784 303111 1.792 0.58333 23145 527387
1.693 0.52650 21031 509983 1.743 0.55560 77665 378839 1.793 0.58389 01946 229958
1.694 0.52709 25962 298627 1.744 0.55618 13254 867879 1.794 0.58444 77636 366044
1.695 0.52768 27408 324136 1.745 0.55675 45556 543905 1.795 0.58500 50219 402422
1.696 0.52827 25373 697113 1.746 0.55732 74574 174105 1.796 Oi58556 19698 800079
1.697 0.52886 19862 520893 1.747 0.55790 00311 519195 1.797 0.58611 86078 014220
1.698 0.52945 10878 891556 1.748 0.55847 22772 333437 1.798 Oi58667 49360 494285
1.699 0.53003 98426 897950 1.749 0.55904 41960 364650 1.799 0.58723 09549 683961
1.700 0.53062 82510 621704 1.750 0.55961 57879 354227 1.800 0.58778 66649 021190
C-58)5
[ 1 [ I
C-5814
[C-5814I
In 10 = 2.30258 50929 940457
112 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X In x x In x 2 In x
1.800 0.58778 66649 021190 1.850 0.61518 56390 902335 1.900 0.64185 38861 723948
1.801 0.58834 20661 938190 1.851 0.61572 60335 913605 1.901 0.64238 00635 062921
1.802 0.58889 71591 861462 1.852 0.61626 61362 239876 1.902 0.64290 59641 231986
1.803 0.58945 19442 211802 1.853 0.61680 59473 032227 1.903 0.64343 15883 140124
1.804 0.59000 64216 404319 1.854 0.61734 54671 436634 1.904 0.64395 69363 691736
1.805 0.59056 05917 848442 1.855 0.61788 46960 593985 1.905 0.64448 20085 786643
1.806 0.59111 44549 947937 1.856 0.61842 36343 640088 1.906 0.64500 68052 320104
1.807 0.59166 80116 100914 1.857 0.61896 22823 705687 1.907 0.64553 13266 182820
1.808 0.59222 12619 699848 1.858 0.61950 06403 916468 1.908' 0.64605 55730 260948
1.809 0.59277 42064 131581 1.859 0.62003 87087 393070 1.909 0.64657 95437 436106
1.810 0.59332 68452 777344 1.860 0.62057 64877 251099 1.910 0.64710 32420 585385
1.811 0.59387 91789 012763 1.861 0.62111 39776 601137 1.911. 0.64762 66652 581360
1.812 0.59443 12076 207876 1.862 0.62165 11788 548753 1.912 0.64814 98146 292095
1.813 0.59498 29317 727140 1.863 0.62218 80916 194514 1.913 0.64867 26904 581158
1.814 0.59553 43516 929449 1.864 0.62272 47162 633994 1.914 0.64919 52930 307625
1.815 0.59608 54677 168141 1.865 0.62326 10530 957789 1.915 0.64971 76226 326093
1.816 0.59663 62801 791016 1.866 0.62379 71024 251521 1.916 0.65023 96795 486688
1.817 0.59718 67894 140341 1.867 0.62433 28645 595856 1.917 0.65076 14640 635074
1.818 0.59773 69957 552871 1.868 0.62486 83398 066509 1.918 0.65128 29764 612465
1.819 0.59828 68995 359852 1.869 0.62540 35284 734258 1.919 0.65180 42170 255629
1.820 0.59883 65010 887040 1.870 0.62593 84308 664953 1.92:O 0.65232 51860 396902
0.59938 58007 454709 1.871 0.62647 30472 919526 1. 9i!l 0.65284 58837 864196
1.872 0.62700 73780 554003 1.9;!2 0.65336 63105 481007
;.;;;'
1:823 0.60048 34956
0.59993 47988 377666
965260 1.873 0.62754 14234 619515 1.9:!3 0.65388 64666 066427
1.824 0.60103 18916 521396 1.874 0.62807 51838 162304 1.924 0.65440 63522 435147
1.825 0.60157 99870 344548 1.875 0.62860 86594 223741 1.925 0.65492 59677 397475
1.826 0.60212 77821 727767 1.876 0.62914 18505 840329 1.926 0.65544 53133 759338
1.827 0.60267 52773 958697 1.877 0.62967 47576 043718 1.927 0.65596 43894 322293
1.828 0.60322 24730 319583 1.878 0.63020 73807 860712 1.928 0.65648 31961 883539
1.829 Oi60376 93694 087286 1.879 0.63073 97204 313283 1,929 0.65700 17339 235920
1.830 0.60431 59668 533296 1.880 0.63127 17768 418578 1.930 0.65752 00029 167942
1.831 0.60486 22656 923737 1.881 0.63180 35503 188933 1.931 0.65803 80034 463774
1.832 0.60540 82662 519385 1.882 0.63233 50411 631879 1.932 0.65855 57357 903263
1.833 0.60595 39688 575680 1.883 0.63286 62496 750154 1.933 0.65907 32002 261938
1.834 0.60649 93738 342731 1.884 0.63339 71761 541713 1.934 0.65959 03970 311026
1.835 0.60704 44815 065336 1.885 0.63392 78208 999741 1.935 0.66010 73264 817451
1.836 0.60758 92921 982987 1.886 0.63445 81842 112658 1.936 0.66062 39888 543853
1.837 0.60813 38062 329886 1.887 0.63498 82663 864132 1.937 0.66114 03844 248588
1.838 0.60867 80239 334953 1.888 0.63551 80677 233089 1.938 0.66165 65134 685745
1.839 0.60922 19456 221840 1.889 0.63604 75885 193725 1.'939 0.66217 23762 605148
1.840 0.60976 55716 208943 1.890 0.63657 68290 715510 1.940 0.66268 79730 752368
1.841 0.61030 89022 509408 1.891 0.63710 57896 763204 1.941 0.66320 33041 868732
1.842 0.61085 19378 331151 1.892 0.63763 44706 296865 0.66371 83698 691332
1.843 0.61139 46786 876862 1.893 0.63816 28722 271858 :* zt; 0.66423 31703 953030
1.844 0.61193 71251 344021 1.894 0.63869 09947 638865 1:944 0.66474 77060 382473
1.845 0.61247 92774 924905 1.895 0.63921 88385 343897 1.945 0.66526 19770 704096
1.846 0.61302 11360 806604 1.896 0.63974 64038 328301 1.946 0.66577 59837 638133
1.847 0.61356 27012 171029 1.897 0.64027 36909 528772 1.947 0.66628 97263 900626
1.848 0.61410 39732 194924 1.898 0.64080 07001 877361 1.948 0.66680 32052 203434
1.849 Or61464 49524 049878 1.899 0.64132 74318 301488 1.949 0.66731 64205 254238
1.850 0.61518 56390 902335 1.900 0.64185 38861 723948 L950 0.66782 93725 756554
[ 1
C-58)4
[ I
‘-;8’4
X In x X In 2 X In x
1.950 0.66782 93725 756554 2.000 0.69314 71805 599453 2.050 0.71783 97931 503168
1.951 Oi66834 20616 409742 2.001 0.69364 70556 015964 2.051 0.71832 74790 902436
1.952 0.66885 44879 909007 2.002 0.69414 66808 930288 2.052 0.71881 49273 085231
1.953 Oi66936 66518 945419 2.003 0.69464 60566 836812 2.053 0.71930 21380 367965
1.954 0.66987 85536 205910 2.004 0.69514 51832 226184 2.054 0.71978 91115 063665
1.955 0.67039 01934 373291 2.005 0.69564 40607 585325 2.055 0.72027 58479 481979
1.956 0.67090 15716 126256 2.006 0.69614 26895 397438 2.056 0.72076 23475 929187
1.957 0.67141 26884 139392 0.69664 10698 142011 2.057 0.72124 86106 708201
1.958 0.67192 35441 083186 %*00:: 0.69713 92018 294828 2.058 0.72173 46374 118579
1.959 0.67243 41389 624037 2:009 0.69763 70858 327974 2.059 0.72222 04280 456524
1.960 0.67294 44732 424259 2.010 0.69813 47220 709844 2.060 0.72270 59828 014897
1.961 Oi67345 45472 142092 2.011 0.69863 21107 905150 2.061 0.72319 13019 083220
1.962 0.67396 43611 431713 2.012 0.69912 92522 374928 2.062 0.72367 63855 947682
1.963 0.67447 39152 943240 2.013 0.69962 61466 576544 2.063 0.72416 12340 891148
1.964 0.67498 32099 322741 2.014 0.70012 27942 963706 2.064 0.72464 58476 193163
1.965 0.67549 22453 212246 0.70061 91953 986463 2.065 0.72513 02264 129961
1.966 0.67600 10217 249748 I* 00:: 0.70111 53502 091222 2.066 0.72561 43706 974468
1.967 0.67650 95394 069220 2:017 0.70161 12589 720747 2.067 0.72609 82806 996312
1.968 0.67701 77986 300617 2.018 0.70210 69219 314172 2.068 0.72658 19566 461827
1.969 0.67752 57996 569885 2.019 0.70260 23393 307004 2.069 0.72706 53987 634060
1.970 0.67803 35427 498971 2.020 0.70309 75114 131134 2.070 0.72754 86072 772777
11971 Oi678ii ib281 7Oii83i 0.70359 24384 214840 2.071 0.72803 15824 134471
1.972 0.67904 82561 804437 :- 8;: 0.70408 71205 982797 2.072 0.72851 43243 972366
11973 Oi67955 52270 404783 2:023 0.70458 15581 856084 2.073 0.72899 68334 536425
1.974 0.68006 19410 112898 2.024 0.70507 57514 252191 2.074 0.72947 91098 073356
1.975 0.68056 83983 530852 2.025 0.70556 97005 585025 2.075 0.72996 11536 826616
1.976 0.68107 45993 256761 27026 0.70606 34058 264916 2.076 0.73044 29653 036422
1.977 0.68158 05441 884799 2.027 0.70655 68674 698630 2.077 0.73092 45448 939753
1.978 0.68208 62332 005204 z*. 82298 0.70705
0.70754 30608
00857 436777
289367 2.078 0.73140 58926 770357
1.979 0.68259 16666 204287 2.079 0.73188 70088 758759
1.980 0.68309 68447 064439 2.030 0.70803 57930 536960 2.080 0.73236 78937 132266
1.981 0.6836$'?7677 164139 2.031 0.70852 82825 982476 2.081 0.73284 85474 114974
1.982 0.68410 64359 077962 2.032 0.70902 05297 162355 2.082 0.73332 89701 927771
1.983 0.68461 08495 376589 2.033 0.70951 25346 462096 2.083 0.73380 91622 788349
1.984 0.68511 50088 626811 2.034 0.71000 42976 263682 2.084 0.73428 91238 911205
1.985 0.68561 89141 391537 2.035 0.71049 58188 945583 2.085 0.73476 88552 507648
1.986 0.68612 25656 229808 2:036 0.71098 70986 882763 2.086 0.73524 83565 785807
1.987 0.68662 59635 696798 2.037 0.71147 81372 446688 2.087 0.73572 76280 950637
1.988 0.68712 91082 343823 2.038 0.71196 89348 005331 2.088 0.73620 66700 203923
1.989 0.68763 19998 718351 2.039 0.71245 94915 923181 2.089 0.73668 54825 744287
1.990 0.68813 46387 364010 0.71294 98078 561250 2.090 0.73716 40659 767196
11991
1.992
Or68863
0.68913
70250
91591
820592
624065
%- FA!
2:042
0.71343
0.71392
98838
97197
277077
424738
2.091
2.092
0.73764
0.73812
24204
05462
464965
026765
1.993 0.68964 10412 306577 2.043 0.71441 93158 354850 2.093 0.73859 84434 638627
1.994 0.69014 26715 396466 2.044 0.71490 86723 414580 2.094 0.73907 61124 483451
1.995 0.69064 40503 418268 2.045 0.71539 77894 947651 2.095 0.73955 35533 741011
i;99i Oi69114 51778 892722 2iO46 0.71588 66675 294347 2.096 0.74003 07664 587957
1.997 0.69164 60544 336782 2.047 0.71637 53066 791525 2.097 0.74050 77519 197829
1.998 0.69214 66802 263618 2.048 0.71686 37071 772614 2.098 0.74098 45099 741054
1.999 0.69264 70555 182630 2.049 0.71735 18692 567627 2.099 0.74146 10408 384959
2.000 0.69314 71805 599453 2.050 0.71783 97931 503168 2.100 0.74193 73447 293773
r (91 [C-i)3 1
L 5 J
For x>2.1 see Example 5.
[ 1 C-i)3
N e-z
0.000 1.00000 00000 00000 000 1.00000 00000 00000 000
0.001 1.00100 05001 66708 342 0.99900 04998 33374 992
0.002 1.00200 20013 34000 267 0.99800 19986 67333 067
0.003 iI 45045 03377 026 0.99700 44955 03372 976
0.004 1.00400 80106 77341 872 0.99600 79893 43991 472
0. 003 1.00501 25208 59401 063 0.99501 24791 92682 313
0.006 1.00601 80360 54064 865 0.99401 79640 53935 265
0.007 1.00702 45572 66848 555 0.99302 44429 33235 105
0.008 1.00803 20855 04273 431 0.99203 19148 37060 630
0.009 1.00904 06217 73867 814 0.99104 03787 72883 662
0.010 1.01005 01670 84168 058 0.99004 ,d337 49168 054
0.011 1.01106 07224 44719 556 0.98906 02787 75368 698
0.012 1.01207 22888 66077 754 0.98807 17128 61930 540
0.013 1.01308 48673 59809 158 0.98708 41350 20287 583
0.014 1.01409 84589 38492 345 0.98609 75442 62861 903
0.015 1.01511 30646 15718 979 0.98511 19396 03062 661
0.016 1.01612 86854 06094 822 0.98412 73200 55285 115
0.017 1.01714 53223 25240 748 0.98314 36846 34909 635
0.018 1.01816 29763 89793 761 0.98216 10323 58300 718
0.019 1.01918 16486 17408 011 0.9811'7 93622 42806 006
0.020 1.02020 13400 26755 810 0.98019 86733 06755 302
0.021 1.02122 20516 37528 653 0.97921 89645 69459 588
0.022 1.02224 37844 70438 235 0.97824 02350 51210 045
0.023 1.02326 65395 47217 475 0.97726 24837 73277 073
0.024 1.02429 03178 90621 534 0.97628 57097 57909 314
0.025 1.02531 51205 24428 841 0.97530 99120 28332 669
0.026 1.02634 09484 73442 115 0.97433 50896 08749 328
0.027 1.02736 78027 63489 392 0.97336 12415 24336 791
0.028 1.02839 56844 21425 045 0.97238 83668 01246 891
0.029 1.02942 45944 75130 820 0.97141 64644 66604 825
0.030 1.03045 45339 53516 856 0.97044 55335 48508 177
0.031 1.03148 55038 86522 716 0.96947 55730 76025 948
0.032 1.03251 75053 05118 420 0.96850 65820 79197 585
0.033 1.03355 05392 41305 472 0.96753 85595 89032 009
0.034 1.03458 46067 28117 894 0.96657 15046 37506 651
0.035 1.03561 97087 99623 260 0.96560 54162 57566 478
0.036 1.03665 58464 90923 727 0.96464 02934 83123 030
0.037 1.03769 30208 38157 074 0.96367 61353 49053 452
0.038 1.03873 12328 78497 733 0.96271 29408 91199 529
0.039 1.03977 04836 50157 831 0.96175 07091 46366 723
0.040 1.04081 07741 92388 227 0.96078 94391 52323 209
0.041 1.04185 21055 45479 549 0.95982 91299 47798 914
0.042 1.04289 44787 50763 238 0.95886 97805 72484 552
0.043 1.04393 78948 50612 586 0.95791 13900 67030 669
0.044 1.04498 23548 88443 779 0.95695 39574 73046 678
0.045 1.04602 78599 08716 943 0.95599 74818 33099 907
0.046 1.04707 44109 56937 184 0.95504 19621 90714 635
0.047 1.04812 20090 79655 638 0.95408 73975 90371 141
0.048 1.04917 06553 24470 516 0.95313 37870 77504 745
0.049 liO5022 03507 40028 148 0.95218 11296 98504 853
0.050 1.05127 10963 76024 040 0. 95122 94245 00714 009
[c-J)1 1 [(-;I1
1
118 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[c-p1
1 [(-l’l1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 119
EXPONENTIAL FUNCTION Table 4.4
X ez e-1
[ (-Y1 [c-y1
120 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[
C-i)2
1 [c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 121
EXPONENTIAL FUNCTION Table 4.4
X ez e-2
0.250 1.28402 54166 87741 484 0.77880 07830 71404 868
0.251 1.26531 00843 31195 317 0.77802 23715 58957 312
0.252 1.28659 60372 84840 591 0.77724 47380 68946 150
0.253 1.28788 32768 34630 366 0.77646 78818 23737 828
0.254 ii28917 18042 67804 299 0.77569 18020 46476 034
[c-p1
0.300 0.74081 82206 81717 866
[1 1
C-l)2
122 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X ez e-2
X e= e-=
0.350 1.41906 75485 93257 248 0.70468 80897 18713 434
0.351 1.42048 73259 12195 200 0.70398 37538 55620 921
0.352 1.42190 85237 18577 438 0.70328 01219 76340 929
0.353 1.42333 11434 33601 886 0.70257 71933 77241 521
0.354 1.42475 51864 79888 380 0.70187 49673 55394 037
0.355 1.42618 06542 81480 082 0.70117 34432 08572 398
0.356 1.42760 75482 63844 915 0.70047 26202 35252 !399
0.357 1.42903 58698 53876 979 0.69977 24977 34611 008
0.358 1.43046 56204 79897 983 0.69907 30750 06525 666
0.359 1.43189 68015 71658 672 0.69837 43513 51573 587
0.360 1.43332 94145 60340 258 0.69767 63260 71031 057
0.361 1.43476 34608 78555 848 0.69697 89984 66872 738
0.362 1.43619 89419 60351 880 0.69628 23678 41770 967
0.363 1.43763 58592 41209 556 0.69558 64334 99095 062
0.364 1.43907 42141 58046 276 0.69489 11947 42910 621
0.365 1.44051 40081 49217 078 0.69419 66508 77978 831
0.366 1.44195 52426 54516 071 0.69350 28012 09755 768
0.367 1.44339 79191 15177 881 0.69280 96450 44391 707
0.368 1.44484 20389 73879 090 0.69211 71816 88730 425
0.369 1.44628 76036 74739 677 0.69142 54104 50308 508
0.370 1.44773 46146 63324 462 0.69073 43306 37354 660
0.371 1.44918 30733 86644 554 0.69004 39415 58789 010
0.372 1.45063 29812 93158 799 0.68935 42425 24222 423
0.373 1.45208 43398 32775 223 0.68866 52328 43955 806
0.374 1.45353 71504 56852 487 0.68797 69118 28979 422
0.375 1.45499 14146 18201 336 0.68728 92787 90972 199
0.376 1.45644 71337 71086 052 0.68660 23330 42301 040
0.377 1.45790 43093 71225 910 0.68591 60738 96020 141
0.378 1.45936 29428 75796 632 0.68523 05006 65870 297
0.379 1.46082 30357 43431 842 0.68454 56126 66278 222
0.380 1.46228 45894 34224 532 0.68386 14092 12355 858
0.381 1.46374 76054 09728 512 0.68317 78896 19899 696
0.382 1.46521 20851 32959 881 0.68249 50532 05390 084
0.383 1.46667 80300 68398 485 0.68181 28992 85990 553
0.384 1.46814 54416 81989 380 0.68113 14271 79547 125
0.385 1.46361 43214 41144 302 0.68045 06362 04587 638
0.386 1.47108 46708 14743 133 0.67977 05256 80321 060
0.387 1.47255 64912 73135 370 0.67909 10949 26636 810
0.388 1.47402 97842 88141 592 0.67841 23432 64104 077
0.389 1.47550 45513 33054 939 0.67773 42700 13971 142
0.390 1.47698 07938 82642 577 0.67705 68744 98164 700
0.391 1.47845 85134 13147 180 0.67638 01560 39289 177
0.392 1.47993 77114 02288 401 0.67570 41139 60626 058
0.393 1.48141 83893 29264 352 0.67502 87475 86133 209
0.394 1.48290 05486 74753 084 0.67435 40562 40444 198
0.395 1.48438 41909 20914 066 0.67368 00392 48867 624
0.396 1.48586 93175 51389 667 0.67300 66959 37386 438
0.397 1.48735 59300 51306 642 0.67233 40256 32657 274
0.398 1.48884 40299 07277 615 0.67166 20276 62009 771
0.399 1.49033 36186 07402 565 0.67099 07013 53445 901
0.400 1.49182 46976 41270 318 0.67032 00460.35639 301
C-l)9
[ 1
c-y
1 1
124 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X ez e+
1 1
t-p
IIc-y1
ELEMENTARY TRANSCENDENTAL FUNCX’IONS 125
EXPONENTIAL FUNCTION Table 4.4
5 ez ecz
1(72 1 [ c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS
X e2 e-x
1 1
c-y
[
C-i)8
1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 127
EXPONENTIAL FUNCTION Table 4.4
X ez e-2
[ 1
c-y
[
C-i)7
1
128 ELEMENTARY TRANSCENDENTAL FUNCTIONS
5 ez e-a
X e-z
0.650 1.91!554 08290 13896 070 0.52204 57767 61016 048
0.651 1.91'745 73279 32661 108 0.52152 39919 20157 530
0.652 1.91'337 57443 08913 867 0.52100 27286 03334 394
0,653 1.92129 60800 61070 883 0.52048 19862 89283 277
0.654 1.92321 83371 09468 067 0.51996 17644 57261 823
0.655 1.92514 25173 76362 630 0.51944 20625 87048 156
0.656 1.92706 86227 85934 997 0.51892 28801 58940 364
0.657 1.92899 66552 64290 740 0.51840 42166 53755 974
0.658 1.92'092 66167 39462 496 0.51788 60715 52831 438
0.659 1.92128585091 41411 902 0.51736 84443 38021 612
0.660 1.9:!479 23344 02031 522 0.51685 13344 91699 238
0.661 1.93672 80944 55146 776 0.51633 47414 96754 426
0.662 1.93866 57912 36517 879 0.51581 86648 36594 140
0.663 1.94060 54266 83841 774 0.51530 31039 95141 674
0.664 1.94254 70027 36754 070 0.51478 80584 56836 146
0.665 1.94449 05213 36830 982 0.51427 35277 06631 974
0.666 1.94643 59844 27591 272 0.51375 95112 29998 365
0.667 1.94838 33939 54498 192 0.51324 60085 12918 798
0.668 1.95033 27518 64961 432 0.51273 30190 41890 516
0.669 1.95228 40601 08339 065 0.51222 05423 03924 002
0.670 1 =1542373206 35939 496 0.51170 85777 86542 478
0.671 1:(:,5619 25354 01023 417 0.51119 71249 77781 383
0.672 1.05814 97063 58805 754 0.51068 61833 66187 865
0.673 1.')6010 88354 66457 630 0.51017 57524 40820 271
0.674 1.96206 99246 83108 314 0.50966 58316 91247 632
0.675 l/36403 29759 69847 187 0.50915 64206 07549 157
0.676 ii 96599 79912 89725 700 0.50864 75186 80313 718
0.677 1.96796 49726 07759 335 0.50813 91254 00639 348
0.678 1.96993 39218 90929 575 0.50763 12402 60132 723
0.679 1.97190 48411 08185 868 0.50712 38627 50908 661
0.680 1.97387 77322 30447 594 0.50661 69923 65589 610
0.681 1.97585 25972 30606 040 0.50611 06285 97305 142
0.682 1.97782 94380 83526 371 0.50560 47709 39691 448
0.683 1.97980 82567 66049 605 0.50509 94188 86890 827
0.684 1.98178 90552 56994 589 0.50459 45719 33551 185
0.685 1.98377 18355 37159 979 0.50409 02295 74825 526
0.686 1: 98575 65995 89326 220 0.50358 63913 06371 449
0.687 1,98774 33493 98?57 531 0.50308 30566 24350 644
0.688 1,98973 20869 50703 885 0.50258 "2250 25428 387
0.689 1.99172 28142 35403 001 0.50207 .3960 06773 037
0.690 1.99371 55332 43082 329 0.50157 60690 66055 534
0.691 1.99571 02459 66461 043 0.50107 47437 01448 895
0.692 1.99770 69544 00252 033 0.50057 39194 11627 713
0.693 1.99970 56605 41163 899 0.50007 35956 95767 658
0.694 2.00170 63663 87902 948 0.49957 37720 53544 971
0.695 Z!.OO37090739 41175 193 0.49907 44479 85i.35 969
0.696 :!.00571 37852 03688 356 0.49857 56229 91216 541
0.697 :!.00772 05021 80153 865 0.49807 72965 72961 653
0.698 2.00972 92268 77288 865 0.49757 94682 32044 844
0.699 2.01173 99613 03818 219 0.49708 21374 70637 732
0.700 2.01375 27074 70476 522 0.49658 53037 91409 515
!Lc-y 1 [I 1
(-;I6
ELEMENTARY TRANSCENDENTAL FVNCTIONS
X ez e-z
X ez e-z
0.750 2.11700 00166 12674 669 0.47236 65527 41014 707
0.751 2.11911 80754 a2217 212 0.47189 44222 92841 982
0.752 2.12123 a2534 70011 a30 0.47142 27637 39130 a75
0.753 2.12336 05526 96236 688 0.47095 15766 08222 791
0.754 2.12548 49752 a3191 190 0.47048 08604 28930 562
0.755 2.12761 15233 55298 098 0.47001 06147 30537 969
0.756 2.12974 0.1990 39105 663 0.46954 08390 42799 274
0.757 2.13187 10044 63289 745 0.46907 15328 95938 749
0.758 2.13400 39417 58655 946 0.46860 26958 20650 211
0.759 2.13613 90130 58141 739 0.46813 43273 48096 543
0.760 2.13827 62204 96818 602 0.46766 64270 09909 234
0.761 2.14041 55662 11894 152 0.46719 89943 38187 907
0.762 2.14255 70523 42714 282 0.46673 20288 65499 852
0.763 2.14470 06810 30765 301 0.46626 55301 24879 557
0.764 2.14684 64544 19676 075 0.46579 94976 49828 242
0.765 2.14899 43746 55220 173 0.46533 39309 74313 393
0.766 2.15114 44438 a5318 010 0.46486 88296 32768 297
0.767 2.15329 66642 60038 993 0.46440 41931 60091 573
0.768 2.15545 10379 31603 678 0.46394 00210 91646 708
0.769 2.15760 75670 54385 916 0.46347 63129 63261 598
0.770 2.15976 62537 a4915 008 0.46301 30683 11228 073
0.771 2.16192 71002 ala77 a66 0.46255 02866 72301 444
0.772 2.16409 01087 06121 167 0.46208 79675 83700 034
0.773 2.16625 52812 20653 514 0.46162 61105 83104 714
0.774 2.16842 26199 90647 604 0.46116 47152 08658 446
0.775 2.17059 21271 a3442 386 0.46070 37809 98965 ala
0.77 2.17276 38049 68545 234 0.46024 33074 93092 580
0.77 t 2.17493 76555 17634 114 0.45978 32942 30565 la9
0.778 2.17711 36810 04559 757 0.45932 37407 51370 344
0.779 2.17929 18836 05347 a30 0.45886 46465 95954 527
0.780 2.18147 22654 98201 117 0.45840 60113 05223 545
0.781 2.18365 48288 63501 691 Oi45794 78344 20542 069
0.782 2.18583 95758 a3813 099 0.45749 01154 a3733 175
0.783 2.18802 65087 43882 545 0.45703 28540 37077 a90
0.784 2.19021 56296 30643 070 0.45657 60496 23314 727
0.785 2.19240 69407 33215 744 0.45611 97017 85639 236
0.786 2.19460 04442 42911 a52 0.45566 38100 67703 540
0.787 2.19679 61423 53235 086 0.45520 83740 13615 885
0.788 2.19899 40372 59883 740 0.45475 33931 67940 176
0.789 2.20119 41311 60752 903 0.45429 88670 75695 532
0.790 2.20339 64262 55936 659 0.45384 47952 82355 822
0.791 2.20560 09247 47730 288 0.45339 11773 33849 215
0.792 2.20780 76288 40632 465 0.45293 80127 76557 724
0.793 2.21001 65407 41347 466 0.45248 53011 57316 754
0.794 2.21222 76626 58787 377 0.45203 30420 23414 649
0.795 2.21444 09968 04074 299 0.45158 12349 22592 237
0.796 2.21665 65453 90542 561 0.45112 98794 03042 379
0.797 2.21887 43106 33740 936 0.45067 89750 13409 518
0.798 2.22109 42947 51434 850 0.45022 a5213 02789 227
0.799 2.22331 64999 63608 607 0.44977 a5178 20727 758
0.800 2.22554 09284 92467 605 0.44932 a9641 17221 591
C-l)3
[ 1 [1 1
(96
132 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X @ ecz
[ C-i)3
1 [ 1
(536
ELEMENTARY TRANSCENDENTAL FUNCTIONS 133
EXPONENTIAL FUNCTION Table 4.4
X ez e-2
[ 1
C-l)3
[
(-;I5
1
ELEMENTARY TRANSCENDENTAL FUNCTIONS
X eI e+
[ 1
C-l)3
[C-l)5 1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 135
EXPONENTIAL FUNCTION Table 4.4
x er e-=
[ C-l)3
1 [ (-!I5
1
ELEMENTARY TRANSCENDENTAL FUrVCTIONS
2 e2 e-l
ez e-2
e* e-=
Ii
1.48413 15910 25766 034 - f- 3j6.73794 69990 85467 097
4.03428 79349 27351 226,--
1.09663 31584 28458 599
2.98095 79870 41728 275 - 3
4 2.47875
3.35462 21766
9.11881
1.23409 62790 66358
80408
96555 25118
45162
66795 423
388
495
080
3)8.10308 39275 75384 008 --
4)2.20264 65794 80671 652- 4.53999 29762 48485 154
:: 5.98741 41715 19781 846 -- 1.67017 00790 24565 931
12 1.62754 79141 90039 208 6.14421 23533 28209 759
13 4.42413 39200 89205 033- 2.26032 94069 81054 326
14 1.20260 42841 64776 778--c 8.31528 71910 35678 841
( 6)3.26901 73724 72110 639 - !- 713.05902 32050 18257 884
Ii
6)8.88611 05205 07872 637
7)2.41549 52.75357529 821
6.56599 69137 33051 114~
1.78482 30036 31872 608 - 7
9 1.12535
8 1.52299 17471
5.60279
4.13993 64375 85166
79744
77187 37267 844
71262
92591 540
660
145
4.85165 19540 97902 780- 2.06115 36224 38557 828
1.31881 57344 83214 697- 7.58256 04279 11906 728
3.58491 28461 31591 562-p 2.78946 80928 68924 808
9.74480 34462 48902 600- 1.02618 79631 70189 030
2.64891 22129 84347 229 3.77513 45442 79097 752
25 7.20048 99337 38587 252--. -11 1.38879 43864 96402 059
26 1.95729 60942 88387 643 -12 5.10908 90280 63324 720
27 5.32048 24060 17986 167 -12 1.87952 88165 39083 295
28 1.44625 70642 91475 174- -13 6.91440 01069 40203 009
29 3.93133 42971 44042 074- I -13 I 2.54366 56473 76922 910
30 1.06864 74581 52446 215, -14 9.35762 29688 40174 605
2.90488 49665 24742 5232 -14 3.44247 71084 69976 458
:: 7.89629 60182 68069 516 -14 1.26641 65549 09417 572
2.14643 57978 59160 646~ -15 4.65888 61451 03397 364
;34 5.83461 74252 74548 814 I -15 I 1.71390 84315 42012 966
(15)1.58601 34523 13430 728 6.30511 67601 46989 386
4.31123 15471 15195 227, 2.31952 28302 43569 388
1.17191 42372 80261 1311 8.53304 76257 44065 794
3.18559 31757 11375 622~ 3.13913 27920 48029 629
(16)8.65934 00423 99374 695 1.15482 24173 01578 599
40 (17)2.35385 26683 70199 854 4.24835 42552 91588 995
6.39843 49353 00549 492 1.56288 21893 34988 768
2 1.73927 49415 20501 047 5.74952 22642 93559 807
43 4.72783 94682 29346 561 2.11513 10375 91080 487
44 (19)1.28516 00114 35930 828 7.78113 22411 33796 516
45 2.86251 85805 49393 644
46 1.05306 17357 55381 238
47 I 19
20I 9.49611
3.49342 28861
2.58131
7.01673 59120 48509
94206
71057 97631
90067
02448 535
396
739
875 3.87399 76286 87187 113
48 1.42516 40827 40935 106
49 (21)1.90734 65724 95099 691 5.24288 56633 63463 937
50 (21)5.18470 55285 87072 464 (-22)1.92874 98479 63917 783
ELEMENTARY TRANSCENDENTAL FUNCTIONS 139
EXPONENTIAL FUNCTION Table 4.4
ez ec2
@O- fl e-2210-n
1.00000 00001 00000 00000 50000 0.99999 99999 00000 00000 50000
1.00000 00002 00000 00002 00000 0.99999 99998 00000 00002 00000
1I00000 00003 00000 00004 50000 0.99999 99997 00000 00004 50000
1.00000 00004 00000 00008 00000 0.99999 99996 00000 00008 00000
1.00000 00005 00000 00012 50000 0.99999 99995 00000 00012 50000
1.00000 00006 00000 00018 00000 0.99999 99994 00000 00018 00000
1.00000 00007 00000 00024 50000 0.99999 99993 00000 00024 50000
1.00000 00008 00000 00032 00000 0.99999 99992 00000 00032 00000
1.00000 00009 00000 00040 50000 0.99999 99991 00000 00040 50000
1.00000 00010 00000 00050 00000 0.99999 99990 00000 00050 00000
1.00000 00020 00000 00200 00000 0.99999 99980 00000 00200 00000
1.00000 00030 00000 00450 00000 0.99999 99970 00000 00450 00000
1.00000 00040 00000 00800 00000 0.99999 99960 00000 00800 00000
1.00000 00050 00000 01250 00000 0.99999 99950 00000 01250 00000
1.00000 00060 00000 01800 00000 0.99999 99940 00000 01800 00000
1.00000 00070 00000 02450 00001 0.99999 99930 00000 02449 99999
1.00000 00080 00000 03200 00001 0.99999 99920 00000 03199 99999
1.00000 00090 00000 04050 00001 0.99999 99910 00000 04049 99999
1.00000 00100 00000 05000 00002 0.99999 99900 00000 04999 99998
1.00000 00200 00000 20000 00013 0.99999 99800 00000 19999 99987
1.00000 00300 00000 45000 00045 0.99999 99700 00000 44999 99955
1.00000 00400 00000 80000 00107 0.99999 99600 00000 79999 99893
1.00000 00500 00001 25000 00208 0.99999 99500 00001 24999 99792
1.00000 00600 00001 80000 00360 0.99999 99400 00001 79999 99640
1.00000 00700 00002 45000 00572 0.99999 99300 00002 44999 99428
1.00000 00800 00003 20000 00853 0.99999 99200 00003 19999 99147
1.00000 00900 00004 05000 -01215 0.99999 99100 00004 04999 98785
1.00000 01000 00005 00000 01667 0.99999 99000 00004 99999 98333
1.00000 02000 00020 00000 13333 0.99999 98000 00019 99999 86667
1.00000 03000 00045 00000 45000 0.99999 97000 00044 99999 55000
1.00000 04000 00080 00001 06667 0.99999 96000 00079 99998 93333
1.00000 05000 00125 00002 08333 0.99999 95000 00124 99997 91667
1.00000 06000 00180 00003 60000 0.99999 94000 00179 99996 40000
1.00000 07000 00245 00005 71667 0.99999 93000 00244 99994 28333
1.00000 08000 00320 00008 53334 0.99999 92000 00319 99991 46667
1.00000 09000 00405 00012 15000 0.94999 91000 00404 99987 85000
1.00000 10000 00500 00016 66667 0.99999 90000 00499 99983 33334
1.00000 20000 02000 00133 33340 0.99999 80000 01999 99866 66673
1.00000 30000 04500 00450 00034 0.99999 70000 04499 99550 00034
1.00000 40000 08000 01066 66773 0.99999 60000 07999 98933 33440
1.00000 50000 12500 02083 33594 0.99999 50000 12499 97916 66927
1.00000 60000 18000 03600 00540 0.99999 40000 17999 96400 00540
1.00000 70000 24500 05716 67667 0.99999 30000 24499 94283 34334
1.00000 80000 32000 08533 35040 0.99999 20000 31999 91466 68373
1.00000 90000 40500 12150 02734 0.99999 10000 40499 87850 02734
Compiled from C. E. Van Orstrand, Tables of the exponential function and of the circular sine
and cosine to radian arguments, Memoirs of the National Academy of Sciences,vol. 14, Fifth
Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission),
ELEMENTARY TRANSCENDENTAL FTJNCTIONS 141
RADIX TABLE OF THE EXPONENTIAL FUNCTION Table 4.5
1.00001 00000 50000 16666 70833 0.99999 00000 49999 83333 37500
1.00002 00002 00001 33334 00000 0.99998 00001 99998 66667 33333
1.00003 00004 50004 50003 37502 0.99997 00004 49995 50003 37498
1.00004 00008 00010 66677 33342 0.99996 00007 99989 33343 99991
1.00005 00012 50020 83359 37526 0.99995 00012 49979 16692 70807
1.00006 00018 00036 00054 00065 0.99994 00017 99964 00053 99935
1.00007 00024 50057 16766 70973 0.99993 00024 49942 83433 37360
1.00008 00032 00085 33504 00273 0.99992 00031 99914 66837 33060
1.00009 00040 50121 50273 37992 0.99991 00040 49878 50273 37008
1.00010 00050 001% 67083 34167 0.99990 00049 99833 33749 99167
1.00020 00200 01333 40000 26668 0.99986 30199 98666 73333 06668
1.00030 00450 04500 33752 02510 0.99970 00449 95500 33747 97512
1.00040 00800 10667 73341 86724 0.99960 00799 89334 39991 46724
1.00050 01250 20835 93776 04384 0.99950 01249 79169 27057 29384
1.00060 01800 36005 40064 80648 0.99940 01799 64005 39935 20648
1.00070 02450 57176 67223 40801 0.99930 02449 42843 33609 95801
1.00080 03200 85350 40273 10308 0.99920 03199 14683 73060 30307
1.00090 04051 21527 34242 14882 0.99910 04048 78527 33257 99880
13 1.00100 05001 66708 34166 80558 0.99900 04998 33374 99166 80554
1.00200 20013 34000 26675 55810 0.99800 19986 67333 06675 55302
1.00300 45045 03377 02601 29341 0.99700 44955 03372 97601 20662
1.00400 80106 77341 87235 88080 0.99600 79893 43991 47235 23064
1.00501 25208 59401 06338 35662 0.99501 24791 92682 31335 25642
1.0060.1 80360 54064 86485 55845 0.99401 79640 53935 26474 44988
1.00702 45572 66848 55523 16000 0.99302 44429 33235 10490 47970
1.00803 20855 04273 43117 20736 0.99203 19148 37060 63033 98697
1.00904 06217 73867 81406 25705 0.99104 03787 72883 66216 45648
1.01005 01670 84168 05754 21655 0.99004 98337 49168 05357 39060
1.02020 13400 26755 81016 01439 0.98019 86733 06755 30222 08141
1.03045 45339 53516 85561 24400 0.97044 55335 48508 17693 25284
1.04081 07741 92388 22675 70448 0.96078 94391 52323 20943 92107
1.05127 10963 76024 03969 75176 0.95122 94245 00714 00909 14253
1.06183 65465 45359 62222 46849 0.94176 45335 84248 70953 71528
1.07250 81812 54216 47905 31039 0.93239 38199 05948 22885 79726
1.08328 70676 74958 55443 59878 0.92311 63463 86635 78291 07598
1.09417 42837 05210 35787 28976 0.91393 11852 7122‘8 18674 73535
1.10517 09180 75647 62481 17078 0.90483 74180 35959 57316 42491
1.22140 27581 60169 83392 10720 0.81873 07530 77981 85866 99355
1.34985 88075 76003 10398 37443 0.74081 82206 81717 86606 68738
1.49182 46976 41270 31782 48530 0.67032 00460 35639 30074 44329
54 ; 1.64872
1.82211
12707
88003
00128
90508
14684
97487
86508
53677
0.60653
0.54881
06597
16360
12633
94026
42360
43262
37995
84589
6 ; 2.01375 27074 70476 52162 45494 0.49658 53037 91409 51470 48001
8' 1 2.22554 09284 92467 60457 95375 0.44932 89641 17221 59143 01024
9 1 2.45960 31111 56949 66380 01266 0.40656 96597 40599 11188 34542
1 0 2.71828 18284 59045 23536 02875 0.36787 94411 71442 32159 55238
142 ELEMENTARY TRANSCENDENTAL FUNCTIONS
I sin z cos x
0.000 0.00000 00000 00000 00000 000 1.00000 00000 00000 00000 000
0.001 0.00099 99998 33333 34166 667 0.99999 95000 00041 66666 528
0.002 0.00199 99986 66666 93333 331 0.99999 80000 00666 66657 778
0.003 0.00299 99955 00002 02499 957 0.99999 55000 03374 99898 750
0.004 0.00399 99893 33341 86666 342 0.99999 20000 10666 66097 778
0.005 0.00499 99791 66692 70831 783 0.99998 75000 26041 64496 529
0.006 0.00599 99640 00064 79994 446 0.99998 20000 53999 93520 004
0.007 0.00699 99428 33473 39150 327 0.99997 55001 00041 50326 542
0.008 0.00799 99146 66939 73291 723 0.99996 80001 70666 30257 El’9
0.009 0.00899 98785 00492 07405 100 0.99995 95002 73374 26188 857
0.010 0.00999 98333 34166 66468 254 0.99995 00004 16665 27778 026
0.011 0.01099 97781 68008 75446 684 0.99993 95006 10039 20617 059
0.01199 97120 02073 59289 053 0.99992 80008 63995 85281 066
00. “0:: 0.01299 96338 36427 42921 659 0.99991 55011 90034 96278 551
0: 014 0.01399 95426 71148 51241 801 0.99990 20016 00656 20901 438
0.015 0.01499 94375 06328 09109 944 0.99988 75021 09359 17975 106
0.016 0.01599 93173 42071 41340 585 0.99987 20027 30643 36508 430
0.017 0.01699 91811 78498 72691 726 0.99985 55034 80008 14243 829
0.018 0.01799 90280 15746 27852 832 0.99983 80043 73952 76107 331
0.019 0.01899 88568 53967 31431 205 0.99981 95054 29976 32558 650
0.020 0.01999 86666 93333 07936 649 0.99980 00066 66577 77841 270
0.021 0.02099 84565 34033 81764 335 0.99977 95081 03255 88132 556
0.022 0.02199 82253 76279 77175 771 0.99975 80097 60509 19593 878
0.023 0.02299 79722 20302 18277 769 0.99973 55116 59836 06320 750
0.024 0.02399 76960 66354 28999 311 0.99971 20138 23734 58193 002
0.025 0.02499 73959 14712 33066 217 0.99968 75162 75702 58624 967
0.026 0.02599 70707 65676 53973 517 0.99966 20190 40237 62215 698
0.027 0.02699 67196 19572 14955 411 0.99963 55221 42836 92299 214
0.028 0.02799 63414 76750 38952 746 0.99960 80256 09997 38394 779
0.029 0.02899 59353 37589 48577 881 0.99957 95294 69215 53557 207
0.030 0.02999 55002 02495 66076 853 0.99955 00337 48987 51627 216
0.031 0.03099 50350 71904 13288 752 0.99951 95384 78809 04381 810
0.032 0.03199 45389 46280 11602 188 0.99948 80436 89175 38584 710
0.033 0.03299 40108 26119 81908 762 0.99945 55494 11581 32936 824
0.034 0.03399 34497 11951 44553 435 0.99942 20556 78521 14926 773
0.035 0.03499 28546 04336 19281 702 0.99938 75625 23488 57581 460
0.036 0.03599 22245 03869 25183 461 0.99935 20699 80976 76116 700
0.037 0.03699 15584 11180 80633 489 0.99931 55780 86478 24487 902
0.038 0.03799 08553 26937 03228 414 0.99927 80868 76484 91840 819
0.039 0.03899 01142 51841 09720 085 0.99923 95963 88487 98862 358
0.040 0.03998 93341 86634 15945 255 0.99920 01066 60977 94031 457
0.041 0.04098 85141 32096 36751 449 0.99915 96177 33444 49770 040
0.042 0.04198 76530 89047 85918 946 0.99911 81296 46376 58494 043
0.043 0.04298 67500 58349 76078 755 0.99907 56424 41262 28564 524
0.044 0.04398 58040 40905 18626 492 0.99903 21561 60588 80138 853
0.045 0.04498 48140 37660 23632 066 0.99898 76708 47842 40921 992
0.046 0.04598 37790 49604 99745 054 0.99894 21865 47508 41817 869
0.047 0.04698 26980 77774 54095 689 0.99889 57033 05071 12480 849
0.048 0.04798 15701 23249 92191 340 ii99884 82211 67013 76767 299
0.049 0.04898 03941 87159 17808 403 0.99879 97401 80818 48087 272
0.050 0.04997 91692 70678 32879 487 0.99875 02603 94966 24656 287
C-79)6
For conversion from degrees to radians see Example 13.
For use and extension of the table see Examples 15-17.
From C. E. Van Orstrand, Tables of the exponential functionand of thecir-
cular sine and cosine to radian arguments,Memoirs of the National Academy of
Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington,
D.C., 1921 (with permission). Known errors have been corrected.
ELEMENTARY TRANSCENDENTAL FUNCTIONS 143
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
2 sin 2 co9 5
0.050 0.04997 91692 70678 32879 487 0.99875 02603 94966 24656 287
0.051 0.05097 78943 75032 37375 800 0.99869 97818 58936 84647 237
0.052 0.05197 65685 01496 29184 649 0.99864 83046 23208 81242 407
0.053 0.05297 51906 51396 03981 925 0.99859 58287 39259 37585 623
0.054 0.05397 37598 26109 55099 505 0.99854 23542 59564 41634 531
0.055 0.05497 22750 27067 73387 446 0.99848 78812 37598 40913 005
0.056 0.05597 07352 55755 47070 891 0.99843 24097 27834 37163 704
0.057 0.05696 91395 13712 61601 567 0.99837 59397 85743 80900 770
0.058 0.05796 74868 02534 99503 794 0.99831 84714 67796 65862 676
0.059 0.05896 57761 23875 40214 896 0.99826 00048 31461 23365 235
0.060 0.05996 40064 79444 59919 909 0.99820 05399 35204 16554 766
0.061 0.06096 21768 71012 31380 500 0.99814 00768 38490 34561 437
0.062 0.06196 02863 00408 23757 982 0.99807 86156 01782 86552 769
0.063 0.06295 83337 69523 02430 343 0.99801 61562 86542 95687 334
0.064 0.06395 63182 80309 28803 166 0.99795 26989 55229 92968 628
0.065 0.06495 42388 34782 60114 361 0.99788 82436 71301 10999 144
0.066 0.06595 20944 35022 49232 601 0.99782 27904 99211 77634 635
0.067 0.06694 98840 83173 44449 361 0.99775 63395 04415 09538 592
0.068 0.06794 76067 81445 89264 458 0.99768 88907 53362 05636 926
0.069 0.06894 52615 32117 22165 004 0.99762 04443 13501 40472 866
0.070 0.06994 28473 37532 76397 655 0.99755 10002 53279 57462 091
0.071 0.07094 03632 00106 79734 071 0.99748 05586 42140 62048 084
0.072 0.07193 78081 22323 54229 480 0.99740 91195 50526 14757 726
0.073 0.07293 51811 06738 15974 250 0.99733 66830 49875 24157 139
0.074 0.07393 24811 55977 74838 360 0.99726 32492 12624 39707 777
0.075 0.07492 97072 72742 34208 684 0.99718 88181 12207 44522 774
0.076 0.07592 68584 59805 90718 980 0.99711 33898 23055 48023 568
0.077 0.07692 39337 20017 33972 485 0.99703 69644 20596 78496 785
0.078 0.07792 09320 56301 46257 015 0.99695 95419 81256 75551 417
0.079 0.07891 78524 71660 02252 478 0.99688 11225 82457 82476 279
0.080 0.07991 46939 69172 68730 688 0.99680 17063 02619 38497 771
0.081 0.08091 14555 51998 04247 389 0.99672 12932 21157 70937 933
0.082 0.08190 81362 23374 58826 394 0.99663 98834 18485 87272 823
0.083 0.08290 47349 86621 73635 718 0.99655 74769 76013 67091 212
0.084 0.08390 12508 45140 80655 638 0.99647 40739 76147 53953 598
0.085 0.08489 76828 02416 02338 544 0.99638 96745 02290 47151 570
0.086 0.08589 40298 62015 51260 514 0.99630 42786 38841 93367 506
0.087 0.08689 02910 27592 29764 492 0.99621 78864 71197 78234 626
0.088 0.08788 64653 02885 29594 973 0.99613 04980 85750 17797 412
0.089 0.08888 25516 91720 31524 112 0.99604 21135 69887 49872 388
0.090 0.08987 85491 98011 04969 125 0.99595 27330 11994 25309 284
0.091 0.09087 44568 25760 07600 919 0.99586 23565 01450 99152 586
0.092 0.09187 02735 79059 84943 819 0.99577 09841 28634 21703 483
0.093 0.09286 59984 62093 69966 323 0.99567 86159 84916 29482 217
0.094 0.09386 16304 79136 82662 751 0.99558 52521 62665 36090 844
0.095 0.09485 71686 34557 29625 724 0.99549 08927 55245 22976 426
0.096 0.09585 26119 32817 03609 347 0.99539 55378 57015 30094 649
0.097 0.09684 79593 78472 83083 006 0.99529 91875 63330 46473 881
0.098 0.09784 32099 76177 31775 683 0.99520 18419 70541 00679 686
0.099 0.09883 83627 30679 98210 683 0.99510 35011 75992 51179 796
0.100 0.09983 34166 46828 15230 681 0.99500 41652 78025 76609 556
144 ELEMENTARY TRANSCENDENTAL FUNCTIONS
2 sin x cos x
0.200 0.19866 93307 95061 21545 941 0.98006 65778 41241 63112 420
0.201 0.19964 92978 74900 91597 545 0.97986 74185 10310 03887 090
0.202 OI20062 90653 05459 37903 151 0.97966 72793 12041 59192 306
0.203 0.20160 86321 06969 25571 640 0.97946 61604 46575 47187 084
0.204 0.20258 79972 99863 82615 083 OI97926 40621 15030 52742 047
0.205 0.20356 71599 04777 97905 397 0.97906 09845 19505 07327 536
0.206 0.20454 61189 42549 19110 856 0.97885 69278 63076 68803 784
0.207 0.20552 48734 34218 50612 330 0.97865 18923 49802 01113 156
0.208 0.20650 34224 01031 51399 175 0.97844 58781 84716 53874 491
0.209 0.20748 17648 64439 32944 665 0.97823 88855 73834 41879 553
0.210 0.20845 98998 46099 57060 871 0.97803 09147 24148 24491 614
0.211 0.20943 78263 67877 33732 895 0.97782 19658 43628 84946 201
0.212 0.21041 55434 51846 18932 346 0.97761 20391 41225 09554 014
0.213 0.21139 30501 20289 12409 982 0.97740 11348 26863 66806 039
0.214 0.21237 03453 95699 55467 398 0.97718 92531 11448 86380 882
0.215 0.21334 74283 00782 28707 677 0.97697 63942 06862 38054 344
0.216 0.21432 42978 58454 49764 905 0.97676 25583 25963 10511 247
0.217 0.21530 09530 91846 71012 439 0.97654 77456 82586 90059 555
0.218 0.21627 73930 24303 77249 851 0.97633 19564 91546 39246 782
0.219 0.21725 36166 79385 83368 434 0.97611 51909 68630 75378 736
0.220 0.21822 96230 80869 31995 179 0.97589 74493 30605 48940 602
0.221 0.21920 54112 52747 91115 124 0.97567 87317 95212 21920 392
0.222 0.22018 09802 19233 51671 977 0.97545 90385 81168 46034 788
0.223 0.22115 63290 04757 25146 920 0.97523 83699 08167 40857 388
0.224 0.22213 i4566 33970 41115 484 0.97501 67259 96877 71849 392
0.225 0.22310 63621 31745 44782 417 0.97479 41070 68943 28292 737
0.226 0.22408 10445 23176 94494 428 0.97457 05133 46983 01125 708
0.227 0.22505 55028 33582 59230 720 0.97434 59450 54590 60681 052
0.228 0.22602 97360 88504 16071 214 0.97412 04024 16334 34326 607
0.229 0.22700 37433 13708 47642 363 0.97389 38856 57756 84008 477
0.230 0.22797 75235 35188 39540 462 0.97366 63950 0.537483696 773
0.231 0.22895 10757 79163 77732 354 0.97343 79306 86678 96733 940
0.232 0.22992 43990 72082 45933 437 0.97320 84929 30133 53085 695
0.233 0.23089 74924 40621 22962 869 0.97297 80819 65176 26494 602
0.234 0.23187 03549 11686 80075 884 0.97274 66980 22218 11536 294
0.235 0.23284 29855 12416 78273 112 0.97251 43413 32643 00578 389
0.236 0.23381 53832 70180 65586 809 0.97228 10121 28807 60642 091
0.237 0.23478 75472 12580 74343 904 0.97204 67106 44041 10166 529
0.238 0.23575 94763 67453 18405 752 0.97181 14371 12644 95675 843
0.239 0.23673 11697 62868 90384 520 0.97157 51917 69892 68349 034
0.240 0.23770 26264 27134 58836 079 0.97133 79748 52029 60492 618
0.241 0.23867 38453 88793 65429 334 0.97109 97865 96272 61916 095
0.242 0.23964 48256 76627 22091 869 0.97086 06272 40809 96210 262
0.243 0.24061 55663 19655 08131 828 0.97062 04970 24800 96928 391
0.244 0.24158 60663 47136 67335 933 0.97037 93961 88375 83670 294
0.245 0.24255 63247 88572 05043 522 0.97013 73249 72635 38069 313
0.246 0.24352 63406 73702 85196 546 0.96989 42836 19650 79682 233
0.247 0.24449 61130 32513 27365 389 0.96965 02723 72463 41782 166
0.248 0.24546 56408 95231 03750 445 0.96940 52914 75084 47054 425
0.249 0.24643 49232 92328 36159 337 0.96915 93411 72494 83195 397
0.250 0.24740 39592 54522 92959 685 0.96891 24217 10644 78414 459
[ 1
c-:)3
1c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 147
CIRCULAR SINES .4ND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin x cos x
0.250 0.24740 39592 54522 92959 685 0.96891 24217 10644 78414 459
0.251 0.24837 27478 12778 86007 332 0.96866 45333 36453 76838 955
0.252 0.24934 12879 98307 67549 922 0.96841 56762 97810 13822 250
0.253 0.25030 95788 42569 27105 742 0.96816 58508 43570 91154 897
0.254 0.25127 76193 77272 88317 722 0.96791 50572 23561 52178 941
0.255 0.25224 54086 34378 05782 506 0.96766 32956 88575 56805 375
0.256 0.25321 29456 46095 61854 486 0.96741 05664 90374 56434 780
0.257 0.25418 02294 44888 63424 714 0.96715 68698 81687 68781 180
0.258 0.25514 72590 63473 38674 587 0.96690 22061 16211 52599 126
0.259 0.25611 40335 34820 33804 209 0.96664 65754 48609 82314 035
0.260 0.25708 05518 92155 09735 339 0.96638 99781 34513 22555 822
0.261 0.25804 68131 68959 38788 820 0.96613 24144 30519 02595 835
0.262 0.25901 28163 98972 01336 401 0.96587 38845 94190 90687 131
0.263 0.25997 85606 16189 82426 844 0.96561 43888 84058 68308 107
0.264 0.26094 40448 54868 68386 239 0.96535 39275 59618 04309 520
0.265 0.26190 92681 49524 43392 399 0.96509 25008 81330 28964 923
0.266 0.26287 42295 34933 86023 278 0.96483 01091 10622 07924 537
0.267 0.26383 89280 46135 65779 278 0.96456 67525 09885 16072 584
0.268 0.26480 33627 18431 39579 372 0.96430 24313 42476 11288 118
0.269 0.26576 75325 87386 48230 942 0.96403 71458 72716 08109 368
0.270 0.26673 14366 88831 12873 229 0.96377 08963 65890 51301 623
0.271 0.26769 50740 58861 31394 301 0.96350 36830 88248 89328 696
0.272 0.26865 84437 33839 74821 451 0.96323 55063 07004 47727 972
0.273 0.26962 15447 50396 83684 915 0.96296 63662 90334 02389 084
0.274 0.27058 43761 45431 64354 828 0.96269 62633 07377 52736 246
0.275 0.27154 69369 56112 85351 302 0.96242 51976 28237 94814 248
0.276 0.27250 92262 19879 73627 557 0.96215 31695 23980 94278 169
0.277 0.27347 12429 74443 10825 981 0.96188 01792 66634 59286 807
0.278 0.27443 29862 57786 29507 043 0.96160 62271 29189 13299 879
0.279 0.27539 44551 08166 09350 952 0.96133 13133 85596 67778 997
0.280 0.27635 56485 64113 73331 967 0.96105 54383 10770 94792 459
0.281 0.27731 65656 64435 83865 270 0.96077 86021 80586 99523 878
0.282 0.27827 72054 48215 38926 293 0.96050 08052 71880 92684 682
0.283 0.27923 75669 54812 68142 411 0.96022 20478 62449 62830 504
0.284 0.28019 76492 23866 28856 909 Oi95994 23302 31050 48581 495
0.285 0.28115 74512 95294 02165 110 0.95966 16526 57401 10746 590
0.286 0.28211 69722 09293 88922 591 0.95938 00154 22179 04351 746
0.287 0.28307 62110 06345 05725 374 0.95909 74188 07021 50572 193
0.288 0.28403 51667 27208 80861 997 0.95881 38630 94525 08568 713
0.289 0.28499 38384 12929 50237 384 0.95852 93485 68245 47227 984
0.290 0.28595 22251 04835 53268 394 0.95824 38755 12697 16807 013
0.291 0.28691 03258 44540 28750 981 0.95795 74442 13353 20481 688
0.292 0.28786 81396 73943 10698 841 0.95767 00549 56644 85799 478
0.293 0.28882 56656 35230 24153 475 0.95738 17080 29961 36036 308
0.294 0.28978 29027 70875 80965 551 0.95709 24037 21649 61457 636
0.295 0.29073 98501 23642 75547 489 0.95680 21423 21013 90483 768
0.296 Oi29169 65067 36583 80597 155 Oi95651 09241 18315 60759 429
0.297 0.29265 28716 53042 42792 582 0.95621 87494 04772 90127 632
0.298 Oi29360 89439 16653 78457 616 Oi95592 56184 72560 47507 858
0.299 0.29456 47225 71345 69198 389 0.95563 15316 14809 23678 590
0.300 0.29552 02066 61339 57510 532 0.95533 64891 25606 01964 231
[C-f)4 1 [ (-;)I1
148 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos x
0.300 0.29552 02066 61339 57510 532 0.95533 64891 25606 01964 231
0.301 0.29647 53952 31151 42357 025 0.95504 04912 99993 28826 414
0.302 0.29743 02873 25592 74716 586 0.95474 35384 33968 84359 763
0.303 0.29838 48819 89771 53102 518 0.95444 56308 24485 52692 116
0.304 0.29933 91782 69093 19051 897 0.95414 67687 69450 92289 242
0.305 0.30029 31752 09261 52585 026 0.95384 69525 67727 06164 084
0.306 0.30124 68718 56279 67635 045 0.95354 61825 19130 11990 559
0.307 0.30220 02672 56451 07447 613 0.95324 44589 24430 12121 945
0.308 0.30315 33604 56380 39950 549 0.95294 17820 85350 63513 878
0.309 0.30410 61505 02974 53093 365 0.95263 81523 04568 47552 001
0.310 0.30505 86364 43443 50156 564 0.95233 35698 85713 39784 281
0.311 0.30601 08173 25301 45030 632 0.95202 80351 33367 79558 038
0.312 0.30696 26921 96367 57464 615 0.95172 15483 53066 39561 711
0.313 0.30791 42601 04767 08284 189 0.95141 41098 51295 95271 383
0.314 0.30886 55200 98932 14579 138 0.95110 57199 35494 94302 111
0.315 0.30981 64712 27602 84860 120 0.95079 63789 14053 25664 080
0.316 i:j1076 71125 39828 14184 658 0.95048 60870 96311 88923 617
0.317 0.31171 74430 84966 79252 234 0.95017 48447 92562 63269 094
0.318 0.31266 74619 12688 33468 402 0.94986 26523 14047 76481 749
0.319 0.31361 71680 72974 01977 833 0.94954 95099 72959 73811 467
0.320 0.31456 65606 16117 76666 176 0.94923 54180 82440 86757 531
0.321 0.31551 56385 92727 11130 659 0.94892 03769 56583 01754 395
0.322 0.31646 44010 53724 15619 332 0.94860 43869 10427 28762 501
0.323 0.31741 28470 50346 51938 844 0.94828 74482 59963 69764 173
0.324 0.31836 09756 34148 28330 674 0.94796 95613 22130 87164 613
0.325 0.31930 87858 57000 94315 718 0.94765 07264 14815 72098 048
0.326 0.32025 62767 71094 35507 128 0.94733 09438 56853 12639 034
0.327 0.32120 34474 28937 68391 319 0.94701 02139 68025 61918 976
0.328 0.32215 02968 83360 35077 048 0.94668 85370 69063 06147 877
0.329 0.32309 68241 87512 98012 460 0.94636 59134 81642 32541 351
0.330 0.32404 30283 94868 34670 020 0.94604 23435 28386 97152 941
0.331 0.32498 89085 59222 32199 224 0.94571 78275 32866 92611 768
0.332 0.32593 44637 34694 82047 011 0.94539 23658 19598 15765 535
0.333 0.32687 96929 75730 74545 756 0.94506 59587 14042 35228 939
0.334 Oij2782 45953 37100 93468 777 0.94473 86065 42606 58837 502
0.335 0.32876 91698 73903 10553 241 0.94441 03096 32643 01006 864
0.336 0.32971 34156 41562 79990 386 0.94408 10683 12448 49997 577
0.337 0.33065 73316 95834 32882 957 0.94375 08829 11264 35085 413
0.338 0.33160 09170 92801 71669 766 0.94341 97537 59275 93637 243
0.339 0.33254 41708 88879 64517 288 0.94308 76811 87612 38092 499
0.340 0.33348 70921 40814 39678 177 0.94275 46655 28346 22850 264
0.341 0.33442 96799 05684 79816 635 0.94242 07071 14493 11062 025
0.342 0.33537 19332 40903 16300 519 0.94208 58062 80011 41330 105
0.343 0.33631 38512 04216 23460 104 0.94174 99633 59801 94311 834
0.344 0.33725 54328 53706 12813 399 0.94141 31786 89707 59229 468
0.345 0.33819 66772 47791 27257 928 0.94107 54526 06513 00285 905
0.346 0.33913 75834 45227 35228 880 0.94073 67854 47944 22986 218
0.347 0.34007 81505 05108 24823 531 0.94039 71775 52668 40365 059
0.348 0.34101 83774 86866 97891 850 0.94005 66292 60293 39119 944
0.349 0.34195 82634 50276 64093 188 0.93971 51409 11367 45650 473
0.350 0.34289 78074 55451 34918 963 0.93937 27128 47378 92003 503
ELEMENTARY TRANSCENDENTAL FUNCTIONS 149
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
sin z cos x
0.350 0.34289 78074 55451 34918 963 0.93937 27128 47378 92003 503
0.351 0.34383 70085 62847 17681 237 0.93902 93454
10755 al724 321
0.352 0.34477 58658 33263 09467 102 0.93868 50389 44865 55613 a41
0.353 0.34571 43783 27841 91058 778 0.93833 97937 94014 57391 a69
0.354 0.34665 25451 08071 20819 319 0.93799 36103 03447 99266 461
0.355 0.34759 03652 35784 28543 852 0.93764 64888 19349 27409 412
0.356 0.34852 78377 73161 09276 237 0.93729 84296 88839 87337 915
0.357 0.34946 49617 82729 17091 064 0.93694 94332 59978 89202 418
0.358 0.35040 17363 58840 a91
27364 0.93659 94998 al762 72980 716
0.359 0.35133 al604 70292 87868 632 0.93624 86299 04124 73578 312
0.360 0.35227 42332 75089 97684 991 0.93589 68236 77934 85835 091
0.361 0.35320 99538 05683 15610 866 0.93554 40815 54999 29438 322
0.362 0.35414 53211 26351 96384 608 0.93519 04038 88060 13742 042
0.363 0.35508 03343 01729 15734 065 0.93483 57910 30795 02492 a55
0.364 0.35601 49923 96801 63913 294 0.93448 02433 37816 78462 165
0.365 0.35694 92944 76911 39203 863 0.93412 37611 64673 07984 897
0.366 0.35788 32396 41380 647
07756 0.93376 63448 67846 05404 739
0.367 0.35881 68268 55391 65142 021 0.93340 79948 04751 97425 922
0.368 0.35975 00552 86229 93504 354 0.93304 a7113 33740 a7371 606
0.369 0.36068 29239 67042 91160 721 0.93268 84948 14096 19348 a71
0.370 0.36161 54319 64961 97803 729 0.93232 73456 06034 42320 381
0.371 0.36254 75783 47479 21412 373 0.93196 52640 70704 74082 737
0.372 0.36347 93621 al3
82448 31502 0.93160 70188 65151 560
22505
0.373 0.36441 07825 38085 52343 006 0.93123 a3054 67499 62553 347
0.374 0.36534 la384 a2970 56131 067 0.93087 34291 26582 73524 125
0.375 0.36627 25290 86047 56137 291 0.93050 76219 12314 29114 948
0.376 0.36720 28534 16625 99809 733 0.93014 08841 90501 47704 265
0.377 0.36813 28105 44381 61843 251 0.92977 32163 27881 98417 211
0.378 0.36906 23995 39357 37211 926 0.92940 46186 92123 64451 a36
0.379 0.36999 16194 71964 34164 758 0.92903 50916 51824 06312 328
0.380 0.37092 04694 12982 67184 549 0.92866 46355 76510 24949 253
0.37184 89484 33562 49909 aal 0.92829 32508 36638 24806 a58
i* 33:: 0.37277 70556 05224 88020 096 0.92792 09378 03592 76777 471
0: 383 0.37370 47899 99862 72083 la4 0.92754 76968 49686 81063 030
0.384 0.37463 21506 a9741 70366 479 0.92717 35283 48161 29943 792
0.385 0.37555 91367 47501 21610 089 0.92679 84326 73184 70454 235
0.386 0.37648 57472 46155 27762 945 0.92642 24101 99852 66966 223
0.37741 19812 59093 46681 397 0.92604 04187 63679 438
54613
2 ;:i 0.37833 78378 60081 a4790 240 0.92566 75863 63138 47019 143
01389 0.37926 33161 23263 89706 110 oI9252a 87857 54580 07941 297
0.390 0.38018 a4151 23161 42823 118 0.92490 90598 57313 04145 068
0.391 0.38111 31339 34675 51860 671 0.92452 84090 51063 22192 776
0.392 0.38203 74716 33087 43373 349 0.92414 68337 16481 39537 314
0.393 0.38296 14272 94059 55222 774 0.92376 43342 35142 86457 070
0.394 0.38388 49999 93636 29011 366 0.92338 09109 a9547 07898 401
0.395 0.38480 ala88 08245 02477 888 0.92299 65643 63117 25225 693
0.396 0.38573 09928 14697 01854 707 0.92261 12947 40199 97879 040
0.38665 34110 90188 34186 658 0.92222 51025 06064 84939 589
it ;z; 0.38757 54427 12300 79611 426 0.92183 79880 46904 06602 584
0:399 0.38849 70867 59002 a3601 363 0.92144 99517 49832 05558 150
0.400 0.38941 a3423 08650 49166 631 0.92106 09940 02885 08279 a53
r(-_sbq
L '_I cC-f)’
1
150 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos x
0.400 0.38941 83423 08650 49166 631 0.92106 09940 02885 08279 853
0.401 0.39033 92084 39988 29019 595 0.92067 11151 95020 86221 075
0.402 0.39125 96842 32150 17700 358 0.92028 03157 16118 16919 248
0.403 0.39217 97687 64660 43663 363 0.91988 85959 56976 45007 979
0.404 0.39309 94611 17434 61324 955 0.91949 59563 09315 43137 110
0.405 0.39401 87603 70780 43071 820 0.91910 23971 65774 72800 745
0.406 0;j9493 76656 05398 71230 202 0.91870 79189 19913 45073 295
0.407 0.39585 61759 02384 29995 816 0.91831 25219 66209 81253 568
0.408 0.39677 42903 43226 97324 356 0.91791 62067 00060 73416 956
0.409 0.39769 20080 09812 36782 508 0.91751 89735 17781 44815 737
0.410 0.39860 93279 84422 89359 380 0.91712 08228 16605 10547 564
0.411 0.39952 62493 49738 65238 251 0.91672 17549 94682 37232 150
0.412 0.40044 27711 88838 35528 558 0.91632 17704 51081 03796 202
0.413 0.40135 88925 85200 23958 010 0.91592 08695 85785 61266 649
0.414 0.40227 46126 22702 98524 766 0.91551 90527 99696 92832 194
0.415 0.40318 99303 85626 63109 550 0.91511 63204 94631 73753 232
0.416 0.40410 48449 58653 49047 645 0.91471 26730 73322 31180 180
0.417 0.40501 93554 26869 06660 654 0.91430 81109 39416 03880 251
0.418 0.40593 34608 75762 96747 939 0.91390 26344 97475 01872 722
0.419 0.40684 71603 91229 82037 655 0.91349 62441 52975 65972 725
0.420 0.40776 04530 59570 18597 279 0.91308 89403 12308 27243 609
0.421 0.40867 33379 67491 47203 546 0.91268 07233 82776 66357 915
0.422 0.40958 58142 02108 84671 703 0.91227 15937 12591 72866 996
0.423 0.41049 78808 50946 15143 980 0.91186 15518 90901 04379 332
0.424 0.41140 95370 01936 81337 201 0.91145 05981 47728 45647 576
0.425 0.41232 07817 43424 75749 435 0.91103 87329 54033 67564 373
0.426 0.41323 16141 64165 31825 593 0[91062 59567 21681 86066 990
0.421 0.41414 20333 53326 15081 889 0.91021 22698 63449 20950 808
0.428 0.41505 20384 00488 14189 067 Oi90979 76727 93022 54591 701
0.429 0.41596 16283 95646 32014 301 0.90938 21659 24998 90577 360
0.430 0.41687 08024 29210 76621 692 0.90896 57496 74885 12247 591
0.431 0.41777 95595 92007 52231 243 0.90854 84244 59097 41143 638
0.432 0.41868 78989 75279 50136 257 0.90813 01906 94960 95366 563
0.433 0.41959 58196 70687 39579 028 0.90771 10488 00709 47844 729
0.434 0.42050 33207 70310 58584 774 0.90729 09991 95484 84510 435
0.435 0.42141 04013 66648 04753 684 0.90687 00422 99336 62385 731
0.436 0,42231 70605 52619 26011 018 Oi90644 81785 33221 67577 465
0.42322 32974 21565 11315 146 0.90602 54083 19003 73181 601
ii* t;; 0.42412 91110 67248 81323 456 0.90560 17320 79452 97096 848
0:439 0.42503 45005 83856 79016 027 0.90517 71502 38245 59741 647
0.440 0.42593 94650 65999 60276 972 0.90475 16632 19963 41716 554
0.441 0.42684 40036 08712 84433 381 0.90432 52714 50093 41286 061
0.442 0.42774 81153 07458 04751 750 0.90389 79753 55027 31889 904
0.443 0.42865 17992 58123 58891 823 0.90346 97753 62061 19473 892
0.444 0.42955 50545 57025 59317 145 0.90304 06718 99394 99766 305
0.445 0.43045 78803 00908 83666 443 0.90261 06653 96132 15457 899
0.446 0.43136 02755 86947 65073 141 0.90217 97562 82279 13291 573
0.447 0.43226 22395 12746 82453 917 0.90174 79449 88745 01061 718
0.448 0.43316 37711 76342 50745 219 0.90131 52319 47341 04523 319
0.449 0.43406 48696 76203 11100 244 0.90088 16175 90780 24210 832
[1(-y 1
0.450 0.90044 71023 52676 92166 884
ELEMENTARY TRANSCENDENTAL FUNCTIONS 151
CIRCULAR SINES AND COSINES FOR RADIAN ARGCMENTS Table 4.6
x sin x cos x
0.450 0.43496 55341 11230 21042 084 0.90044 71023 52676 92166 884
0.451 0.43586 57635 80759 44573 567 0.90001 16866 67546 28580 847
0.452 0.43676 55571 84561 42243 681 0.89957 53709 70803 98337 319
0.453 0;43766 49140 22842 61170 507 Oi89913 81556 98765 67474 569
0.454 0.43856 38331 96246 25020 568 0.89870 00412 88646 59552 965
0.455 0.43946 23138 05853 23944 492 0.89826 10281 78561 11933 463
0.456 0.44036 03549 53183 04468 918 0.89782 11168 07522 31966 167
0.457 0.44125 79557 40194 59344 542 0.89738 03076 15441 53089 030
0.458 0.44215 51152 69287 17350 215 0.89693 86010 43127 90836 721
0.459 0.44305 18326 43301 33053 008 0.89649 59975 32287 98759 714
0.460 0.44394 81069 65519 76524 151 0.89605 24975 25525 24253 639
0.461 0.44484 39373 39668 23010 752 0.89560 81014 66339 64298 937
0.462 0.44573 93228 69916 42563 218 0.89516 28097 99127 21110 867
0.463 0.44663 42626 60878 89618 275 0.89471 66229 69179 57699 908
0.464 0.44752 87558 17615 92537 506 0.89426 95414 22683 53342 602
0.465 0.44842 28014 45634 43101 319 0.89382 15656 06720 58962 873
0.466 0.44931 63986 50888 85958 244 Oi89337 26959 69266 52423 883
0.467 0.45020 95465 39782 08029 479 0.89292 29329 59190 93730 459
0.468 0745110 22442 19166 27868 603 Oi89247 22770 26256 80142 134
0.469 0.45199 44907 96343 84976 342 0.89202 07286 21120 01196 857
0.470 0.45288 62853 79068 29070 327 0.89156 82881 95328 93645 402
0.471 0.45377 76270 75545 09309 736 0.89111 49562 01323 96296 541
0.472 0.45466 85149 94432 63474 735 0.89066 07330 92437 04773 005
0.473 0.45555 89482 44843 07100 635 0.89020 56193 22891 26178 292
0.474 0.45644 89259 36343 22566 671 0.88974 96153 47800 33674 367
0.475 0.45733 84471 78955 48139 307 0.88929 27216 23168 20970 288
0.476 0.45822 75110 83158 66969 994 0.88883 49386 05888 56721 822
0.477 0.45911 61167 59888 96047 279 0.88837 62667 53744 38842 074
0.478 0.46000 42633 20540 75103 180 0.88791 67065 25407 48723 197
0.479 0.46089 19498 76967 55473 739 0.88745 62583 80438 05369 212
0.480 0.46177 91755 41482 88913 664 0.88699 49227 79284 19439 995
0.481 0.46266 59394 26861 16364 968 0.88653 27001 83281 47206 469
0.482 0.46355 22406 46338 56679 522 0.88606 95910 54652 44417 051
0.483 0.46443 80783 13613 95295 430 0.88560 55958 56506 20075 401
0.484 0.46532 34515 42849 72867 132 0.88514 07150 52837 90129 517
0.485 0.46620 83594 48672 73849 162 0.88467 49491 08528 31072 223
0.486 0.46709 28011 46175 15033 451 0.88420 82984 89343 33453 094
0.487 0.46797 67757 50915 34040 104 0.88374 07636 61933 55301 874
0.488 0.46886 02823 78918 77761 558 0.88327 23450 93833 75463 416
0.489 Oi46974 33201 46678 90760 024 0.88280 30432 53462 46844 214
0.490 0.47062 58881 71158 03618 136 0.88233 28586 10121 49570 547
0.491 0.47150 79855 69788 21242 715 0.88186 17916 33995 44058 307
0.492 0.47238 96114 60472 11121 556 0.88138 98427 96151 23994 541
0.493 0.47327 07649 61583 91533 149 0.88091 70125 68537 69230 763
0.494 Oi47415 14451 91970 19709 261 0.88044 33014 23984 98588 075
0.49'5 0.47503 16512 70950 79950 264 0.87996 87098 36204 22574 157
0.496 0.47591 13823 18319 71693 150 0.87949 32382 79786 96012 154
0.497 0.47679 06374 54345 97532 118 0.87901 68872 30204 70581 529
0.498 0.47766 94157 99774 51191 668 0.87853 96571 63808 47270 917
0.499 0.47854 77164 75827 05452 099 0.87806 15485 57828 28743 023
c(6;W 1 l’-:“l
152 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos 2
0.500 0.47942 55386 04203 00027 329 0.87758 25618 90372 71611 628
0.501 0.48030 28813 07080 29394 947 0.87710 26976 40428 38630 733
0.502 0.48117 97437 07116 30578 414 0.87662 19562 87859 50795 903
0.503 0.48205 61249 27448 70881 314 0.87614 03383 13407 39357 847
0.504 0.48293 20240 91696 35573 583 0.87565 78441 98689 97748 295
0.505 0.48380 74403 23960 15529 617 0.87517 44744 26201 33418 203
0.506 0.48468 23727 48823 94818 170 0.87469 02294 79311 19588 355
0.507 0.48555 68204 91355 38243 967 0.87420 51098 42264 46912 391
0.508 0.48643 07826 77106 78840 928 0.87371 91160 00180 75052 318
0.509 0.48730 42584 32116 05316 931 0.87323 22484 39053 84166 561
0.510 0.48817 72468 82907 49450 013 0.87274 45076 45751 26310 581
0.511 0.48904 97471 56492 73435 934 0.87225 58941 08013 76750 129
0.512 0.48992 17583 80371 57187 006 0.87176 64083 14454 85187 176
0.513 0.49079 32796 82532 85582 104 0.87127 60507 54560 26898 565
0.514 0.49166 43101 91455 35667 778 0.87078 48219 18687 53787 441
0.515 0.49253 48490 36108 63810 364 0.87029 27222 98065 45347 504
0.516 0.49340 48953 45953 92799 025 0.86979 97523 84793 59540 132
0.517 0.49427 44482 50944 98899 617 0.86930 59126 71841 83584 429
0.518 0.49514 35068 81528 98859 309 0.86881 12036 53049 84660 240
0.519 0.49601 20703 68647 36861 855 0.86831 56258 23126 60524 189
0.520 0.49688 01378 43736 71433 446 0.86781 91796 77649 90038 785
0.521 0.49774 77084 38729 62299 043 0.86732 18657 13065 83614 647
0.522 0.49861 47812 86055 57189 109 0.86682 36844 26688 33565 898
0.523 0.49948 13555 18641 78596 658 0.86632 46363 16698 64378 779
0.524 0.50034 74302 69914 10484 518 0.86582 47218 82144 82893 524
0.525 0.50121 30046 73797 84942 748 0.86532 39416 22941 28399 561
0.526 0.50207 80778 64718 68796 092 0.86482 22960 39868 22644 077
0.527 0.50294 26489 77603 50161 411 0.86431 97856 34571 19753 996
0.528 0.50380 67171 47881 24954 981 0.86381 64109 09560 56071 436
0.529 0.50467 02815 11483 83349 596 0.86331 21723 68210 99902 671
0.530 0.50553 33412 04846 96181 366 0.86280 70705 14761 01380 670
0.531 0.50639 58953 64911 01306 143 0.86230 11058 54312 41041 248
0.532 0.50725 79431 29121 89905 473 0.86179 42788 92829 81312 894
0.533 0.50811 94836 35431 92741 999 0.86128 65901 37140 13920 311
0.534 0.50898 05160 22300 66364 220 0.86077 80400 94932 10201 726
0.535 0.50984 10394 28695 79260 534 0.86026 86292 74755 70140 025
0.536 0.51070 10529 94093 97962 456 0.85975 83581 86021 71507 760
0.537 0.51156 05558 58481 73096 946 0.85924 72273 39001 18926 068
0.538 0.51241 95471 62356 25387 754 0.85873 52372 44824 92837 581
0.539 0.51327 80260 46726 31605 686 0.85822 23884 15482 98393 339
0.540 0.51413 59916 53113 10467 728 0.85770 86813 63824 14253 797
0.541 0.51499 34431 23551 08484 914 0.85719 41166 03555 41303 947
0.542 0.51585 03796 00588 85758 874 0.85667 86946 49241 51282 623
0.543 0.51670 68002 27290 01726 969 0.85616 24160 16304 35326 032
0.544 0.51756 27041 47234 00855 920 0.85564 52812 21022 52425 567
0.545 0.51841 80905 04516 98283 861 0.85512 72907 80530 77799 957
0.546 0.51927 29584 43752 65410 714 0.85460 84452 12819 51181 787
0.547 0.52012 73071 10073 15436 812 0.85408 87450 36734 25018 472
0.548 0.52098 11356 49129 88849 675 0.85356 81907 71975 12587 703
0.549 0.52183 44432 07094 38858 868 Oi85304 67829 39096 36027 442
0.550 0.52268 72289 30659 16778 838 0.85252 45220 59505 74280 498
C-f)7
1 1 C-J)1
[ 1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 153
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin 2 cos x
0.550 0.52268 72289 30659 16778 838 0.85252 45220 59505 74280 498
0.551 0.52353 94919 67038 57359 653 0.85200 14086 55464 10953 761
0.552 0.52439 12314 63969 64065 565 0.85147 74432 50084 82092 114
0.553 0.52524 24465 69712 94301 297 0.85095 26263 67333 23867 il0
0.554 0.52609 31364 33053 44585 976 0.85042 69585 32026 20180 431
0.555 0.52694 33002 03301 35674 635 0.84990 04402 69831 50182 218
0.556 0.52779 29370 30292 97627 180 0.84937 30721 07267 35704 287
0.557 0.52864 20460 64391 54824 757 0.84884 48545 71701 88608 318
0.558 0.52949 06264 56488 10933 415 0.84831 57881 91352 58049 047
0.559 0.53033 86773 58002 33815 002 0.84778 58734 95285 77652 517
0.560 0.53118 61979 20883 40385 187 0.84725 51110 13416 12609 452
0.561 0.53203 31872 97610 81418 533 0.84672 35012 76506 06683 799
0.562 0.53287 96446 41195 26300 543 0.84619 10448 16165 29136 481
0.563 0.53372 55691 05179 47726 585 0.84565 77421 64850 21564 438
0.564 0.53457 09598 43639 06347 607 0.84512 35938 55863 44654 991
0.565 0.53541 58160 11183 35362 572 0.84458 86004 23353 24855 579
0.566 0.53626 01367 62956 25057 521 0.84405 27624 02313 00958 945
0.567 0.53710 39212 54637 07291 168 0.84351 60803 28580 70603 796
0.568 0.53794 71686 42441 39926 969 0.84297 85547 38838 36691 011
0.569 0.53878 98780 83121 91211 553 0.84244 01861 70611 53715 445
0.570 0.53963 20487 33969 24099 446 0.84190 09751 62268 74013 376
0.571 0.54047 36797 52812 80524 005 0.84136 09222 53020 93925 658
0.572 0.54131 47702 98021 65614 465 0.84082 00279 82920 99876 632
0.573 0.54215 53195 28505 31859 028 0.84027 82928 92863 14368 839
0.574 0.54299 53266 03714 63213 905 0.83973 57175 24582 41893 605
0.575 0.54383 47906 83642 59158 222 0.83919 23024 20654 14757 543
0.576 0.54467 37109 28825 18694 718 0.83864 80481 24493 38825 019
0.577 0;54551 20865 00342 24296 136 0.83810 29551 80354 39176 658
0.578 0.54634 99165 59818 25797 231 0.83755 70241 33330 05683 918
0.579 0.54718 72002 69423 24232 321 0.83701 02555 29351 38499 807
0.580 0.54802 39367 91873 55618 270 0.83646 26499 15186 93465 789
0.581 0.54886 01252 90432 74682 851 0.83591 42078 38442 27434 927
0.582 0.54969 57649 28912 38538 382 0.83536 49298 47559 43511 337
0.583 0.55053 08548 71672 90300 563 0.83481 48164 91816 36205 988
0.584 0.55136 53942 83624 42652 424 0.83426 38683 21326 36508 907
0.585 0.55219 93823 30227 61353 309 0.83371 20858 87037 56877 861
0.586 0.55303 28181 77494 48692 799 0.83315 94697 40732 36143 543
0.587 0.55386 57009 91989 26889 504 0.83260 60204 35026 84331 337
0.588 0.55469 80299 40829 21434 637 0.83205 17385 23370 27399 720
0.589 0.55552 98041 91685 44380 278 0.83149 66245 60044 51895 332
0.590 0.55636 10229 12783 77572 254 0.83094 06791 00163 49524 800
0.591 0.55719 16852 72905 55827 556 0.83038 39026 99672 61643 346
0.592 0.55802 17904 41388 50056 192 Oi82982 62959 15348 23660 255
0.593 0.55885 13375 88127 50327 409 0.82926 78593 04797 09361 243
0.594 0.55968 03258 83575 48880 201 Oi82870 85934 26455 75147 786
0.595 0.56050 87544 98744 23078 004 0.82814 84988 39590 04193 468
0.596 0.56133 66226 05205 18307 516 0.82758 75761 04294 50517 407
0.597 0.56216 39293 75090 30821 541 0.82702 58257 81491 82974 799
0.598 0.56299 06739 81092 90525 792 0.82646 32484 32932 29164 660
0.599 0.56381 68555 96468 43709 545 Oi82589 98446 21193 19254 799
0.600 0.56464 24733 95035 35720 095 0.82533 56149 09678 29724 095
C-f)7
[ 1
154 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos x
0.600 0.56464 24733 95035 35720 095 0.82533 56149 09678 29724 095
0.601 O;i6546 75265 51175 93580 897 0.82477 05598 62617 27022 123
0.602 0.56629 20142 39837 08553 336 0.82420 46800 45065 11146 193
0.603 0.56711 59356 36531 18642 028 0.82363 79760 22901 59135 858
0.604 0.56793 92899 17336 91043 574 0:82307 04483 62830 68484 934
0.605 0.56876 20762 58900 04538 687 0.82250 20976 32380 00471 116
0.606 Oi56958 42938 38434 31827 607 0.82193 29243 99900 23403 216
0.607 0.57040 59418 33722 21808 719 0.82136 29292 34564 55786 102
0.608 0.57122 70194 23115 81800 299 0.82079 21127 06368 09403 380
0.609 0.57204 75257 85537 59705 300 0.82022 04753 86127 32317 893
0.610 0.57286 74601 00481 26119 098 0.81964 80178 45479 51790 075
0.611 0:$7368 68215 48012 56380 111 0.81907 47406 56882 17114 225
0.612 0.57450 56093 08770 12563 221 0.81850 06443 93612 42372 770
0.613 O:i7532 38225 63966 25415 904 0.81792 57296 29766 49108 549
0.614 0.57614 14604 95387 76236 989 0.81734 99969 40259 08915 198
0.615 0.57695 85222 85396 78697 975 0.81677 34469 00822 85945 685
0.616 0.57777 50071 16931 60606 809 0.81619 60800 88007 79339 051
0.617 0.57859 09141 73507 45614 047 0.81561 78970 79180 65565 411
0.618 0.57940 62426 39217 34861 330 0.81503 88984 52524 40689 288
0.619 0.58022 09916 98732 88572 073 0.81445 90847 87037 62551 318
0.620 0.58103 51605 37305 07584 296 0.81387 84566 62533 92868 400
0.621 0.58184 87483 40765 14825 522 0.81329 70146 59641 39252 335
0.622 0.58266 17542 95525 36729 641 0.81271 47593 59801 97147 027
0.623 0.58347 41775 88579 84595 681 0.81213 16913 45270 91684 290
0.624 0.58428 60174 07505 35888 387 0.81154 78111 99116 19458 331
0.625 0.58509 72729 40462 15480 540 0.81096 31195 05217 90218 953
0.626 0.58590 79433 76194 76836 923 0.81037 76168 48267 68483 556
0.627 0.58671 80279 04032 83139 861 0.80979 13038 13768 15067 973
0.628 0.58752 75257 13891 88356 252 0.80920 41809 88032 28536 214
0.629 0.58833 64359 96274 18246 006 0.80861 62489 58182 86569 178
0.630 0.58914 47579 42269 51311 811 0.80802 75083 12151 87252 371
0.631 0.58995 24907 43555 99690 151 0.80743 79596 38679 90282 722
0.632 0.59075 96335 92400 89983 484 0.80684 76035 27315 58094 522
0.633 0.59156 61856 81661 44033 509 0.80625 64405 68414 96904 569
0.634 0.59237 21462 04785 59635 440 0.80566 44713 53140 97676 566
0.635 0.59317 75143 55812 91193 198 0.80507 16964 73462 77004 837
0.636 0.59398 22893 29375 30315 454 0.80447 81165 22155 17917 411
0.637 0.59478 64703 20697 86352 425 0.80388 37320 92798 10598 548
0.638 0.59559 00565 25599 66873 364 0.80328 85437 79775 93030 752
0.639 0.59639 30471 40494 58084 641 0.80269 25521 78276 91556 338
0.640 0.59719 54413 62392 05188 355 0.80209 57578 84292 61358 611
0.641 0.59799 72383 88897 92681 375 0.80149 81614 94617 26862 715
0.642 0.59879 84374 18215 24594 757 0.80089 97636 06847 22056 216
0.643 0.59959 90376 49145 04673 426 0.80030 05648 19380 30729 469
0.644 0.60039 90382 81087 16496 070 0.79970 05657 31415 26635 842
0.645 0.60119 84385 14041 03535 151 0.79909 97669 42951 13571 848
0.646 0.60199 72375 48606 49156 949 0.79849 81690 54786 65377 243
0.647 0.60279 54345 85984 56561 576 0.79789 57726 68519 65855 159
0.648 0.60359 30288 27978 28662 868 0.79729 25783 86546 48612 327
0.649 0.60439 00194 76993 47908 070 0.79668 85868 12061 36819 444
0.650 0.60518 64057 36039 56037 252 0.79608 37985 49055 82891 760
ELEMENTARY TRANSCENDENTAL FUNCTIONS 155
CIRCULAR SINES AND COSINES FOR RADIAN ARGKMENTS Table 4.6
X sin x cos x
0.650 0.60518 64057 36039 56037 252 0.79608 37985 49055 a2891 760
0.651 0.60598 21868 08730 33782 358 0.79547 a2142 02318 08089 927
0.652 0.60677 73618 99284 80505 ala 0.79487 la343 77432 42041 la3
0.653 0.60757 19302 12527 93778 646 0.79426 46596 80778 62180 929
0.654 0.60836 58909 53891 48897 929 0.79365 66907 19531 33114 757
0.655 0.60915 92433 29414 78343 652 0.79304 79281 01659 45900 987
0.656 0.60995 19865 45745 51174 755 0.79243 83724 35925 57253 785
0.657 0.61074 41198 10140 52364 359 0.79182 80243 31885 28666 909
0.658 0.61153 56423 30466 62074 073 0.79121 68843 99886 65458 154
0.659 0.61232 65533 15201 34867 307 0.79060 49532 51069 55734 550
0.660 0.61311 68519 73433 78861 515 0.78999 22314 97365 09278 382
0.661 0.61390 65375 14865 34819 272 0.78937 a7197 51494 96354 080
0.662 0.61469 56091 49810 55178 137 0.78876 44186 26970 a6436 061
0.663 0.61548 40660 a9197 a3019 la6 0.78814 93287 38093 86857 558
0.664 0.61627 19075 44570 30974 165 0.78753 34506 99953 81380 523
0.665 0.61705 91327 28086 60071 171 0.78691 67851 28428 68686 643
0.666 0.61784 57408 52521 58518 785 0.78629 93326 40184 00789 551
0.667 0.61863 17311 31267 20428 576 0.78568 10938 52672 21368 279
0.668 0.61941 71027 78333 24475 901 0.78506 20693 a4132 04022 017
0.669 0.62020 la550 08348 12498 919 0.78444 22598 53587 90446 244
0.670 0.62098 59870 36559 68035 744 0.78382 16658 80849 28530 294
0.671 0.62176 94980 78835 94799 654 0.78320 02880 86510 10376 414
0.672 0.62255 23873 51665 95092 281 0.78257 81270 91948 10240 374
0.673 0.62333 46540 72160 48154 700 0.78195 51835 19324 22393 698
0.674 0.62411 62974 58052 a8456 349 0.78133 14579 91581 98907 578
0.675 0.62489 73167 27699 a3921 682 0.78070 69511 32446 a7358 526
0.676 0.62567 77111 00082 14094 496 0.78008 16635 66425 68455 a30
0.677 0.62645 74797 94805 48239 849 0.77945 55959 18805 93590 a77
0.678 0.62723 66220 32101 23383 477 0.77882 a7488 15655 22308 414
0.679 0.62801 51370 32827 22288 658 0.77820 11228 a3820 59699 786
0.680 0.62879 30240 18468 51370 418 0.77757 27187 50927 93718 239
0.681 0.62957 02822 11138 la547 018 0.77694 35370 45381 32416 339
0.682 0.63034 69108 33578 11028 644 0.77631 35783 96362 41105 566
0.683 0.63112 29091 09159 73043 207 0.77568 28434 33829 79438 156
0.684 0.63189 a2762 61884 a3499 197 0.77505 13327 88518 38411 247
0.685 0.63267 30115 16386 33585 498 0.77441 90470 91938 77293 390
0.686 0.63344 71140 97929 04308 084 0.77378 59869 76376 60473 500
0.687 0.63422 05832 32410 43963 542 0.77315 21530 74891 94232 293
0.688 0.63499 34181 46361 45549 306 0.77251 75460 21318 63436 286
0.689 0.63576 56180 66947 24110 566 0.77188 21664 50263 68154 418
0.690 0.63653 71822 21967 94023 743 0.77124 60149 97106 60197 354
0.691 0.63730 al098 39859 46216 467 0.77060 90922 97998 79579 541
0.692 0.63807 84001 49694 25323 984 0.76997 13989 89862 90904 069
0.693 0.63884 80523 81182 06781 a99 0.76933 29357 10392 19670 418
0.694 0.63961 70657 64670 73855 200 0.76869 37030 98049 @SO5 132
0.695 0.64038 54395 31146 94603 464 0.76805 37017 92068 53315 502
0.696 0.64115 31729 12236 98782 Ii5 0.76741 29324 32449 39366 321
0.697 0.64192 02651 40207 54600 136 0.76677 13956 59961 77279 757
0.698 0.64268 67154 47966 i5a92 698 0.76612 90921 16142 38958 434
0.699 0.64345 25230 69063 48031 063 0.76548 60224 43294 73431 759
0.700 0.64421 76872 37691 05367 261 0.76484 21872 84488 42625 586
[ c-y1 [c-y 1
156 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin 5 cos x
0.700 0.64421 76872 37691 05367 261 0.76484 21872 84488 42625 586
0.701 0.64498 22071 88685 07414 902 0.76419 75872 83558 57055 252
0.702 0.64574 60821 57525 65445 583 Oi76355 22230 85105 i1442 075
0.703 0.64650 93113 80337 88940 870 0.76290 60953 34492 20253 368
0.704 0.64727 18940 93892 61979 783 0.76225 92046 77847 53166 023
0.705 0.64803 38295 35607 19561 705 0.76161 15517 62061 70453 752
0.706 0.64879 51169 43546 23864 641 0.76096 31372 34787 58298 030
0.707 0.64955 57555 56422 40438 747 0.76031 39617 44439 64022 815
0.708 0.65031 57446 13597 14335 062 0.75966 40259 40193 31253 107
0.709 0.65107 50833 55081 46169 354 0.75901 33304 71984 34997 406
0.710 0.65183 37710 21536 68121 013 0.75836 18759 90508 16654 146
0.711 0.65259 18068 54275 19866 915 0.75770 96631 47219 18942 159
0.712 0.65334 91900 95261 24450 173 0.75705 66925 94330 20755 235
0.713 0.65410 59199 87111 64083 709 0.75640 29649 84811 71940 852
0.714 0.65486 19957 73096 55888 565 0.75574 84809 72391 28003 128
0.715 0.65561 74166 97140 27566 883 0.75509 32412 11552 84730 074
0.716 0.65637 21820 03821 93009 463 0.75443 72463 57536 12745 203
0.717 0.65712 62909 38376 27837 851 0.75378 04970 66335 91983 563
0.718 0.65787 97427 46694 44880 853 0.75312 29939 94701 46092 263
0.719 0.65863 25366 75324 69585 417 0.75246 47378 00135 76755 558
0.720 0.65938 46719 71473 15361 800 0.75180 57291 40894 97944 549
0.721 0.66013 61478 83004 58862 952 0.75114 59686 75987 70091 576
0.722 0.66088 69636 58443 15198 027 0.75048 54570 65174 34189 363
0.723 0.66163 71185 46973 13079 967 0.74982 41949 68966 45814 983
0.724 0.66238 66117 98439 69907 065 0.74916 21830 48626 09078 707
0.725 0.66313 54426 63349 66778 441 0.74849 94219 66165 10497 806
0.726 0.66388 36103 92872 23443 354 0.74783 59123 84344 52795 369
0.727 0.66463 11142 38839 73184 280 0.74717 16549 66673 88624 209
0.728 0.66537 79534 53748 37633 666 0.74650 66503 77410 54215 910
0.729 0.66612 41272 90759 01524 309 0.74584 08992 81559 02955 103
0.730 0.66686 96350 03697 87373 259 0.74517 44023 44870 38879 013
0.731 0.66761 44758 47057 30099 195 0.74450 71602 33841 50102 364
0.732 0.66835 86490 75996 51573 181 0.74383 91736 15714 42167 693
0.733 0.66910 21539 46342 35102 739 0.74317 04431 58475 71321 153
0.734 0.66984 49897 14589 99849 159 0.74250 09695 30855 77713 862
0.735 0.67058 71556 37903 75177 973 0.74183 07534 02328 18528 866
0.736 0.67132 86509 74117 74942 523 0.74115 97954 43109 01033 791
0.737 0.67206 94749 81736 71700 537 0.74048 80963 24156 15559 237
0.738 0.67280 96269 19936 70863 650 0.73981 56567 17168 68402 998
0.739 0.67354 91060 48565 84779 796 0.73914 24772 94586 14660 158
0.740 0.67428 79116 28145 06748 388 0.73846 85587 29587 90979 142
0.741 0.67502 60429 19868 84968 216 0.73779 39016 96092 48243 787
0.742 0.67576 34991 85605 96417 996 0.73711 85068 68756 84181 492
0.743 0.67650 02796 87900 20669 485 0.73644 23749 22975 75897 532
0.744 0.67723 63836 89971 13633 096 0.73576 55065 34881 12335 582
0.745 0.67797 18104 55714 81235 936 0.73508 79023 81341 26664 537
0.746 0.67870 65592 49704 53032 193 0.73440 95631 39960 28591 681
0.747 0.67944 06293 37191 55745 803 0.73373 04894 89077 36602 285
0.748 0.68017 40199 84105 86745 313 0.73305 06821 07766 10125 695
0.749 0.68090 67304 57056 87450 880 0.73237 01416 75833 81627 975
0.750 0.68163 87600 23334 16673 324 0.73168 88688 73820 88631 184
[(-;GJ
1 [(-;)I1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 157
CIRCULAR SINES AND COSINES FOR RADIAN .4RGUMENTS Table 4.6
X sin x cos x
0.750 0.68163 87600 23334 16673 324 0.73168 88688 73820 88631 184
0.751 0.68237 01079 50908 23885 163 0.73100 68643 83000 05659 342
0.752 0.68310 07735 08431 22423 554 0.73032 41288 85375 76111 160
0.753 0.68383 07559 65237 62625 080 0.72964 06630 63683 44059 608
0.754 0.68456 00545 91345 04892 285 0.72895 64676 01388 85978 367
0.755 0.68528 86686 57454 92691 917 0.72827 15431 82687 42395 268
0.756 0.68601 65974 34953 25484 772 0.72758 58904 92503 49472 750
0.757 0.68674 38401 95911 31587 089 0.72689 95102 16489 70515 436
0.758 0.68747 03962 13086 40963 419 0.72621 24030 41026 27404 8k7
0.759 0.68819 62647 59922 57950 885 0.72552 45696 53220 31961 494
0.760 0.68892 14451 10551 33914 776 0.72483 60107 40905 17233 969
0.761 0.68964 59365 39792 39835 383 0.72414 67269 92639 68715 814
0.762 0.69036 97383 23154 38826 030 0.72345 67190 97707 55489 548
0.763 0.69109 28497 36835 58582 200 0.72276 59877 46116 61298 318
0.764 0.69181 52700 57724 63761 700 0.72207 45336 28598 15545 123
0.765 0.69253 69985 63401 28295 794 0.72138 23574 36606 24219 693
0.766 0.69325 80345 32137 07631 223 0.72068 94598 62317 00753 084
0.767 0.69397 83772 42896 10903 039 0.71999 58415 98627 96800 072
0.768 0.69469 80259 75335 73038 195 0.71930 15033 39157 32949 410
0.769 0.69541 69800 09807 26789 802 0.71860 64457 78243 29362 010
0.770 0.69613 52386 27356 74701 988 0.71791 06696 10943 36337 129
0.771 0.69685 28011 09725 61005 296 0.71721 41755 33033 64806 626
0.772 0.69756 96667 39351 43442 524 0.71651 69642 41008 lb757 355
0.773 0.69828 58347 99368 65024 972 0.71581 90364 32078 15581 770
0.774 0.69900 13045 73609 25718 983 0.71512 03928 04171 36356 807
0.775 0.69971 60753 46603 54062 747 0.71442 10340 55931 36051 117
0.776 0.70043 01464 03580 78713 256 0.71372 09608 86716 83660 709
0.777 0.70114 35170 30469 99923 379 0.71302 01739 96600 90273 093
0.778 0.70185 61865 13900 60948 949 0.71231 86740 86370 39059 972
0.779 0.70256 81541 41203 19385 818 0.71161 64618 57525 15198 564
0.780 0.70327 94192 00410 18436 790 0.71091 35380 12277 35721 626
0.781 0.70398 99809 80256 58108 374 0.71020 99032 53550 79296 239
0.782 0.70469 98387 70180 66337 280 0.70950 55582 84980 15931 435
0.783 0.70540 89918 60324 70046 581 0.70880 05038 10910 36614 737
0.784 0.70611 74395 41535 66131 480 0.70809 47405 36395 82877 671
0.785 0.70682 51811 05365 92374 614 0.70738 82691 67199 76290 330
0.786 0.70753 22158 44073 98290 801 0.70668 10904 09793 47885 059
0.787 0.70823 85430 50625 15901 193 0.70597 32049 71355 67509 330
0.788 0.70894 41620 18692 30436 730 0.70526 46135 59771 73107 880
0.789 0.70964 90720 42656 50970 857 0.70455 53168 83632 99934 173
0.790 0.71035 32724 17607 80981 403 0.70384 53156 52236 09691 278
0.791 0.71105 67624 39345 88841 574 0.70313 46105 75582 19602 208
0.792 0.71175 95414 04380 78239 979 0.70242 32023 64376 31409 812
0.793 0.71246 16086 09933 58529 620 0.70171 10917 30026 60306 275
0.794 0.71316 29633 53937 15005 776 0.70099 82793 84643 63792 314
0.795 0.71386 36049 35036 79112 713 0.70028 47660 41039 70466 123
0.796 0.71456 35326 52590 98579 148 0.69957 05524 12728 08742 151
0.797 0.71526 27458 06672 07482 391 0.69885 56392 13922 35499 779
0.798 0.71596 12436 98066 96241 109 0.69814 00271 59535 64661 971
0.799 0.71665 90256 28277 81536 630 0.69742 37169 65179 95703 964
0.800 0.71735 60908 99522 76162 718 0.69670 67093 47165 42092 075
[c-y1 [ C-f,9
1
158 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x co5 x
0.800 0.71735 60908 99522 76162 718 0.69670 67093 47165 42092 075
0.801 0.71805 24388 14736 58803 753 0.69598 90050 22499 59652 695
0.802 0.71874 80686 77571 43741 255 0.69527 06047 08886 74871 538
0.803 0.71944 29797 92397 50488 651 0.69455 15091 24727 13123 218
0.804 0.72013 71714 64303 73354 263 0.69383 17189 89116 26831 236
0.805 0.72083 06429 99098 50932 396 0.69311 12350 21844 23558 425
0.806 0.72152 33937 03310 35522 503 0.69239 00579 43394 94027 956
0.807 0.72221 54228 84188 62476 322 0.69166 81884 74945 40074 951
0.808 0.72290 67298 49704 19472 935 0.69094 56273 38365 02528 784
0.809 0.72359 73139 08550 15721 677 0.69022 23752 56214 89026 151
0.810 0.72428 71743 70142 51092 818 0.68949 84329 51747 01754 964
0.811 Oi72497 63105 44620 85175 959 0.68877 38011 48903 65129 158
0.812 0.72566 47217 42849 06266 069 0.68804 84805 72316 53394 472
0.813 0.72635 24072 76416 00277 085 0.68732 24719 47306 18165 280
0.814 0.72703 93664 57636 19583 027 0.68659 57759 99881 15892 545
0.815 0.72772 55985 99550 51786 534 0.68586 83934 56737 35262 969
0.816 Oi72841 11030 15926 884L4 775 0.68514 03250 45257 24529 414
0.817 0.72909 58790 21260 93542 651 0.68441 15714 93509 18772 652
0.818 Oi72977 99259 30776 72343 223 0.68368 21335 30246 67094 544
0.819 0.73046 32430 60427 39565 302 0.68295 20118 84907 59742 692
0.820 0.73114 58297 26895 87938 131 0.68222 12072 87613 55166 656
0.821 0.73182 76852 47595 56503 084 0.68148 97204 69169 07005 802
0.822 0.73250 88089 40670 98872 320 0.68075 75521 61060 91008 857
0.823 0.73318 92001 24998 51414 329 0.68002 47030 95457 31885 232
0.824 0.73386 88581 20187 01366 283 0.67929 11740 05207 30088 213
0.825 0.73454 77822 46578 54873 150 0.67855 69656 23839 88530 058
0.826 0.73522 59718 25249 04953 477 0.67782 20786 85563 39229 106
0.827 0.73590 34261 78008 99391 793 0.67708 65139 25264 69888 949
0.828 0.73658 01446 27404 08557 557 0.67635 02720 78508 50409 750
0.829 0.73725 61264 96715 93150 579 0.67561 33538 81536 59331 781
0.830 0.73793 13711 09962 71872 858 0.67487 57600 71267 10211 246
0.831 0.73860 58777 91899 89026 752 0.67413 74913 85293 77928 481
0.832 0.73927 96458 68020 82039 434 0.67339 85485 61885 24928 580
0.833 0.73995 26746 64557 48913 544 0.67265 89323 39984 27394 537
0.834 0.74062 49635 08481 15603 989 0.67191 86434 59207 01352 983
0.835 0.74129 65117 27503 03320 808 0.67117 76826 59842 28712 570
0.836 0.74196 73186 50074 95758 049 0.67043 60506 82850 83235 098
0.837 0.74263 73836 05390 06248 576 0.66969 37482 69864 56439 445
0.838 0.74330 67059 23383 44844 755 0.66895 07761 63185 83438 385
0.839 0.74397 52849 34732 85324 932 0.66820 71351 05786 68708 357
0.840 0.74464 31199 70859 32125 657 0.66746 28258 41308 11792 267
0.841 0.74531 02103 63927 87199 577 0.66671 78491 14059 32935 396
0.842 0.74597 65554 46848 16798 923 0.66597 22056 69016 98654 482
0.843 0.74664 21545 53275 18184 539 0.66522 58962 51824 47240 065
0.844 0.74730 70070 17609 86260 385 0.66447 89216 08791 14192 152
0.845 0.74797 11121 74999 80133 429 0.66373 12824 86891 57589 286
0.846 0.74863 44693 61339 89598 886 0.66298 29796 33764 83391 100
0.847 0.74929 70779 13273 01550 724 0.66223 40137 97713 70674 409
0.848 0.74995 89371 68190 66317 368 0.66148 43857 27703 96802 946
0.849 0.75062 00464 64233 63922 547 0.66073 40961 73363 62530 783
[c-w1
0.850 0.75128 04051 40292 70271 207 0.65998 31458 84982 17039 542
[ 71
i-8)9
7
ELEMENTARY TRANSCENDENTAL FUNCTIONS 159
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin z cos x
0.850 0.75128 04051 40292 70271 207 0.65998 31458 84982 17039 542
0.851 0.75194 00125 36009 23260 432 0.65923 15356 13509 82909 449
0.852 0.75259 88679 91775 88815 295 0.65847 92661 10556 81024 321
0.853 0.75325 69708 48737 26849 594 0.65772 63381 28392 55410 547
0.854 0.75391 43204 48790 57151 380 0.65697 27524 19944 98010 152
0.855 0.75457 09161 34586 25193 237 0.65621 85097 38799 73388 013
0.856 0.75522 67572 49528 67867 227 0.65546 36108 39199 43373 300
0.857 0.75588 18431 37776 79144 450 0.65470 80564 76042 91635 218
0.858 0.75653 61731 44244 75659 143 0.65395 18474 04884 48193 134
0.859 0.75718 97466 14602 62217 260 0.65319 49843 81933 13861 148
0.860 0.75784 25628 95276 97229 459 0.65243 74681 64051 84627 203
0.861 0.75849 46213 33451 58068 441 0.65167 92995 08756 75966 794
0.862 0.75914 59212 77068 06350 566 0.65092 04791 74216 47091 357
0.863 0.75979 64620 74826 53141 684 0.65016 10079 19251 25131 418
0.864 0.76044 62430 76186 24087 122 0.64940 08865 03332 29254 574
0.865 0.76109 52636 31366 24465 750 0.64864 01156 86580 94718 373
0.866 0.76174 35230 91346 04168 673 0.64787 86962 29767 96858 196
0.867 0.76239 10208 07866 22598 272 0.64711 66288 94312 75010 176
0.868 0.76303 77561 33429 13500 144 0.64635 39144 42282 56369 276
0.869 0.76368 37284 21299 49706 858 0.64559 05536 36391 79782 561
0.870 0.76432 89370 25505 07814 480 0.64482 65472 40001 19477 766
0.871 0.76497 33813 00837 32779 191 0.64406 18960 17117 08727 234
0.872 0.76561 70606 02852 02438 134 0.64329 66007 32390 63447 280
0.873 0.76625 99742 87869 91953 834 0.64253 06621 51117 05733 091
0.874 0.76690 21217 12977 38182 114 0.64176 40810 39234 87329 202
0.875 0.76754 35022 36027 03963 458 0.64099 68581 63325 13035 656
0.876 0.76818 41152 15638 42337 736 0.64022 89942 90610 64049 903
0.877 0.76882 39600 11198 60682 252 0.63946 04901 88955 21244 528
0.878 0.76946 30359 82862 84773 027 0.63869 13466 26862 88380 872
0.879 0.77010 13424 91555 22769 271 0.63792 15643 73477 15258 639
0.880 0.77073 88788 98969 29120 965 0.63715 11441 98580 20801 550
0.881 0.77137 56445 67568 68399 506 0.63638 00868 72592 16079 131
0.882 0.77201 16388 60587 79051 337 0.63560 83931 66570 27264 710
0.883 0.77264 68611 42032 37074 497 0.63483 60638 52208 18529 695
0.884 0.77328 13107 76680 19618 049 0.63406 30997 01835 14874 218
0.885 0.77391 49871 30081 68504 290 0.63328 95014 88415 24894 213
0.886 0.77454 78895 68560 53673 706 0.63251 52699 85546 63485 020
0.887 0.77518 00174 59214 36552 600 0.63174 04059 67460 74481 571
0.888 0.77581 13701 69915 33343 321 0.63096 49102 09021 53235 256
0.889 0.77644 19470 69310 78237 045 0.63018 87834 85724 69127 530
0.890 0.77707 17475 26823 86549 033 0.62941 20265 73696 88020 355
0.891 0.77770 07709 12654 17776 316 0.62863 46402 49694 94643 540
0.892 0.77832 90165 97778 38577 722 0.62785 66252 91105 14919 057
0.893 0.77895 64839 53950 85676 211 0.62707 79824 75942 38222 428
0.894 0.77958 31723 53704 28683 432 0.62629 87125 82849 39581 242
0.895 0.78020 90811 70350 32846 443 0.62551 88163 91096 01810 880
0.896 0.78083 42097 77980 21716 548 0.62473 82946 80578 37587 545
0.897 0.78145 85575 51465 39740 163 0.62395 71482 31818 11458 656
0.898 0.78208 21238 66458 14771 667 0.62317 53778 25961 61790 683
0.899 0.78270 49080 99392 20508 171 0.62239 29842 44779 22654 524
0.900 0.78332 69096 27483 38846 138 0.62160 99682 70664 45648 472
[c-p1
1 [ (-f)S
1
160 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos x
0.900 0.78332 69096 27483 38846 138 0.62160 99682 70664 45648 472
0.901 0.78394 81278 28730 22159 796 0.62082 63306 86633 21658 870
0.902 0.78456 85620 81914 55501 279 0.62004 20722 76323 02558 530
0.903 0.78518 82117 66602 18722 439 0.61925 71938 23992 22842 983
0.904 0.78580 70762 63143 48518 260 Oib1847 16961 14519 21204 658
0.905 0.78642 51549 52674 00391 817 0.61768 55799 33401 62045 040
0.906 0.78704 24472 17115 10540 713 0.61689 88460 66755 56924 921
0.907 0.78765 89524 39174 57664 940 0.61611 14953 01314 85952 792
0.908 0.78827 46700 02347 24696 094 0.61532 35284 24430 19111 466
0.909 0.78888 95992 90915 60447 888 0.61453 49462 24068 37523 020
0.910 0.78950 37396 89950 41187 896 0.61374 57494 88811 54652 118
0.911 0.79011 70905 85311 32130 474 0.61295 59390 07856 37447 803
0.912 0.79072 96513 63647 48850 789 0.61216 55155 71013 27423 839
0.913 0.79134 14214 12398 18619 897 0.61137 44799 68705 61677 674
0.914 0.79195 24001 19793 41660 812 0.61058 28329 91968 93848 110
0.915 0.79256 25868 74854 52325 499 0.60979 05754 32450 15011 758
0.916 0.79317 19810 67394 80192 738 0.60899 77080 82406 74518 350
0.917 0.79378 05820 88020 11086 785 0.60820 42317 34706 00764 999
0.918 0.79438 83893 28129 48016 785 0.60741 01471 82824 21909 476
0,919 0.79499 54021 79915 72036 860 0.60661 54552 20845 86522 589
0.920 0.79560 16200 36366 03026 828 0.60582 01566 43462 84179 741
0.921 0.79620 70422 91262 60393 471 0.60502 42522 45973 65991 745
0.922 0.79681 16683 39183 23692 319 0.60422 77428 24282 65074 984
0.923 0.79741 54975 75501 93169 858 0.60343 06291 74899 16960 980
0.924 0.79801 85293 96389 50226 129 0.60263 29120 94936 79945 468
0.925 0.79862 07631 98814 17797 639 0.60183 45923 82112 55377 043
0.926 0.79922 21983 80542 20660 537 0.60103 56708 34746 07885 466
0.927 0.79982 28343 40138 45653 978 0.60023 61482 51758 85549 703
0.928 0.80042 26704 76967 01823 638 0.59943 60254 32673 40005 791
0.929 0.80102 17061 91191 80485 294 0.59863 53031 77612 46494 584
0.930 0.80161 99408 83777 15208 432 0.59783 39822 87298 23849 491
0.931 0.80221 73739 56488 41719 806 0.59703 20635 63051 54424 260
0.932 0.80281 40048 11892 57726 899 0.59622 95478 06791 03960 905
0.933 0.80340 98328 53358 82661 218 0.59542 64358 21032 41397 846
0.934 0.80400 48574 85059 17341 371 0.59462 27284 08887 58618 345
0.935 0.80459 90781 11969 03555 863 0.59381 84263 74063 90139 324
0.936 0.80519 24941 39867 83565 545 0.59301 35305 20863 32740 634
0.937 0.80578 51049 75339 59525 671 0.59220 80416 54181 65034 867
0.938 0.80637 69100 25773 52827 488 0.59140 19605 79507 66977 785
0.939 0.80696 79086 99364 63359 313 0.59059 52881 02922 39319 443
0.940 0.80755 81004 05114 28687 022 0.58978 80250 31098 22996 099
0.941 0.80814 74845 52830 83153 915 0.58898 01721 71298 18462 976
0.942 0.80873 60605 53130 16899 872 0.58817 17303 31375 04967 973
0.943 0.80932 38278 17436 34799 758 0.58736 27003 19770 59766 388
0.944 0.80991 07857 57982 15321 017 0.58655 30829 45514 77276 748
0.945 0.81049 69337 87809 69300 383 0.58574 28790 18224 88177 827
0.946 0.81108 22713 20770 98639 669 0.58493 20893 48104 78446 913
0.947 0.81166 67977 71528 54920 560 0.58412 07147 45944 08339 436
0.948 0.81225 05125 55555 97938 351 0.58330 87560 23117 31310 012
0.949 0.81283 34150 89138 54154 591 0.58249 62139 91583 12874 994
0.950 0.81341 55047 89?73 75068 542 0.58168 30894 63883 49416 618
1 1
c-y
[ !-;I81
ELEMENTARY TRANSCENDENTAL FUNCTIONS 161
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin x cos x
0:950 0.81341 55047 89373 75068 542 0.58168 30894 63883 49416 618
0.951 0.81399 67810 74171 95507 433 0.58086 93832 53142 86928 810
0.952 0.81457 72433 62256 91835 411 0.58005 50961 73067 39704 748
0.953 0.81515 68910 73166 40081 165 0.57924 02290 37944 08966 251
0.954 0.81573 57236 27252 73984 145 Oi57842 47826 62640 01435 096
0.955 0.81631 37404 45683 42959 322 0.57760 87578 62601 47846 300
0.956 0.81689 09409 50441 69980 433 0.57679 21554 53853 21403 511
0.957 0.81746 73245 64327 09381 654 0.57597 49762 52997 56176 536
0.958 0.81804 28907 10956 04577 644 0.57515 72210 77213 65441 113
0.959 0.81861 76388 14762 45701 891 0.57433 88907 44256 59961 007
0.960 0.81919 15683 00998 27163 322 0.57351 99860 72456 66212 505
0.961 0.81976 46785 95734 05121 101 0.57270 05078 80718 44551 395
0.962 0.82033 69691 25859 54877 569 0.57188 04569 88520 07322 513
0.963 0.82090 84393 19084 28189 263 0.57105 98342 15912 36911 940
0.964 0.82147 90886 03938 10495 962 0.57023 86403 83518 03741 923
0.965 0.82204 89164 09771 78067 694 0.56941 68763 12530 84208 614
0.966 0.82261 79221 66757 55069 656 0.56859 45428 24714 78562 699
0.967 0.82318 61053 05889 70544 986 0.56777 16407 42403 28733 004
0.968 0.82375 34652 58985 15315 328 0.56694 81708 88498 36093 162
0.969 0.82432 00014 58683 98799 136 0.56612 41340 86469 79171 417
0.970 0.82488 57133 38450 05747 662 0.56529 95311 60354 31303 653
0.971 0.82545 06003 32571 52898 564 0.56447 43629 34754 78229 727
0.972 0.82601 46618 76161 45547 087 0.56364 86302 34839 35633 190
0.973 0.82657 78974 05158 34034 750 0.56282 23338 86340 66624 480
0.974 0.82714 03063 56326 70155 495 0.56199 54747 15554 99167 663
0.975 0.82770 18881 67257 63479 226 0.56116 80535 49341 43450 813
0.976 0.82826 26422 76369 37592 699 0.56034 00712 15121 09200 110
0.977 0.82882 25681 22907 86257 689 0.55951 15285 40876 22937 736
0.978 0.82938 16651 46947 29486 397 0.55868 24263 55149 45183 654
0.979 0.82993 99327 89390 69534 022 0.55785 27654 87042 87601 358
0.980 0.83049 73704 91970 46808 453 0.55702 25467 66217 30087 666
0.981 0.83105 39776 97248 95697 028 0.55619 17710 22891 37806 645
0.982 0.83160 97538 48619 00310 290 0.55536 04390 87840 78167 757
0.983 0.83216 46983 90304 50142 703 0.55452 85517 92397 37748 295
0.984 0.83271 88107 67360 95650 254 0.55369 61099 68448 39160 207
0.985 0.83327 20904 25676 03744 902 0.55286 31144 48435 57861 376
0.986 0.83382 45368 11970 13205 801 0.55202 95660 65354 38911 453
0.987 0.83437 61493 73796 90007 262 0.55119 54656 52753 13672 322
0.988 0.83492 69275 59543 82563 379 0.55036 08140 44732 16453 272
0.989 0.83547 68708 18432 76889 279 0.54952 56120 75943 01100 969
0.990 0.83602 59786 00520 51678 926 0.54868 98605 81587 57534 313
0.991 0.83657 42503 56699 33299 444 0.54785 35603 97417 28224 252
0.992 0.83712 16855 38697 50701 883 0.54701 67123 59732 24618 647
0.993 0.83766 82835 99079 90248 385 0.54617 93173 05380 43512 268
0.994 0.83821 40439 91248 50455 694 0.54534 13760 71756 83362 006
0.995 0.83875 89661 69442 96654 953 0.54450 28894 96802 60547 375
0.996 0.83930 30495 88741 15567 733 0.54366 38584 19004 25576 412
0.997 0.83984 62937 05059 69798 245 0.54282 42836 77392 79237 026
0.998 0.84038 86979 75154 52241 668 0.54198 41661 11542 88693 907
0.999 0.84093 02618 56621 40408 555 0.54114 35065 61572 03531 067
1.000 0.84147 09848 07896 50665 250 0.54030 23058 68139 71740 094
IIC-f)1
1 [c-p7 1
162 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.6 CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS
X sin x cos x
1.000 0.84147 09848 07896 50665 250 0.54030 23058 68139 71740 094
1. 001 0.84201 08662 88256 92390 268 0.53946 05648 72446 55654 214
1.002 0.84254 99057 57821 22046 578 0.53861 82844 16233 47828 237
1.003 0.84308 81026 77549 97169 747 0.53777 54653 41780 86864 465
1.004 0.84362 54565 09246 30271 873 0.53693 21084 91907 73184 669
1.005 0.84416 19667 15556 42661 273 0.53608 82147 09970 84748 188
1.006 0.84469 76327 59970 18177 851 0.53524 37848 39863 92716 262
1.007 0.84523 24541 06821 56844 116 0.53439 88197 26016 77062 668
1.008 0.84576 64302 21289 28431 774 0.53355 33202 13394 42130 747
1.009 0.84629 95605 69397 25943 853 0.53270 72871 47496 32136 904
1.010 0.84683 18446 18015 19012 310 0.53186 07213 74355 46620 673
1.011 0.84736 32818 34859 07211 051 0.53101 36237 40537 55841 426
1.012 0.84789 38716 88491 73284 331 0.53016 59950 93140 16121 808
1.013 0.84842 36136 48323 36290 466 0.52931 78362 79791 85137 984
1.014 0.84895 25071 84612 04660 810 0.52846 91481 48651 37156 798
1.015 0.84948 05517 68464 29173 940 0.52761 99315 48406 78219 896
1.016 0.85000 77468 71835 55845 003 0.52677 01873 28274 61274 932
1.017 0.85053 40919 67530 78730 164 0.52591 99163 37999 01253 921
1.018 0.85105 95865 29204 92646 111 0.52506 91194 27850 90098 832
1.019 0.85158 42300 31363 45804 549 0.52421 77974 48627 11734 503
1.020 0.85210 80219 49362 92361 655 0.52336 59512 51649 56988 961
1.021 0.85263 09617 59411 44882 415 0.52251 35816 88764 38461 245
1.022 0.85315 30489 38569 26719 808 0.52166 06896 12341 05336 792
1.023 0.85367 42829 64749 24308 778 0.52080 72758 75271 58150 502
1.024 0.85419 46633 16717 39374 945 0.51995 33413 30969 63497 542
1.025 0.85471 41894 74093 41057 997 0.51909 88868 33369 68691 985
1.026 0.85523 28609 17351 17949 715 0.51824 39132 36926 16373 373
1.027 0.85575 06771 27819 30046 586 0.51738 84213 96612 59061 276
1.028 0.85626 76375 87681 60616 931 0.51653 24121 67920 73657 956
1.029 0.85678 37417 79977 67982 525 0.51567 58864 06859 75899 186
1.030 0.85729 89891 88603 37214 627 0.51481 88449 69955 34753 350
1.031 0.85781 33792 98311 31744 398 0.51396 12887 14248 86768 878
1.032 0.85832 69115 94711 44887 626 0.51310 32184 97296 50370 116
1.033 0.85883 95855 64271 51283 734 0.51224 46351 77168 40101 715
1.034 0.85935 14006 94317 58248 998 0.51138 55396 12447 80821 625
1.035 0.85986 23564 73034 57043 938 0.51052 59326 62230 21842 776
1.036 0.86037 24523 89466 7.4054 819 0.50966 58151 86122 51023 535
1.037 0.86088 16879 33518 21889 224 0.50880 51880 44242 08807 028
1.038 0.86139 00625 95953 50385 634 0.50794 40520 97216 02209 404
1.039 0.86189 75758 68397 97536 975 0.50708 24082 06180 18757 138
1.040 0.86240 42272 43338 40328 079 0.50622 02572 32778 40373 447
1.041 0.86291 00162 14123 45486 997 0.50535 76000 39161 57213 919
1.042 0.86341 49422 74964 20150 131 0.50449 44374 87986 81451 427
1.043 0.86391 90049 20934 62441 124 0.50363 07704 42416 61010 426
1.044 0.86442 22036 47972 11963 456 0.50276 65997 66117 93250 711
1.045 0.86492 45379 52878 00206 699 0.50190 19263 23261 38600 728
1.046 0.86542 60073 33318 00866 385 0.50103 67509 78520 34140 520
1.047 0.86592 66112 87822 80077 424 0.50017 10745 97070 07134 396
1.048 0.86642 63493 15788 46561 037 0.49930 48980 44586 88513 415
1.049 0.86692 52209 17477 01685 140 0.49843 82221 87247 26307 756
1.050 0.86742 32255 94016 89438 141 0.49757 10478 91726 99029 085
[(-;)I1 c-8)7
II 1
7
ELEMENTARY TRANSCENDENTAL FUNCTIONS 163
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
5 sin z cos x
1.050 0.86742 32255 94016 89438 141 0.49757 10478 91726 99029 085
1.051 0.86792 03628 47403 46316 092 0.49670 33760 25200 29002 975
1.052 0.86841 66321 80499 51123 146 0.49583 52074 55338 95651 499
1.053 0.86891 20330 97035 74685 276 0.49496 65430 50311 48726 051
1.054 0.86940 65651 01611 29477 198 0.49409 73836 78782 21490 510
1.055 0.86990 02276 99694 19162 460 0.49322 77302 09910 43854 806
1.056 0.87039 30203 97621 88046 624 0.49235 75835 13349 55459 008
1.057 0.87088 49427 02601 70443 529 0.49148 69444 59246 18707 979
1.058 0.87137 59941 22711 39954 543 0.49061 58139 18239 31756 732
1.059 0.87186 61741 66899 58660 794 0.48974 41927 61459 41446 534
1.060 0.87235 54823 44986 26228 295 0.48887 20818 60527 56191 864
1.061 0.87284 39181 67663 28925 947 0.48799 94820 87554 58818 317
1.062 0.87333 14811 46494 88556 345 0.48712 63943 15140 19351 528
1.063 0.87381 81707 93918 11299 356 0.48625 28194 16372 07757 202
1.064 0.87430 39866 23243 36468 402 0.48537 87582 64825 06632 362
1.065 0.87478 89281 48654 85179 424 0.48450 42117 34560 23847 867
1.066 0.87527 29948 85211 08932 453 0.48362 91807 00124 05142 311
1.067 0.87575 61863 48845 38105 753 0.48275 36660 36547 46667 387
1.068 0.87623 85020 56366 30362 492 0.48187 76686 19345 07484 800
1.069 0.87671 99415 25458 18969 874 0.48100 11893 24514 22014 811
1.070 0.87720 05042 74681 61030 706 0.48012 42290 28534 12436 509
1.071 0.87768 01898 23471 85627 336 0.47924 67886 08365 01039 904
1.072 0.87815 89976 92149 41877 919 0.47836 88689 41447 22529 904
1.073 0.87863 69274 01900 46904 963 0.47749 04709 05700 36282 289
1.074 0.87911 39784 74797 33716 111 0.47661 15953 79522 38551 762
1.075 0.87959 01504 33788 98997 101 0.47573 22432 41788 74632 160
1.076 0.88006 54428 02703 50816 869 0.47485 24153 71851 50968 911
1.077 0.88053 98551 06248 56244 731 0.47397 21126 49538 47223 840
1.078 0.88101 33868 70011 88879 619 0.47309 13359 55152 28292 396
1.079 0.88148 60376 20461 76291 297 0.47221 00861 69469 56273 392
1.080 0.88195 78068 84947 47373 533 0.47132 83641 73740 02391 353
1.081 0.88242 86941 91699 79609 169 0.47044 61708 49685 58871 547
1.082 0.88289 86990 69831 46247 031 0.46956 35070 79499 50767 810
1.083 0.88336 78210 49337 63390 660 0.46868 03737 45845 47743 217
1.084 0.88383 60596 61096 36998 790 0.46779 67717 31856 75803 727
1.085 0.88430 34144 36869 09797 534 0.46691 27019 21135 28984 862
1.086 Oi88476 98849 09301 08104 243 0.46602 81651 97750 80991 522
1.087 0.88523 54706 11921 88562 972 0.46514 31624 46239 96791 014
1.088 0.88570 01710 79145 84791 522 0.46425 76945 51605 44159 401
1.089 0.88616 39858 46272 53940 000 0.46337 17623 99315 05181 235
1.090 0.88662 69144 49487 23160 860 0.46248 53668 75300 87702 790
1.091 0.88708 89564 25861 35990 371 0.46159 85088 65958 36738 852
1.092 0.88755 01113 13352 98641 470 0.46071 11892 58145 45833 190
1.093 0.88801 03786 50807 26207 951 0145982 34089 39181 68372 764
1.094 0.88846 97579 77956 88779 948 0.45893 51687 96847 28855 783
1.095 0.88892 82488 35422 57470 660 0.45804 64697 19382 34113 686
1.096 0.88938 58507 64713 50354 274 0.45715 73125 95485 84487 142
1.097 0.88984 25633 08227 78315 047 0.45626 76983 14314 84956 158
1.098 0.89029 83860 09252 90807 488 0.45537 76277 65483 56224 382
1.099 0.89075 33184 11966 21527 609 0.45448 71018 39062 45757 688
0.89120 73600 61435 33995 180 0.45359 61214 25577 38777 137
c(-;I1
1.100
1 C-78)6
c 1
164 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.6 CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS
2 sin x cos x
1.100 0.89120 73600 61435 33995 180 0.45359 61214 25577 38777 137
1.101 0.89166 05105 03618 67046 971 0.45270 46874 16008 69206 400
1.102 0.89211 27692 85365 80240 901 0.45181 28007 01790 30573 730
1.103 0.89256 41359 54417 99171 080 0.45092 04621 74808 86868 576
1.104 0.89301 46100 59408 60693 678 0.45002 76727 27402 83352 928
1.105 0.89346 41911 49863 58063 585 0.44913 44332 52361 57327 478
1.106 0.89391 28787 76201 85981 812 0.44824 07446 42924 48852 689
1.107 0.89436 06724 89735 85553 594 0.44734 66077 92780 11424 866
1.108 0.89480 75718 42671 89157 146 0.44645 20235 96065 22607 305
1.109 0.89525 35763 88110 65223 027 0.44555 69929 47363 94616 628
1.110 0.89569 86856 80047 62924 063 0.44466 15167 41706 84864 374
1.111 0.89614 28992 73373 56775 801 0.44376 55958 74570 06453 951
1.112 0.89658 62167 23874 91147 427 0.44286 92312 41874 38633 030
1.113 0.89702 86375 88234 24683 120 0.44197 24237 39984 37201 474
1.114 0.89747 01614 24030 74633 785 0.44107 51742 65707 44874 890
1.115 0.89791 07877 89740 61099 138 0.44017 74837 16293 01603 891
1.116 0.89835 05162 44737 51180 079 0.43927 93529 89431 54849 166
1.117 0.89878 93463 49293 03041 321 0.43838 07829 83253 69812 438
1.118 0.89922 72776 64577 09884 230 0.43748 17745 96329 39623 410
1.119 0.89966 43097 52658 43829 826 0.43658 23287 27666 95482 777
1.120 0.90010 04421 76504 99711 910 0.43568 24462 76712 16761 399
1.121 0.90053 56744 99984 38780 263 0.43478 21281 43347 41055 736
1.122 0.90097 00062 87864 32313 880 0.43388 13752 27890 74199 612
1.123 0.90140 34371 05813 05144 201 0.43298 01884 31095 00232 420
1.124 0.90183 59665 20399 79088 276 0.43207 85686 54146 91323 845
1.125 0.90226 75940 99095 16291 842 0.43117 65167 98666 17655 197
1.126 0.90269 83194 10271 62482 258 0.43027 40337 66704 57257 452
1.127 0.90312 81420 23203 90131 256 0.42937 11204 60745 05806 078
1.128 0.90355 70615 08069 41527 464 0.42846 77777 83700 86372 749
1.129 0.90398 50774 35948 71758 658 0142756 40066 38914 59134 030
1.130 0.90441 21893 78825 91603 708 0.42665 98079 30157 31037 122
1.131 0.90483 83969 09589 10334 160 0.42575 51825 61627 65422 763
1.132 0.90526 36996 02030 78425 425 0.42485 01314 37950 91605 376
1.133 0.90568 80970 30848 30177 523 0.42394 46554 64178 14410 540
1.134 0.90611 15887 71644 26245 348 0.42303 87555 45785 23669 902
1.135 0.90653 41744 00926 96078 401 0.42213 24325 88672 03673 585
1.136 0.90695 58534 96110 80269 960 0.42122 56874 99161 42580 219
1.137 0.90737 66256 35516 72815 632 0.42031 85211 83998 41784 656
1.138 0.90779 64903 98372 63281 260 0.41941 09345 50349 25243 478
1.139 0.90821 54473 64813 78880 126 0.41850 29285 05800 48758 379
1.140 0.90863 34961 15883 26459 422 0.41759 45039 58358 09217 519
1.141 0.90905 06362 33532 34395 940 0.41668 56618 16446 53794 933
1.142 0.90946 68673 00620 94400 939 0.41577 64029 88907 89108 094
1.143 0.90988 21889 00918 03234 153 0.41486 67283 85000 90333 707
1.144 0.91029 66006 19102 04326 885 0.41395 66389 14400 10281 852
1.145 0.91071 01020 40761 29314 164 0.41304 61354 87194 88428 529
1.146 0.91112 26927 52394 39475 912 0.41213 52190 13888 59906 732
1.147 0.91153 43723 41410 67087 073 0.41122 38904 05397 64456 120
1.148 0.91194 51403 96130 56676 684 0.41031 21505 73050 55331 381
1.149 0.91235 49965 05786 06195 821 0.40940 00004 28587 08169 395
1.150 0.91276 39402 60521 08094 403 0.40848 74408 84157 29815 258
ELEMENTARY TRANSCENDENTAL FUNCTIONS 165
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
5 sin x cos x
1.150 0.91276 39402 60521 08094 403 0.40848 74408 84157 29815 258
1.151 0.91317 19712 51391 90306 792 0.40757 44728 52320 67107 284
1.152 0.91357 90890 70367 57146 165 0.40666 10972 46045 15621 071
1.153 Oi91398 52933 10330 30107 602 0.40574 73149 78706 28372 706
1.154 0.91439 05835 65075 88579 865 0.40483 31269 64086 24481 224
1.155 0.91479 49594 29314 10465 816 0.40391 85341 16372 97790 397
1.156 0.91519 84204 98669 12711 431 0.40300 35373 50159 25449 945
1.157 0.91560 09663 69679 91743 383 0.40208 81375 80441 76456 266
1.158 0.91600 25966 39800 63815 143 0.40117 23357 22620 20152 779
1.159 0.91640 33109 07401 05261 556 0.40025 61326 92496 34689 958
1.160 0.91680 31087 71766 92661 866 0.39933 95294 06273 15445 164
1.161 0.91720 19898 33100 42911 136 0.39842 25267 80553 83402 355
1.162 0.91759 99536 92520 53200 023 0.39750 51257 32340 93491 775
1.163 0.91799 69999 52063 40902 883 0.39658 73271 79035 42889 706
1.164 0.91839 31282 14682 83374 147 0.39566 91320 38435 79278 377
1.165 0.91878 83380 84250 57652 941 0.39475 05412 28737 09066 125
1.166 0.91918 26291 65556 80075 906 0.39383 15556 68530 05567 898
1.167 0.91957 60010 64310 45798 178 0.39291 21762 76800 17146 187
1.168 0.91996 84533 87139 68222 492 0.39199 24039 72926 75312 486
1.169 0.92035 99857 41592 18336 360 0.39107 22396 76682 02789 366
1.170 0.92075 05977 36135 63957 301 0.39015 16843 08230 21533 266
1.171 0.92114 02889 80158 08886 071 0.38923 07387 88126 60718 072
1.172 0.92152 90590 83968 31967 851 0.38830 94040 37316 64679 599
1.173 0.92191 69076 58796 26061 369 0.38738 76809 77135 00821 054
1.174 0.92230 38343 16793 36915 902 0.38646 55705 29304 67479 575
1.175 0.92268 98386 71033 01956 127 0.38554 30736 15936 01753 942
1.176 0.92307 49203 35513 88974 783 0.38462 01911 59525 87293 547
1.177 0.92345 90789 25145 34733 097 0.38369 69240 82956 62048 718
1.178 0.92384 23140 55777 83468 944 0.38277 32733 09495 25982 487
1.179 0.92422 46253 44173 25312 701 0.38184 92397 62792 48743 902
1.180 0.92460 60124 08020 34610 754 0.38092 48243 66881 77302 960
1.181 0.92498 64748 65932 08156 619 0.38000 00280 46178 43547 271
1.182 0.92536 60123 37446 03329 642 0.37907 48517 25478 71840 534
1.183 0.92574 46244 43024 76141 242 0.37814 92963 29958 86542 917
1.184 0.92612 23108 04056 19188 645 0.37722 33627 85174 19493 444
1.185 0.92649 90710 42853 99516 095 0.37629 70520 17058 17454 471
1.186 0.92687 49047 82657 96383 480 0.37537 03649 51921 49518 342
1.187 0.92724 98116 47634 38942 352 0.37444 33025 16451 14476 334
1.188 0.92762 37912 62876 43819 290 0.37351 58656 37709 48149 962
1.189 0.92799 68432 54404 52606 588 0.37258 80552 43133 30684 752
1.190 0.92836 89672 49166 69260 202 0.37165 98722 60532 93806 568
1.191 0.92874 01628 75038 97404 950 0.37073 13176 18091 28040 589
1.192 0.92911 04297 60825 77546 899 0.36980 23922 44362 89893 026
1.193 0.92947 97675 36260 24192 928 0.36887 30970 68273 08995 672
1.194 0.92984 81758 32004 62877 403 0.36794 34330 19116 95213 382
1.195 0.93021 56542 79650 67095 956 0.36701 34010 26558 45714 570
1.196 0.93058 22025 11719 95146 303 0.36608 30020 20629 52004 819
1.197 0.93094 78201 61664 26876 083 0.36515 22369 31729 06923 698
1.198 0.93131 25068 63866 00337 679 0.36422 11066 90622 11604 876
1.199 0.93167 62622 53638 48349 974 0.36328 96122 28438 82399 631
1.200 0.93203 90859 67226 34967 013 0.36235 77544 76673 57763 837
[C-f)1
1 p-y51
L ’ J
166 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.6 CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS
X sin x cos x
1.200 0.93203 90859 67226 34967 013 0.36235 77544 76673 57763 837
1.201 0.93240 09776 41805 91853 542 0.36142 55343 67184 05108 539
1.202 0.93276 19369 15485 54567 367 0.36049 29528 32190 27614 189
1.203 0.93312 19634 27305 98748 519 0.35956 00108 04273 71008 651
1.204 0.93348 10568 17240 76215 175 0.35862 67092 16376 30309 065
1.205 0.93383 92167 26196 50966 302 0.35769 30490 01799 56527 660
1.206 0.93419 64427 96013 35090 992 0.35675 90310 94203 63341 607
1.207 Oi93455 27346 69465 24584 444 0.35582 46564 27606 33727 018
1.208 0.93490 80919 90260 35070 567 0.35488 99259 36382 26557 166
1.209 0.93526 25144 03041 37431 162 0.35395 48405 55261 83165 039
1.210 0.93561 60015 53385 93341 646 0.35301 94012 19330 33870 301
1.211 0.93596 85530 87806 90713 291 0.35208 36088 64027 04470 775
1.212 0.93632 01686 53752 79041 926 0.35114 74644 25144 22698 521
1.213 0.93667 08478 99608 04663 095 0.35021 09688 38826 24640 616
1.214 0.93702 05904 74693 45913 598 0.34927 41230 41568 61124 730
1.215 0.93736 93960 29266 48199 416 0.34833 69279 70217 04069 578
1.216 0.93771 72642 14521 58969 959 0.34739 93845 61966 52800 358
1.217 0.93806 41946 82590 62598 617 0.34646 14937 54360 40329 260
1.218 0.93841 01870 86543 15169 574 0.34552 32564 85289 39601 140
1.219 0.93875 52410 80386 79170 848 0.34458 46736 92990 69704 455
1.220 0.93909 93563 19067 58093 524 0.34364 57463 16047 02047 552
1.221 0.93944 25324 58470 30937 151 0.34270 64752 93385 66500 405
1.222 0.93978 47691 55418 86621 257 0.34176 68615 64277 57501 890
1.223 0.94012 60660 67676 58302 957 0.34082 69060 68336 40132 702
1.224 0.94046 64228 53946 57600 622 0.33988 66097 45517 56153 996
1.225 0.94080 58391 73872 08723 559 0.33894 59735 36117 30011 855
1.226 0.94114 43146 88036 82507 685 0.33800 49983 80771 74807 668
1.227 0.94148 18490 57965 30357 157 0.33706 36852 20455 98234 533
1.228 0.94181 84419 46123 18091 912 0.33612 20349 96483 08479 750
1.229 0.94215 40930 15917 59701 104 0.33518 00486 50503 20093 523
1.230 0.94248 88019 31697 51002 382 0.33423 77271 24502 59823 955
1.231 0.94282 25683 58754 03206 998 0.33329 50713 60802 72418 427
1.232 0.94315 53919 63320 76390 684 0.33235 20823 02059 26391 462
1.233 0.94348 72724 12574 12870 299 0.33140 87608 91261 19759 164
1.234 0.94381 82093 74633 70486 175 0.33046 51080 71729 85740 328
1.235 0.94414 82025 18562 55790 164 0.32952 11247 87117 98424 316
1.236 0.94447 72515 14367 57139 322 0.32857 68119 81408 78405 786
1.237 0.94480 53560 32999 77695 223 0.32763 21705 98914 98386 387
1.238 0.94513 25157 46354 68328 851 0.32668 72015 84277 88743 487
1.239 0.94545 87303 27272 60431 046 0.32574 19058 82466 43066 054
1.240 0.94578 39994 49538 98628 471 0.32479 62844 38776 23657 769
1.241 0.94610 83227 87884 73405 063 0.32385 03381 98828 67007 475
1.242 0.94643 17000 17986 53628 942 0.32290 40681 08569 89227 042
1.243 0.94675 41308 16467 18984 738 0.32195 74751 14269 91456 764
1.244 0.94707 56148 60895 92311 309 0.32101 05601 62521 65238 364
1.245 0.94739 61518 29788 71844 815 0.32006 33242 00239 97855 712
1.246 0.94771 57414 02608 63367 118 0.31911 57681 74660 77643 341
1.247 0.94803 43832 59766 12259 472 0.31816 78930 33339 99262 871
1.248 0.94835 20779 82619 35461 479 0.31721 96997 24152 68947 423
1.249 0.94866 88225 53474 53335 262 0.31627 11891 95292 09714 116
1.250 0.94898 46193 55586 21434 849 0.31532 23623 95268 66544 754
[c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 167
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin x cos x
1.250 0.94898 46193 55586 21434 849 0.31532 23623 95268 66544 754
1.251 0.94929 94671 73157 62180 713 0.31437 32202 72909 11534 791
1.252 0.94961 33656 91340 96439 444 0.31342 37637 77355 49010 665
1.253 0.94992 63145 96237 75008 528 0.31247 39938 58064 20615 601
1.254 0.95023 83135 74899 10006 196 0.31152 39114 64805 10363 979
1.255 0.95054 93623 15326 06166 303 0.31057 35175 47660 49664 355
1.256 0.95085 94605 06469 92038 225 0.30962 28130 57024 22311 242
1. 257 0.95116 86078 38232 51091 729 0.30867 17989 43600 69445 729
1.258 0.95147 68040 01466 52726 783 0.30772 04761 58403 94485 052
1.259 0.95178 40486 87975 83188 287 0.30676 88456 52756 68021 196
1.260 0.95209 03415 90515 76385 682 0.30581 69083 78289 32688 634
1.261 0.95239 56824 02793 44617 416 0.30486 46652 86939 08001 291
1.262 0.95270 00708 19468 09200 227 0.30391 21173 30948 95158 833
1.263 0.95300 35065 36151 31003 222 0.30295 92654 62866 81822 373
1.264 0.95330 59892 49407 40886 709 0.30200 61106 35544 46859 693
1.265 0.95360 75186 56753 70045 767 0.30105 26538 02136 65060 070
1.266 0.95390 80944 56660 80258 512 0.30009 88959 16100 11818 814
1.267 0.95420 77163 48552 94039 032 0.29914 48379 31192 67791 595
1.268 0.95450 63840 32808 24694 963 0.29819 04808 01472 23518 675
1.269 0.95480 40972 10759 06289 671 0.29723 58254 81295 84019 121
1.270 0.95510 08555 84692 23509 018 0.29628 08729 25318 73355 114
1.271 0.95539 66588 57849 41432 673 0.29532 56240 88493 39166 425
1.272 0.95569 15067 34427 35209 944 0.29437 00799 26068 57175 182
1.273 0.95598 53989 19578 19640 104 0.29341 42413 93588 35661 000
1.274 0.95627 83351 19409 78657 170 0.29245 81094 46891 19906 579
1.275 0.95657 03150 40985 94719 118 0.29150 16850 42108 96613 869
1.276 0.95686 13383 92326 78101 497 0.29054 49691 35665 98290 890
1.277 0.95715 14048 82408 96095 419 0.28958 79626 84278 07609 308
1.278 0.95744 05142 21166 02109 886 0.28863 06666 44951 61732 860
1.279 0.95772 86661 19488 64678 437 0.28767 30819 74982 56616 726
1.280 0.95801 58602 89224 96370 075 0.28671 52096 31955 51277 939
1.281 0.95830 20964 43180 82604 453 0.28575 70505 73742 72036 934
1.282 0.95858 73742 95120 10371 286 0.28479 86057 58503 16730 332
1.283 0.95887 16935 59764 96853 962 0.28383 98761 44681 58895 050
1.284 0.95915 50539 52796 17957 320 0.28288 08626 91007 51923 831
1.285 0.95943 74551 90853 36739 577 0.28192 15663 56494 33192 303
1.286 0.95971 88969 91535 31748 357 0.28096 19881 00438 28157 651
1.287 0.95999 93790 73400 25260 814 0.28000 21288 82417 54428 993
1.288 0.96027 89011 55966 11427 805 0.27904 19896 62291 25809 577
1.289 0.96055 74629 59710 84322 094 Oi27808 15714 00198 56310 871
1.290 0.96083 50642 06072 65890 556 0.27712 08750 56557 64138 661
1.291 0.96111 17046 17450 33810 354 0.27615 99015 92064 75651 234
1.292 0.96138 73839 17203 49249 056 0.27519 86519 67693 29289 769
1.293 0.96166 21018 29652 84528 675 3.27423 71271 44692 79480 997
1.294 0.96193 58580 80080 50693 590 0.27327 53280 84588 00512 263
1.295 0.96220 86523 94730 24982 339 0.27231 32557 49177 90379 053
1.296 0.96248 04845 00807 78203 231 0.27135 09111 00534 74605 108
1.297 0.96275 13541 26481 02013 782 0.27038 82951 01003 10035 206
1.298 0.96302 12610 00880 36103 915 0.26942 54087 13198 88600 711
1.299 0.96329 02048 54098 95282 920 0.26846 22529 00008 41057 992
1.300 0.96355 81854 17192 96470 135 0.26749 88286 24587 40699 798
[c-y1
/
L 1
C-7834
168 ELEMENTARY TRANSCENDENTAL FUNCTIONS
x sin 2: CO8 2
1.300 0.96355 81854 17192 96470 135 0.26749 88286 24587 40699 798
1.301 0.96382 52024 22181 85589 331 0.26653 51368 50360 07039 695
1.302 0.96409 12556 02048 64366 761 0.26557 11785 41018 09469 650
1.303 0.96435 63446 90740 17032 855 0.26460 69546 60519 70890 877
1.304 0796462 04694 23167 36927 537 0.26364 24661 73088 71318 016
1.305 0.96488 36295 35205 53009 126 0.26267 77140 43213 51456 761
1.306 0.96514 58247 63694 56266 806 0.26171 26992 35646 16255 031
1.307 0.96540 70548 46439 26036 635 0.26074 74227 15401 38427 774
1.308 0.96566 73195 22209 56221 061 0.25978 18854 47755 61955 494
1.309 0.96592 66185 30740 81411 924 0.25881 60883 98246 05556 626
1.310 0.96618 49516 12734 02916 926 0.25785 00325 32669 66133 818
1.311 0.96644 23185 09856 14689 520 0.25688 37188 17082 22194 242
1.312 0.96669 87189 64740 29162 218 0.25591 71482 17797 37244 030
1.313 0.96695 41527 20986 02983 276 0.25495 03217 01385 63156 911
1.314 0.96720 86195 23159 62656 736 0.25398 32402 34673 43517 173
1.315 0.96746 21191 16794 3gO85 794 0.25301 59047 84742 16937 022
1.316 0.96771 46512 48390 48019 478 0.25204 83163 18927 20348 457
1.317 0.96796 62156 65416 05402 607 0.25108 04758 04816 92269 738
1.318 0.96821 68121 16306 62628 991 0.25011 23842 10251 76046 556
1.319 0.96846 64403 50465 76697 879 0.24914 40425 03323 23067 996
1.320 0.96871 51001 18265 26273 590 0.24817 54516 52372 95957 398
1.321 0.96896 27911 71045 36648 340 0.24720 66126 25991 71738 199
0.96920 95132 61115 04608 211 0.24623 75263 93018 44974 865
:* 2: 0.96945 52661 41752 23202 252 0.24526 81939 22539 30889 004
1:324 0.96970 00495 67204 06414 685 0.24429 86161 83886 68450 760
1.325 0.96994 38632 92687 13740 188 0.24332 87941 46638 23445 582
1.326 0.97018 67070 74387 74662 236 0.24235 87287 80615 91516 463
1.327 0.97042 85806 69462 13034 465 0.24138 84210 55885 01181 759
1.328 0.97066 94838 36036 71365 051 0.24041 78719 42753 16828 662
1.329 0.97090 94163 33208 35004 060 0.23944 70824 11769 41682 448
1.330 0.97114 83779 21044 56233 768 0.23847 60534 33723 20751 578
1.331 0.97138 63683 60583 78261 900 0.23750 47859 79643 43748 768
1.332 0.97162 33874 13835 59117 786 0.23653 32810 20797 47988 097
1.333 0.97185 94348 43780 95451 405 0.23556 15395 28690 21258 288
1.334 0.97209 45104 14372 46235 282 0.23458 95624 75063 04672 221
1.335 0.97232 86138 90534 56369 230 0.23361 73508 31892 95492 805
1.336 0.97256 17450 38163 80187 900 0.23264 49055 71391 49935 286
1.337 0.97279 39036 24129 04871 129 0.23167 22276 66003 85946 099
1.338 0.97302 50894 16271 73757 0146 0.23069 93180 88407 85958 358
1.339 0.97325 53021 83406 09557 931 0.22972 61778 11512 99624 085
1.340 0.97348 45416 95319 37478 787 0.22875 28078 08459 46523 264
1.341 0.97371 28077 22772 08238 616 0.22777 92090 52617 18849 831
1.342 0.97394 01000 37498 N994 365 0.22680 53825 17584 84074 691
1.343 0.97416 64184 12205 46167 522 0.22583 13291 77188 87585 859
1.344 0.97439 17626 20575 48173 349 0.22485 70500 05482 55305 819
1.345 0.97461 61324 37264 08052 713 0.22388 25459 76744 96286 212
1.346 0.97483 95276 37901 46006 501 0.22290 78180 65480 05279 929
1.347 0.97506 19479 99092 43832 603 0.22193 28672 46415 65290 729
1.348 0.97528 33932 98416 67265 423 0.22095 76944 94502 50100 463
1.349 0.97550 38633 14428 88217 916 0.21998 23007 84913 26774 007
1.350 0.97572 33578 26659 06926 111 0.21900 66870 93041 58142 002
[ 1
(-;I1 c-p3
II 1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 169
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
x sin x cos x
1.350 0.97572 33578 26659 06926 111 0.21900 66870 93041 58142 002
1.351 0.97594 18766 15612 73996 110 0.21803 08543 94501 05261 504
1.352 0.97615 94194 62771 12353 536 0.21705 48036 65124 29854 627
1.353 0.97637 59861 50591 39095 407 0.21607 85358 80961 96725 291
1.354 0.97659 15764 62506 87244 418 0.21510 20520 18281 76154 163
1.355 0.97680 61901 82927 27405 609 0.21412 53530 53567 46271 899
1.356 0.97701 98270 97238 89325 386 0.21314 84399 63517 95410 772
1.357 0.97723 24869 91804 83352 894 0.21217 13137 25046 24434 790
1.358 0.97744 41696 53965 21803 706 0.21119 39753 15278 49048 406
1.359 0.97765 48748 72037 40225 805 0.21021 64257 11553 02083 908
1.360 0.97786 46024 35316 18567 849 0.20923 86658 91419 35767 598
1.361 0.97807 33521 34074 02249 690 0.20826 06968 32637 23964 842
1.362 0.97828 11237 59561 23135 125 0.20728 25195 13175 64404 112
1.363 0.97848 79171 04006 20406 864 0.20630 41349 11211 80880 089
1.364 0.97869 37319 60615 61343 685 0.20532 55440 05130 25435 952
1.365 0.97889 85681 23574 61999 774 0.20434 67477 73521 80524 932
1.366 0.97910 24253 88047 07786 196 0.20336 77471 95182 61151 240
1.367 0.97930 53035 50175 73954 516 0.20238 85432 49113 16990 457
1.368 0.97950 72024 07082 45982 521 0.20140 91369 14517 34489 495
1.369 0.97970 81217 56868 39862 027 0.20042 95291 70801 38946 217
1.370 0.97990 80613 98614 22288 769 0.19944 97209 97572 96568 820
1.371 0.98010 70211 32380 30754 328 0.19846 97133 74640 16515 079
1.372 0.98030 50007 59206 93540 094 0.19748 95072 82010 52911 545
1.373 0.98050 20000 81114 49613 233 0.19650 91036 99890 06852 798
1.374 0.98069 80189 01103 68424 652 0.19552 85036 08682 28380 853
1.375 0.98089 30570 23155 69608 920 0.19454 77079 88987 18444 822
1.376 0.98108 71142 52232 42586 155 0.19356 67178 21600 30840 918
1.377 0.98128 01903 94276 66065 826 0.19258 55340 87511 74132 912
1.378 0.98147 22852 56212 27452 479 0.19160 41577 67905 13553 129
1.379 0.98166 33986 45944 42153 343 0.19062 25898 44156 72884 094
1.380 0.98185 35303 72359 72787 813 0.18964 08312 97834 36320 915
1.381 0.98204 26802 45326 48298 791 0.18865 88831 10696 50314 508
1.382 0.98223 08480 75694 82965 850 0.18767 67462 64691 25395 757
1.383 0.98241 80336 75296 95320 221 0.18669 44217 41955 37980 715
1.384 0.98260 42368 56947 26961 571 0.18571 19105 24813 32156 930
1.385 0.98278 94574 34442 61276 561 0.18472 92135 95776 21451 016
1.386 0.98297 36952 22562 42059 162 0.18374 63119 37540 90577 542
1,387 0.98315 69500 37068 92032 708 0.18276 32665 32988 97169 360
1.388 0.98333 92216 94707 31273 673 0.18178 00183 65185 73489 451
1.389 0.98352 05100 13205 95537 148 0.18079 65884 17379 28124 404
1.390 0.98370 08148 11276 54484 004 0.17981 29776 72999 47659 616
1.391 0.98388 01359 08614 29809 722 0.17882 91871 15656 98336 311
1.392 0.98405 84731 25898 13274 870 0.17784 52177 29142 27690 484
1.393 0.98423 58262 84790 84637 207 0.17686 10704 97424 66173 860
1.394 0.98441 21952 07939 29485 405 0.17587 67464 04651 28756 976
1. 39-5 0.98458 75797 18974 56974 360 0.17489 22464 35146 16514 467
1.396 0.98476 19796 42512 17462 083 0.17390 75715 73409 18192 681
1.397 0.98493 53948 04152 20048 145 0.17292 27228 04115 11759 690
1.398 0.98510 78250 30479 50013 670 0.17193 77011 12112 65937 830
1.399 0.98527 92701 49063 86162 846 0.17095 25074 82423 41718 833
[ (-/I11
1.400 0.98544 97299 88460 18065 947 0.16996 71429 00240 93861 675
II(-;j31
170 ELEMENTARY TRANSCENDENTAL FUNCTIONS
1: sin z cos x
1.400 0.98544 97299 88460 18065 947 0.16996 71429 00240 93861 675
1.401 0.98561 92043 78208 63203 840 0.16898 16083 50929 72373 233
1.402 0.98578 76931 48834 84013 966 0.16799 59048 20024 23971 842
1.403 0.98595 51961 31850 04837 776 0.16701 00332 93227 93533 854
1.404 0.98612 17131 59751 28769 609 0.16602 39947 56412 25523 303
1.405 0.98628 72440 66021 54406 982 0.16503 77901 95615 65404 770
1.406 0.98645 17886 85129 92502 294 0.16405 14205 97042 61039 544
1.407 0.98661 53468 52531 82515 912 0.16306 48869 47062 64065 184
1.408 0.98677 79184 04669 09070 631 0.16207 81902 32209 31258 571
1.409 0.98693 95031 78970 18307 486 0.16109 13314 39179 25882 568
1.410 0.98710 01010 13850 34142 909 0.16010 43115 54831 19016 356
1.411 0.98725 97117 48711 74427 198 0.15911 71315 66184 90869 577
1.412 0.98741 83352 23943 67004 304 0.15812 97924 60420 32080 359
1.413 0.98757 59712 80922 65672 895 0.15714 22952 24876 44997 336
1.414 0.98773 26197 62012 66048 706 0.15615 46408 47050 44945 751
1.415 0.98788 82805 10565 21328 142 0.15516 68303 14596 61477 752
1.416 0.98804 29533 70919 57953 120 0.15417 88646 15325 39606 967
1.417 0.98819 66381 88402 91177 144 0.15319 07447 37202 41027 471
1.418 0.98834 93348 09330 40532 586 0.15220 24716 68347 45317 231
1.419 0.98850 10430 81005 45199 170 0.15121 40463 97033 51126 135
1.420 0.98865 17628 51719 79273 627 0.15022 54699 11685 77348 698
1.421 0.98880 14939 70753 66940 521 0.14923 67432 00880 64281 559
1.422 0.98895 02362 88375 97544 222 Oil4824 78672 53344 74765 840
1.423 0.98909 79896 55844 40562 021 0.14725 88430 57953 95314 499
1.424 0.98924 47539 25405 60478 351 0.14626 96716 03732 37224 747
1.425 0.98939 05289 50295 31560 129 0.14528 03538 79851 37675 648
1.426 0.98953 53145 84738 52533 174 0.14429 08908 75628 60810 986
1.427 0.98967 91106 83949 61159 714 0.14330 12835 80526 98807 514
1.428 0.98982 19171 04132 48716 941 0.14231 15329 84153 72928 666
1.429 0.98996 37337 02480 74376 619 0.14132 16400 76259 34563 848
1.430 0.99010 45603 37177 79485 729 0.14033 16058 46736 66253 390
1.431 0.99024 43968 67397 01748 121 0.13934 14312 85619 82699 275
1.432 0.99038 32431 53301 89307 176 0.13835 11173 83083 31761 733
1.433 0.99052 10990 56046 14729 460 0.13736 06651 29440 95441 799
1.434 0.99065 79644 37773 88889 346 0.13637 00755 15144 90849 940
1.435 0.99079 38391 61619 74754 605 0.13537 93495 30784 71160 849
1.436 0.99092 87230 91709 01072 941 0.13438 84881 67086 26554 495
1.437 0.99106 26160 93157 75959 459 0.13339 74924 14910 85143 546
1.438 0.99119 55180 32073 00385 060 0.13240 63632 65254 13887 244
1.439 0.99132 74287 75552 81565 735 0.13141 51017 09245 19491 852
1.440 0.99145 83481 91686 46252 760 0.13042 37087 38145 49297 752
1.441 0.99158 82761 49554 53923 766 0.12943 21853 43347 92153 306
1.442 0.99171 72125 19229 09874 676 0.12844 05325 16375 79275 576
1.443 0.99184 51571 71773 78212 505 0.12744 87512 48881 85098 002
1.444 0.99197 21099 79243 94748 990 0.12645 68425 32647 28105 135
1.445 0.99209 80708 14686 79795 055 0.12546 48073 59580 71654 525
1.446 0.99222 30395 52141 50856 088 0.12447 26467 21717 24785 871
1.447 0.99234 70160 66639 35228 024 0.12348 03616 11217 43017 513
1.448 0.99247 00002 34203 82494 216 0.12248 79530 20366 29130 391
1.449 0.99259 19919 31850 76923 086 0.12149 54219 41572 33939 548
1.450 0.99271 29910 37588 49766 535 0.12050 27693 67366 57053 287
1c-y1 [ 1
(-92
ELEMENTARY TRANSCENDENTAL FTJNCTIONS 171
CIRCULAR SINES .4ND COSINES FOR RADIAN ARGUMENTS Table 4.6
X sin x cos x
1.450 0.99271 29910 37588 49766 535 0.12050 27693 67366 57053 287
1.451 0.99283 29974 30417 91459 118 0.11950 99962 90401 47620 080
1.452 0.99295 20109 90332 63717 946 0.11851 71037 03450 05063 327
1.453 0.99307 00315 98319 11543 325 0.11752 40925 99404 79804 068
1.454 0.99318 70591 36356 75120 114 0.11653 09639 71276 73971 735
1.455 0.99330 30934 87418 01619 777 0.11553 77188 12194 42103 061
1.456 0.99341 81345 35468 56903 143 0.11454 43581 15402 91829 237
1.457 0.99353 21821 65467 37123 830 0.11355 08828 74262 84551 407
1.458 0.99364 52362 63366 80232 355 0.11255 72940 82249 36104 618
1.459 0.99375 72967 16112 77380 893 0.11156 35927 32951 17410 313
1.460 0.99386 83634 11644 84228 683 0.11056 97798 20069 55117 465
1.461 0.99397 84362 38896 32148 075 0.10957 58563 37417 32232 463
1.462 0.99408 75150 87794 39331 194 0.10858 18232 78917 88737 835
1.463 0.99419 55998 49260 21797 223 0.10758 76816 38604 22199 915
1.464 0.99430 26904 15209 04300 286 0.10659 34324 10617 88365 556
1.465 0.99440 87866 78550 31137 923 0.10559 90765 89208 01747 983
1.466 0.99451 38885 33187 76860 141 0.10460 46151 68730 36201 884
1.467 0.99461 79958 74019 56879 043 0.10361 00491 43646 25487 846
0.99472 11085 96938 37979 012 0.10261 53795 08521 63826 230
: . tb6: 0.99482 32265 98831 48727 437 0.10162 06072 58026 06440 584
1.470 0.99492 43,49777580 89785 993 0.10062 57333 86931 70090 698
1.471 0.99502 44780 32063 44122 430 0.09963 07588 90112 33595 391
1.472 0.99512 36112 62150 87122 898 0.09863 56847 62542 38345 147
1.473 0.99522 17493 68709 96604 762 0.09764 05119 99295 88804 678
1.474 0.99531 88922 53602 62729 932 OiO9664 52415 95545 53005 525
1.475 0.99541 50398 19685 97818 664 0.09564 98745 46561 63028 806
1.476 0.99551 01919 70812 46063 854 0.09465 44118 47711 15478 186
1.477 0.99560 43486 11829 93145 787 0.09365 88544 94456 71943 189
1.478 0.99569 75096 48581 75747 356 0.09266 32034 82355 59452 948
1.479 0.99578 96749 87906 90969 720 0.09166 74598 07058 70920 484
1.480 0.99588 08445 37640 05648 408 0.09067 16244 64309 65577 623
1.481 0.99597 10182 06611 65569 851 0.08967 56984 49943 69400 641
1.482 0.99606 01959 04648 04588 337 0.08867 96827 59886 75526 752
1.483 0.99614 83775 42571 53643 374 0.08768 35783 90154 44661 519
1.484 0.99623 55630 32200 49677 461 0.08668 73863 36851 05477 303
1.485 0.99632 17522 86349 44454 246 0.08569 11075 96168 55002 845
1.486 0.99640 69452 18829 13277 079 0.08469 47431 64385 59004 070
1.487 0.99649 11417 44446 63607 933 0.08369 82940 37866 52356 240
1.488 0.99657 43417 79005 43586 693 0.08270 17612 13060 39407 518
1.489 0.99665 65452 39305 50450 815 0.08170 51456 86499 94334 076
1.490 0.99673 77520 43143 38855 320 0.08070 84484 54800 61486 832
1.491 0.99681 79621 09312 29093 143 0:0797i i6705 i4659 55729 907
1.492 0.99689 71753 57602 15215 811 0.07871 48128 62854 62770 926
1.493 0.99697 53917 08799 73054 448 0.07771 78764 96243 39483 234
1.494 0.99705 26110 84688 68141 099 0.07672 08624 11762 14220 152
1.495 0.99712 88334 08049 63530 364 0.07572 37716 06424 87121 354
1.496 ii99720 40586 02660 27521 334 0.07472 66050 77322 30411 478
1.497 0.99727 62865 93295 41279 821 0.07372 93638 21620 88691 060
0.99735 15173 05727 06360 877 0.07273 20488 36561 79219 898
:*. t;: 0.99742 37506 66724 52131 595 0.07173 46611 19459 92192 943
1.500 0.99749 49866 04054 43094 172 0.07073 72016 67702 91008 819
r’-,7”1
L ’ -I [ 1
C-7812
172 ELEMENTARY TRANSCENDENTAL FUNCTIONS
X sin x cos x
1.500 0.99749 49866 04054 43094 172 0.07073 72016 67702 91008 819
1.501 0.99756 52250 46480 86109 251 0.06973 96714 78750 12531 065
1.502 0799763 44659 23765 37519 509 0.06874 20715 50131 67342 208
1.503 0.99770 27091 66667 10173 501 0.06774 44028 79447 39990 761
1.504 0.99776 99547 06942 80349 750 0.06674 66664 64365 89231 245
1.505 0.99783 62024 77346 94581 063 0.06574 88633 02623 48257 343
1.506 0.99790 14524 11631 76379 092 0.06475 09943 92023 24928 268
1.507 0.99796 57044 44547 32859 104 0.06375 30607 30434 01988 470
1.508 0.99802 89585 11841 61264 976 0.06275 50633 15789 37280 758
1.509 0.99809 12145 50260 55394 397 0.06175 70031 46086 63952 953
1.510 0.99815 24724 97548 11924 274 0.06075 88812 19385 90658 160
1.511 0.99821 27322 92446 36636 332 0.05976 06985 33809 01748 769
1.512 0.99827 19938 74695 50542 912 0.05876 24560 87538 57464 281
2.513 0.99833 02571 85033 95912 947 0.05776 41548 78816 94113 053
1.514 0.99838 75221 65198 42198 118 0.05676 57959 05945 24248 072
1.515 0.99844 37887 57923 91859 188 0.05576 73801 67282 36836 851
1.516 0.99849 90569 06943 86092 495 0.05476 89086 61243 97425 545
1.517 0.99855 33265 56990 10456 612 0.05377 03823 86301 48297 399
1.518 0.99860 65976 53793 00399 163 0.05277 18023 40981 08625 609
1.519 0.99865 88701 44081 46683 784 0.05177 31695 23862 74620 716
1.520 0.99871 01439 75583 00717 231 0.05077 44849 33579 19672 613
1.521 0.99876 04190 97023 79776 634 0.04977 57495 68814 94487 284
1.522 0.99880 96954 58128 72136 872 0.04877 69644 28305 27218 360
1.523 0.99885 79730 09621 42098 089 0.04777 81305 10835 23593 598
1.524 0.99890 52517 03224 34913 328 0.04677 92488 15238 67036 388
1.525 0.99895 15314 91658 81616 285 0.04578 03203 40397 18782 371
1.526 0.99899 68123 28645 03749 180 0.04478 13460 85239 17991 291
1.527 0.99904 10941 68902 17990 729 0.04378 23270 48738 81854 166
1.528 0.99908 43769 68148 40684 234 0.04278 32642 29915 05695 871
1.529 0.99912 66606 83100 92265 762 0.04178 41586 27830 63073 262
i.530 0.99916 79452 71476 01592 427 0.04078 50112 41591 05868 899
1.531 0.99920 82306 91989 10170 755 0.03978 58230 70343 64380 513
1.532 0.99924 75169 04354 76285 152 0.03878 65951 13276 47406 277
1.533 0.99928 58038 69286 79026 436 0.03778 73283 69617 42326 008
1.534 0.99932 30915 48498 22220 463 0.03678 80238 38633 15178 390
1.535 0;99935 93799 04701 38256 819 0.03578 86825 19628 10734 312
1.536 0.99939 46689 01607 91817 592 0.03478 93054 11943 52566 435
1.537 0.99942 89585 03928 83506 202 0.03378 98935 14956 43115 073
1.538 0.99946 22486 77374 53376 306 0.03279 04478 28078 63750 505
1.539 0.99949 45393 88654 84360 752 0.03179 09693 50755 74831 796
1.540 0.99952 58306 05479 05600 596 0.03079 14590 82466 15762 248
1.541 0:99955 61222 96555 95674 180 0.02979 19180 22720 05041 568
1.542 0.99958 54144 31593 85726 242 0.02879 23471 71058 40314 858
1.543 0.99961 37069 81300 62497 095 0.02779 27475 27051 98418 526
1.544 0.99964 09999 17383 71251 832 0.02679 31200 90300 35423 217
1.545 0.99966 72932 12550 18609 586 0.02579 34658 60430 86673 867
1.546 0.99969 25868 40506 75272 821 0.02479 37858 37097 66826 971
1.547 0.99971 68807 75959 78656 660 0.02379 40810 19980 69885 184
1.548 0.99974 01749 94615 35418 249 0.02279 43524 08784 69229 328
1.549 0.99976 24694 73179 23886 150 0.02179 46010 03238 17647 934
1.550 0.99978 37641 89356 96389 761 0.02079 48278 03092 47364 391
[(-;)I1 [(-;I91
ELEMENTARY TRANSCENDENTAL FUNCTIONS 173
CIRCULAR SINES AND COSINES FOR RADIAN ARGUMENTS Table 4.6
.c sin x cos 2
1.550 0.99978 37641 89356 96389 761 0.02079 48278 03092 47364 391
1.551 0.99980 40591 21853 81488 767 0.01979 50338 08120 70061 827
1.552 0.99982 33542 50374 86102 606 0.01879 52200 18116 76905 802
1.553 0.99984 16495 55624 97539 966 0.01779 53874 32894 38564 929
1.554 0.99985 89450 19308 85428 298 0.01679 55370 52286 05229 507
1.555 0.99987 52406 24131 03543 342 0.01579 56698 76142 06628 284
1,556 0.99989 05363 53795 91538 676 0.01479 57869 04329 52043 433
1.557 0.99990 48321 93 07 76575 277 0.01379 58891 36731 30323 849
1.558 0.99991 81281 279470 74851 093 0.01279 59775 73245 09896 874
1.559 0.99993 04241 43888 93030 623 0.01179,60532 13782 38778 533
1.560 0.99994 17202 29966 29574 517 0.01079 61170 58267 44582 392
1.561 0.99995 20163 74406 75969 172 0.00979 61701 06636 34527 146
1.562 0.99996 13125 66914 17856 344 0.00879 62133 58835 95443 014
1.563 0.99996 96087 98192 36062 758 0.00779 62478 14822 93777 062
1.564 0.99997 69050 59945 07529 731 0.00679 62744 74562 75597 546
1.565 0.99998 32013 44876 06142 794 0.00579 62943 38028 66597 372
1.566 0.99998 84976 46689 03461 318 0.00479 63084 05200 72096 784
1.567 0.99999 27939 60087 69348 142 0.00379 63176 76064 77045 359
1.568 0.99999 60902 80775 72499 201 0.00279 63231 50611 46023 436
1.569 0.99999 83866 05456 80873 162 0.00179 63258 28835 23243 059
1.570 0.99999 96829 31834 62021 053 +0.00079 63267 10733 32548 541
1.571 0.99999 99792 58612 83315 895 -0.00020 36732 03695 22583 254
1.572 0.99999 92755 85495 12082 337 -0.00120 36729 14450 59042 804
1.573 0.99999 75719 13185 15626 285 -0.00220 36714 21533 14087 901
1.574 0.99999 48682 43386 61164 539 -0.00320 36677 24944 45343 613
1.575 0.99999 11645 78803 15654 423 -0.00420 36608 24688 30802 109
1.576 0.99998 64609 23138 45523 419 -0.00520 36497 20771 68822 280
1.577 0.99998 07572 81096 16298 798 -0.00620 36334 13205 78129 029
1.578 0.99997 40536 58379 92137 261 -0.00720 36109 02006 97812 142
1.579 0.99996 63500 61693 35254 568 -0.00820 35811 87197 87324 647
1.580 0.99995 76464 98740 05255 179 -0.00920 35432 68808 26480 539
1.581 0.99994 79429 78223 58361 895 -0.01020. 34961 46876 15451 796
1.582 0.99993 72395 09847 46545 499 -0.01120 34388 21448 74764 568
1.583 0.99992 55361 04315 16554 408 -0.01220 33702 92583 45294 454
1.584 0.99991 28327 73330 08844 324 -0.01320 32895 60348 88260 743
1.585 0.99989 91295 29595 56407 893 -0.01420 31956 24825 85219 553
1.586 0.99988 44263 86814 83504 374 -0.01520 30874 86108 38055 737
1.587 0.99986 87233 59691 04289 313 -0.01620 29641 44304 68973 475
1.588 0.99985 20204 63927 21344 232 -0.01720 28245 99538 20485 440
1.589 0.99983 43177 16226 24106 322 -0.01820 26678 51948 55400 452
1.590 0.99981 56151 34290 87198 158 -0.01920 24929 01692 56809 503
1.591 0.99979 59127 36823 68657 422 -0.02020 22987 48945 28070 065
1.592 0.99977 52105 43527 08066 646 -0.02120 20843 93900 92788 583
1.593 0.99975 35085 75103 24582 972 -0.02220 18488 36773 94801 039
1.594 0.99973 08068 53254 14867 933 -0.02320 15910 77799 98151 502
1.595 0.99970 71054 00681 50917 259 -0.02420 13101 17236 87068 552
1.596 0.99968 24042 41086 77790 702 -0.02520 10049 55365 65939 492
1.597 0.99965 67033 99171 11241 891 -0.02620 06745 92491 59282 234
1.598 0.99963 00029 00635 35248 219 -0.02720 03180 28945 11714 764
1.599 0.99960 23027 72179 99440 759 -0.02819 99342 65082 87922 093
-0.02919 95223 01288 72620 577
c(-;I11
1.600 0.99957 36030 41505 16434 211
;=I.57079 63267 94896 61923 132 u=3.14159 26535 89793 23846 264
174 ELEMENTARY TRANSCENDENTAL FUNCTIONS
0.00009 99999 99833 33333 34167 0.99999 99950 00000 0041b 66667
0.00019 99999 98666 bbbbb 93333 0.99999 99800 00000 06666 66666
0.00029 99999 95500 00002 02500 0.99999 99550 00000 33749 99990
0.00039 99999 89333 33341 86667 0.99999 99200 00001 06666 66610
0.00049 99999 79166 bbb92 70833 0.99999 98750 00002 60416 66450
0.00059 99999 64000 00064 80000 0.99999 98200 00005 39999 99352
0.00069 99999 42833 33473 39167 0.99999 97550 OOOlD 00416 65033
0.00079 99999 14666 bb939 73333 0.99999 96800 00017 Obbbb 63026
0.00089 99998 78500 00492 07499 0.99999 95950 00027 33749 92619
0.00099 99998 33333 34166 bbbb5 0.99999 95000 00041 bb6bb 52118
For )I >lO, sin ,110-n = .rlO-n; cos .rlO-n = 1 -i .c210-2n; to 25~.
From C. E. Van Orstrand, Tables of ttie exponential function and of the circular sine and cosine to
radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S.
Government Printing Office, Washington, D.C., 1921 (with permission).
ELEMENTARY TRANSCENDENTAL FUNCTIONS 175
CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS Table 4.8
X sin 5 cos x
0.00000 00000 00000 00000 000 1.00000 00000 00000 00000 000
+0.84147 09848 07896 50665 250 +0.54030 23058 68139 71740 094
+0.90929 74268 25681 69539 602 -0.41614 68365 47142 38699 757
+0.14112 00080 59867 22210 074 -0.98999 24966 00445 45727 157
-0.75680 24953 07928 25137 264 -0.65364 36208 63611 91463 917
-0.95892 42746 63138 46889 315 +0.28366 21854 63226 26446 664
-0.27941 54981 98925 87281 156 +0.96017 02866 50366 02054 565
+0.65698 65987 18789 09039 700 +0.75390 22543 43304 63814 120
+0.98935 82466 23381 77780 812 -0.14550 00338 08613 52586 884
+0.41211 84852 41756 56975 627 -0.91113 02618 84676 98836 829
-0.54402 11108 89369 81340 475 -0.83907 15290 76452 45225 886
:: -0I99999 02065 50703 45705 156 +0.00442 56979 88050 78574 836
12 -0.53657 29180 00434 97166 537 +0.84385 39587 32492 10465 396
13 +0:42016 70368 26640 92186 896 +0.90744 67814 50196 21385 269
14 +0.99060 73556 94870 30787 535 +0.13673 72182 07833 59424 893
+0.65028 78401 57116 86582 974 -0.75968 79128 58821 27384 815
-0.28790 33166 65065 29478 446 -0.95765 94803 23384 64189 964
-0.96139 74918 79556 85726 164 -0.27516 33380 51596 92222 034
-0.75098 72467 71676 10375 016 +0.66031 67082 44080 14481 610
+0.14987 72096 62952 32975 424 +0.98870 46181 86669 25289 835
20 +0.91294 52507 27627 65437 610 +0.40808 20618 13391 98606 227
+0.83665 56385 36056 03186 648 -0.54772 92602 24268 42138 427
t: -0.00885 13092 90403 87592 169 -0I99996 08263 94637 12645 417
23 -0.84622 04041 75170 63524 133 -0.53283 30203 33397 55521 576
24 -0.90557 83620 06623 84513 579 +0.42417 90073 36996 97593 705
-0.13235 17500 97773 02890 201 +0.99120 28118 63473 59808 329
+0.76255 84504 79602 73751 582 +0.64691 93223 28640 34272 138
+0.95637 59284 04503 01343 234 -0:29213 88087 33836 19337 140
+0.27090 57883 07869 01998 634 -0.96260 58663 13566 60197 545
-0.66363 38842 12967 50215 117 -0.74805 75296 89000 35176 519
-0.98803 16240 92861 78998 775 +0.15425 14498 87584 05071 866
310 -0.40403 76453 23065 00604 877 +0.91474 23578 04531 27896 244
+0.55142 66812 41690 55066 156 +0.83422 33605 06510 27221 553
;23 +0.99991 18601 07267 14572 808 -0.01327 67472 23059 47891 522
34 +0.52908 26861 20023 82083 249 -0.84857 02747 84605 18659 997
-0.42818 26694 96151 00440 675 -0.90369 22050 91506 75984 730
;56 -0.99177 88534 43115 73683 529 -0.12796 36896 27404 68102 833
-0.64353 81333 56999 46068 567 +0.76541 40519 45343 35649 108
;78 +0.29636 85787 09385 31739 230 +0.95507 36440 47294 85758 654
39 +0.96379 53862 84087 75326 066 +0.26664 29323 59937 25152 683
40 +0.74511 31604 79348 78698 771 -0.66693 80616 52261 84438 409
-0.15862 26688 04708 98710 332 -0.98733 92775 23826 45822 883
t: -0.91652 15479 15633 78589 899 -0.39998 53149 88351 29395 471
-0.83177 47426 28598 28820 958 +0.55511 33015 20625 67704 483
ti +0.01770 19251 05413 57780 795 +0.99984 33086 47691 22006 901
+0.85090 35245 34118 42486 238 +0.52532 19888 17729 69604 746
442 +0.90178 83476 48809 18503 329 -0.43217 79448 84778 29495 278
47 +0.12357 31227 45224 00406 153 -0.99233 54691 50928 71827 975
48 -0.76825 46613 23666 79904 497 -0.64014 43394 69199 73131 294
49 -0.95375 26527 59471 81836 042 +0.30059 25437 43637 08368 703
50 -0.26237 48537 03928 78591 439 +0.96496 60284 92113 27406 896
From C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to
radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S.
Government Printing Office, Washington, D.C., 1921 (with permission) for x_<lOO.
176 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.8 CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS
sin x co5 x
-0.26237 48537 03928 78591 439 +0.96496 60284 92113 27406 896
+0.67022 91758 43374 73449 435 +0.74215 41968 13782 53946 738
+0.98662 75920 40485 29658 757 -0.16299 07807 95705 48100 333
+0,39592 51501 81834 18150 339 -0.91828 27862 12.118 89119 973
-0.55878 90488 51616 24581 787 -0.82930 98328 63150 14772 785
-0.99975 51733 58619 83659 863 +0.02212 67562 61955 73456 356
-0.52155 10020 86911 88018 741 +0.85322 01077 22584 11396 968
+0.43616 47552 47824 95908 053 +0.89986 68269 69193 78650 300
+0.99287 26480 84537 11816 509 +0.11918 01354 48819 28543 584
+0.63673 80071 39137 88077 123 -0.77108 02229 75845 22938 744
-0.30481 06211 02216 70562 565 -0.95241 29804 15156 29269 382
-0.96611 77700 08392 94701 829 -0.25810 lb359 38267 44570 121
-0.73918 06966 49222 86727 602 +0.67350 71623 23586 25288 783
+0.16735 57003 02806 92152 784 +0.98589 65815 82549 69743 864
+0.92002 60381 96790 68335 154 +0.39185 72304 29550 00516 171
+0.82682 86794 90103 46771 021 -0.56245 38512 38172 03106 212
-0.02655 11540 23966 79446 384 -0.99964 74559 66349 96483 045
-0.85551 99789 75322 25899 683 -0.51776 97997 89505 06565 339
-0.89792 76806 89291 26040 073 +O, 44014 30224 96040 70593 105
-0.11478 48137 83187 22054 507 +0.99339 03797 22271 63756 155
+0.77389 06815 57889 09778 733 +0.63331 92030 86299 83233 201
+0.95105 46532 54374 63665 657 -0.30902 27281 66070 70291 749
+0.25382 33627 62036 27306 903 -0.96725 05882 73882 48729 171
-0.67677 19568 87307 62215 498 -0.73619 27182 27315 96016 815
-0.98514 62604 68247 37085 189 +0.17171 73418 30777 55609 845
-0.38778 lb354 09430 43773 094 +0.92175 12697 24749 31639 230
+0.56610 76368 98180 32361 028 +0.82433 13311 07557 75991 501
+0.99952 01585 80731 24386 610 -0.03097 50317 31216 45752 196
+0.51397 84559 87535 21169 609 -0.85780 30932 44987 85540 835
-0.44411 26687 07508 36850 760 -0.89597 09467 90963 14833 703
-0.99388 86539 23375 18973 081 -0.11038 72438 39047 55811 787
-0.62988 79942 74453 87856 521 +0.77668 59820 21631 15768 342
+0.31322 87824 33085 15263 353 +0.94967 76978 82543 20471 326
+0.96836 44611 00185 40435 015 +0.24954 01179 73338 12437 735
+0.73319 03200 73292 lb636 321 -0.68002 34955 87338 79542 720
-0.17607 56199 48587 07696 212 -0.98437 66433 94041 89491 821
-0.92345 84470 04059 80260 163 -0.38369 84449 49741 84477 893
-0.82181 78366 30822 54487 211 +0.56975 03342 65311 92000 851
+0.03539 83027 33660 68362 543 +0.99937 32836 95124 65698 442
+0.86006 94058 12453 22683 685 +0.51017 70449 41668 89902 379
+0.89399 66636 00557 89051 827 -0.44807 36161 29170 15236 548
+0.10598 75117 51156 85002 021 -ii99436 74609 28201 52610 672
-0.77946 60696 15804 68855 400 -0.62644 44479 10339 06880 027
-0.94828 21412 69947 23213 104 +0.31742 87015 19701 64974 551
-0.24525 19854 67654 32522 044 +0.96945 93666 69987 60380 439
+0.68326 17147 36120 98369 958 +0.73017 35609 94819 66479 352
+0.98358 77454 34344 85760 773 -Oil8043 04492 91083 95011 850
+0.37960 77390 27521 69648 192 -0.92514 75365 96413 89170 475
-0.57338 18719 90422 88494 922 -Or81928 82452 91459 25267 566
-0.99920 68341 86353 69443 272 +0.03982 08803 93138 89816 180
-0.50636 56411 09758 79365 656 +0.86231 88722 87683 93410 194
ELEMENTARY TRANSCENDENTAL FUNCTIONS 177
CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS Table 4.8
Table 4.8 CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS
325 -0.98803 627 -0.15422 167 375 -0.91295 755 -0.40805 454
326 -0.66361 133 +O. 74807 753 376 -0.83663 913 +O. 54775 448
327 +O. 27093 481 +O. 96259 770 377 +o. 00888 145 +O. 99996 056
328 +O. 95638 473 +O. 29210 998 378 +O. 84623 647 +O. 53280 751
329 +O. 76253 895 -0.64694 231 379 +O. 90556 557 -0.42420 631
330 -0.13238 163 -0.99119 882 380 +O. 13232 187 -0.99120 680
331 -0.90559 115 -0.42415 171 381 -0.76257 795 -0.64689 634
332 -0.84620 434 +O. 53285 853 382 -0.95636 712 +O. 29216 764
333 -0.00882 117 +O. 99996 109 383 -0.27087 677 +O. 96261 403
334 +O. 83667 215 +o. 54770 404 384 +O. 66365 643 +O. 74803 752
335 +O. 91293 295 -0.40810 958 385 +O. 98802 697 -0.15428 123
336 +O. 14984 741 -0.98870 914 386 +o. 40401 007 -0.91475 454
337 -0.75100 715 -0.66029 407 387 -0.55145 183 -0.83420 674
338 -0.96138 920 +O. 27519 232 388 -0.99991 146 +O. 01330 689
339 -0.28787 445 +O. 95766 816 389 -0.52905 711 +O. 84858 622
340 +O. 65031 074 +O. 75966 831 390 +O. 42820 991 +O. 90367 930
341 +O. 99060 323 -0.13676 708 391 +O. 99178 271 +O. 12793 379
342 +O. 42013 968 -0.90745 945 392 +O. 64351 506 -0.76543 345
343 -0.53659 836 -0.84383 778 393 -0.29639 737 -0.95506 471
344 -0.99999 034 -0.00439 555 394 -0.96380 342 -0.26661 388
345 -0.54399 582 +O. 83908 793 395 -0.74509 306 +O. 66696 052
346 +O. 41214 595 +O. 91111 784 396 +O. 15865 243 +O. 98733 450
347 +O. 98936 263 +o. 14547 021 397 +O. 91653 361 +O. 39995 769
348 +O. 65696 387 -0.75392 206 398 +O. 83175 801 -0.55513 837
349 -0.27944 444 -0.96016 186 399 -0.01773 206 -0.99984 277
350 -0.95893 283 -0.28363 328 400 -0.85091 936 -0.52529 634
180 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.8 CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS
400 -0.85091 936 -0.52529 634 450 -0.68328 373 -0.73015 296
401 -0.90177 532 +0.43220 513 451 -0.98358 231 +0.18046 010
402 -0.12354 321 +0.99233 919 452 -0.37957 985 +0.92515 898
403 +0.76827 396 +0.64012 118 453 +0.57340 657 +0.81927 096
404 +0.95374 359 -0.30062 129 454 +0.99920 563 -0.03985 100
405 +0.26234 577 -0.96497 394 455 +0.50633 965 -0.86233 414
406 -0.67025 155 -0.74213 399 456 -0.45205 268 -0.89199 124
407 -0.98662 268 +0.16302 052 457 -0.99482 985 -0.10155 572
408 -0.39589 747 +0.91829 472 458 -0.62296 505 +0.78224 967
409 +0.55881 405 +0.82929 299 459 +0.32165 095 +0.94685 832
410 +0.99975 451 -0.02215 689 460 +0.97054 255 +0.24092 979
411 +0.52152 528 -0.85323 583 461 +0.72712 181 -0.68650 847
412 -0.43619 188 -0.89985 368 462 -0.18481 137 -0.98277 401
413 -0.99287 624 -0Ii1915 021 463 -0.92682 982 -0.37548 166
414 -0.63671 476 +0.77109 942 464 -0.81672 521 +0.57702 680
415 +0.30483 933 +0.95240 379 465 +0.04427 279 +0.99901 948
416 +0.96612 555 +0.25807 251 466 +0.86456 660 +0.50251 826
417 +0.73916 039 -0.67352 944 467 +0.88998 186 -0.45599 593
418 -0.16738 542 -0.98589 154 468 +0.09715 190 -0.99526 957
419 -0.92003 785 -0.39182 950 469 -0.78499 906 -0.61949 695
420 -0.82681 172 +0.56247 878 470 -0.94542 551 +0.32583 830
421 +0.02658 129 +0.99964 666 471 -0.23663 211 +0.97159 932
422 +0.85553 559 +0.51774 401 472 +0.68971 977 +0.72407 641
423 +0.89791 441 -0.44017 009 473 +0.98194 647 -0.18915 902
424 +0.11475 487 -0.99339 384 474 +0.37137 611 -0.92848 252
425 -0.77390 977 -0.63329 587 475 -0.58063 573 -0.81416 347
426 -0.95104 534 +0.30905 140 476 -0.99881 376 +0.04869 372
427 -0.25379 421 +0.96725 824 477 -0.49868 703 +0.86678 212
428 +0.67679 415 +0.73617 232 478 +0.45993 026 +0.88795 504
429 +0.98514 108 -0.17174 704 479 +0.99568 978 +0.09274 619
430 +0.38775 385 -0.92176 296 480 +0.61601 671 -0.78773 308
431 -0.56613 249 -0.82431 427 481 -0.33001 928 -0.94397 419
432 -0.99951 922 +0.03100 516 482 -0.97263 707 -0.23232 978
433 -0.51395 260 +0.85781 859 483 -0.72101 682 +0:69291 756
434 +0.44413 968 +0.89595 756 484 +0.19350 297 +0.98109 969
. 435 +0.99389 198 +0.11035 728 485 +0.93dll 702 +0.36726 329
436 +0.62986 458 -0.77670 497 486 +0.81158 578 -0.58423 328
437 -0.31325 741 -0.94966 826 487 -0.05311 369 -0.99858 847
438 -0.96837 198 -0.24951 093 488 -0.86898 067 -0.49484 603
439 -0.73316 982 +0.68004 560 489 -0.88591 083 +0.46385 557
440 +0.17610 529 +0.98437 134 490 -0.08833 866 +0.99609 050
441 +0.92347 001 +0.38367 061 491 +0.79045 167 +0.61252 441
442 +0.82180 066 -0.56977 511 492 +0.94250 438 -0.33419 379
443 -0.03542 843 -0.99937 222 493 +0.22802 291 -0.97365 577
444 -0.86008 478 -0.51015 112 494 -0.69610 177 -0.71794 312
445 -0.89398 316 +0.44810 056 495 -0.98023 370 +0.19784 312
446 -0.10595 754 +0.99437 066 496 -0.36314 328 +0.93173 331
447 +0.77948 495 +0.62642 095 497 +0.58781 939 +0.80899 219
448 +0.94827 257 -0.31745 729 498 +0.99834 363 -0.05753 262
449 +0.24522 276 -0.96946 676 499 +0.49099 533 -0.87116 220
450 -0.68328 373 -0.73015 296 500 -0.46777 181 -0.88384 927
ELEMENTARY TRANSCENDENTAL FUNCTIONS 181
CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS Table 4.8
Table 4.8 CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS
600 +0.04418 245 -0.99902 348 650 +0.30475 320 -0.95243 136
601 -0.81677 739 -0.57695 294 651 -0.63678 449 -0;77104 183
602 -0.92679 586 +0.37556 547 652 -0.99286 546 +0.11923 999
603 -0.18472 249 +0.98279 072 653 -0.43611 050 +0:89989 312
604 +0.72718 389 +0.68644 271 654 +0.52160 244 +0.85318 866
605 +0.97052 075 -0.24101 756 655 +0.99975 651 +0.02206 648
606 iO:32156 532 -0.94688 740 656 +0.55873 905 -0.82934 352
607 -0.62303 579 -0.78219 333 657 -0.39598 051 -0.91825 891
608 -0i99482 067 +0.10164 568 658 -0.98663 742 -0.16293 130
609 -0.45197 201 +0.89203 212 659 -0.67018 443 +0.74219 460
610 +0.50641 763 +0.86228 834 660 +0.26243 303 +0.96495 021
611 +0.99920 923 +0.03976 064 661 +0.95377 077 +0.30053 504
612 +0.57333 248 -0.81932 281 662 +0.76821 607 -0.64019 066
613 -0.37966 351 -0.92512 465 663 -0.12363 295 -0.99232 802
614 -0.98359 862 -0.18037 115 664 -0.90181 440 -0.43212 358
615 -0.68321 769 +0.73021 475 665 -0.85087 185 +0.52537 329
616 +0.24531 043 +0.96944 458 666 -0.01764 165 +0:99984 437
617 +0.94830 128 +0.31737 153 667 +0.83180 821 +0.55506 315
618 +0.77942 830 -0.62649 144 668 +0.91649 743 -0.40004 057
619 -0.10604 746 -0.99436 107 669 +0.15856 314 -0.98734 884
620 -0.89402 368 -0.44801 972 670 -0.74515 337 -0.66689 314
621 -0.86003 865 +0.51022 890 671 -0.96377 931 +0.26670 104
622 -0.03533 805 +0.99937 542 672 -0.29631 100 +0.95509 151
623 +0.82185 218 +0.56970 079 673 +0.64358 428 +0.76537 525
624 +0.92343 531 -0.38375 412 674 +0.99177 114 -0.12802 348
625 +0.17601 627 -0.98438 726 675 +0.42812 819 -0.90371 802
626 -0.73323 132 -0.67997 929 676 -0.52913 384 -0.84853 838
627 -0.96834 941 +0.24959 850 677 -0.99991 266 -0.01321 646
628 -0.31317 153 +0.94969 658 678 -0.55137 639 +0.83425 660
629 +0.62993 482 +0.77664 801 679 +0.40409 279 +0.91471 800
630 +0.99388 200 -0.11044 716 680 +0.98804 092 +0.15419 188
631 +0.44405 865 -0.89599 772 681 +0.66358 878 -0.74809 754
632 -0.51403 017 -0.85777 210 682 -0.27096 382 -0.96258 953
633 -0.99952 202 -0.03091 477 683 -0.95639 354 -0.29208 115
634 -0.56605 794 +0.82436 546 684 -0.76251 945 +0.64696 529
635 +0.38783 721 +0.92172 789 685 +0.13241 151 +0.99119 483
636 +0.98515 661 +0.17165 795 686 +0.90560 393 +0.42412 441
637 +0.67672 757 -0.73623 352 687 +0.84618 828 -0.53288 404
638 -0.25388 168 -0.96723 528 688 +0.00879 102 -0.99996 136
639 -0.95107 328 -0.30896 539 689 -0.83668 866 -0.54767 882
640 -0.77385 250 +0.63336 586 690 -0.91292 065 +0.40813 710
641 +0.11484 470 +0.99338 346 691 -0.14981 760 +0.98871 365
642 +0.89795 421 +0:44008 889 692 +0.75102 706 +0.66027 143
643 +0.85548 876 -0.51782 138 693 +0.96138 090 -0.27522 130
644 +0.02649 089 -0.99964 905 694 +0.28784 558 -0.95767 684
645 -0.82686 259 -0.56240 400 695 -0.65033 364 -0.75964 871
646 -0.92000 241 +0.39191 270 696 -0I99059 911 +0.13679 694
647 -0.16729 626 +0.98590 667 697 -0.42011 233 +0.90747 211
648 +0.73922 130 +0.67346 260 698 +0.53662 379 +0.84382 161
649 +0.96610 221 -0.25815 988 699 +0.99999 047 +0.00436 541
650 +0.30475 320 -0.95243 136 700 +0.54397 052 -0.83910 433
ELEMENTARY TRANSCENDENTAL FUNCTIONS 183
CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS Table 4.8
x sin x cos x X sin z cos x
700 +0.54397 052 -0.83910 433 750 +0.74507 295 -0.66698 298
701 -0.41217 342 -0.91110 541 751 -0.15868 219 -0.98732 971
702 -0.98936 702 -0.14544 039 752 -0.91654 566 -0.39993 006
703 -0.65694 115 +0.75394 186 753 -0.83174 127 +0.55516 345
704 +0.27947 339 +0.96015 344 754 +0.01776 220 +0.99984 224
705 +0.95894 137 +0.28360 437 755 +0.85093 519 +0.52527 069
706 +0.75676 309 -0.65368 925 756 +0.90176 229 -0.43223 231
707 -0.14117 969 -0.98998 399 757 +0.12351 330 -0.99234 292
708 -0.90932 251 -0.41609 202 758 -0.76829 325 -0.64009 802
709 -0.84143 841 +0.54035 304 759 -0.95373 453 +0.30065 004
710 +O.OOOOb029 +1.00000 000 760 -0.26231 668 +0.96498 184
711 +0.84150 356 +0.54025 157 761 +0:67027 392 +0.74211 379
712 +0.90927 234 -0.41620 166 762 +0.98661 776 -0.16305 026
713 +0.14106 032 -0.99000 100 763 +0.39586 979 -0.91830 665
714 -0.75684 190 -0.65359 799 764 -0.55883 905 -0.82927 614
715 -0.95890 717 +0.28372 000 765 -0.99975 384 +0.02218 703
716 -0.27935 761 +0.96018 713 766 -0.52149 956 +0.85325 155
717 +0.65703 205 +0.75386 264 767 +0.43621 901 +0.89984 053
718 +0.98934 947 -0.14555 968 768 +0.99287 983 +0.11912 028
719 +0.41206 355 -0.91115 511 769 +0.63669 152 -0.77111 861
720 -0.54407 170 -0.83903 873 770 -0.30486 804 -0.95239 460
721 -0.99998 994 +0.00448 599 771 -0.96613 333 -0.25804 339
722 -0.53652 204 +0.84388 631 772 -0.73914 009 +0.67355 173
723 +0.42022 174 +0.90742 145 773 +0.16741 514 +0.98588 649
724 +0.99061 560 +0.13667 750 774 +0.92004 966 +0.39180 176
725 +0.65024 204 -0.75972 712 775 +0.82679 477 -0.56250 370
726 -0.28796 105 -0.95764 212 776 -0.02661 142 -0.99964 585
727 -0.96141 408 -0.27510 538 777 -0.85555 119 -0.51771 822
728 -0.75094 744 +0.66036 198 778 -0.89790 114 +0.44019 716
729 +0.14993 682 +0.98869 558 779 -Oil1472 492 +0.99339 730
730 +0.91296 985 +0.40802 702 780 +0.77392 886 +0.63327 255
731 +0.83662 262 -0.54777 970 781 +0.95103 602 -0.30908 007
732 -0.00891 160 -0.99996 029 782 +0.25376 505 -0.96726 589
733 -0.84625 253 -0.53278 200 783 -0.67681 634 -0.73615 192
734 -0.90555 279 +0.42423 360 784 -0.98513 591 +0.17177 673
735 -0.13229 199 +0.99121 079 785 -0.38772 606 +0.92177 465
736 +0.76259 745 +0:64687 335 786 +0.56615 733 +0.82429 720
737 +0.95635 831 -0.29219 647 787 +0.99951 829 -0.03103 529
738 +0.27084 775 -0.96262 220 788 +0.51392 674 -0.85783 408
739 -0.66367 898 -0.74801 752 789 -0.44416 668 -0.89594 417
740 -0.98802 232 +0.15431 102 790 -0.99389 531 -0.11032 732
741 -0.40398 250 +0.91476 672 791 -0.62984 117 +0.77672 396
742 +0.55147 697 +0.83419 011 792 +0.31328 604 +0.94965 881
743 +0.99991 106 -0.01333 703 793 +0.96837 950 +0.24948 174
744 +0.52903 153 -0.84860 217 794 +0.73314 932 -0.68006 770
745 -0.42823 715 -0.90366 639 795 -0.17613 497 -0.98436 603
746 -0.99178 657 -0.12790 390 796 -0.92348 158 -0.38364 277
747 -0.64349 199 +0.76545 285 797 -0.82178 349 +0.56979 988
748 +0.29642 616 +0.95505 577 798 +0.03545 855 +0.99937 115
749 +0.96381 146 +0.26658 483 799 +0.86010 016 +0.51012 519
750 +0.74507 295 -0.66698 298 800 +0.89396 965 -0.44812 751
184 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.8 CIRCULAR SINES AND COSINES FOR LARGE RADIAN ARGUMENTS
Table 4.9
[ Y21 [(-$11 1 1
c-412
5
ELEMENTARY TRANSCENDENTAL FUNCTIONS
* [t--8)75 1 t-l)4
c 1
*see page Il.
ELEMENTARY TRANSCENDENTAL FUNCTIONS 191
CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE Table 4.10
* [(-712
5 1 cC-i)4 1
‘See page n.
ELEMENTARY TRANSCENDENTAL FUNCTIONS
*See page n.
ELEMENTARY ZRANSCENDENTAL FUNCTIONS 195
CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE Table 4.10
* [(-7)25 1 [IC-57131
ELEMENTARY TRANSCENDENTAL FUNCTIONS 197
CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE Table 4.10
[(-;I11
90°-0
[c-y1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 199
CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS Table 4.11
TO FIVE TENTHS OF A DEGREE
e tan 8 cot e set H csc e 9o”-s
22.5O 0.41421 35623 73095 2.41421 35623 73095 1.08239 220 2.61312 593 67.5'
23.0 0.42447 48162 09604 2.35585 23658 23753 1.08636 038 2.55930 467 67.0
23.5 0.43481 23749 60933 2.29984 25472 36257 1.09044 110 2.50784 285 66.5
24. 0 0.44522 86853 08536 2.24603 67739 04216 1.09463 628 2.45859 334 66.0
24.5 0.45572 62555 32584 2.19429 97311 65038 1.09894 787 2.41142 102 65.5
25.0 0.46630 76581 54998 2.14450 69205 09558 1.10337 792 2.36620 158 65.0
25.5 0.47697 55326 98160 2109654 35990 88174 1.10792 854 2.32282 050 64.5
26.0 0.48773 25885 65861 2.05030 38415 79296 1.11260 194 2.28117 203 64.0
26.5 0.49858 16080 53431 2.00568 97082 59020 1.11740 038 2.24115 845 63.5
27.0 0.50952 54494 94429 1.96261 05055 05150 1.12232 624 2.20268 926 63.0
27.5 0.52056 70505 51746 1.92098 21269 71166 1.12738 195 2.16568 057 62.5
28.0 0.53170 94316 61479 1.88072 64653 46332 1.13257 005 2.13005 447 62.0
28.5 0.54295 56996 38437 1.84177 08860 33458 1.13789 318 2.09573 853 61.5
29. 0 0.55430 90514 52769 1.80404 77552 71424 1.14335 407 2.06266 534 61.0
29.5 0.56577 27781 87770 1.76749 40162 42891 1.14895 554 2.03077 204 60.5
30.0 0.57735 02691 89626 1.73205 08075 68877 1.15470 054 2.00000 000 60.0
30.5 0.58904 50164 20551 1.69766 31193 26089 1.16059 210 1.97029 441 59.5
31.0 0.60086 06190 27560 1.66427 94823 50518 1.16663 340 1.94160 403 59.0
31.5 0.61280 07881 39932 1.63185 16871 28789 1.17282 770 1.91388 086 58.5
32.0 0.62486 93519 09327 1.60033 45290 41050 1.17917 840 1.88707 991 58.0
32.5 0.63707 02608 07493 1.56968 55771 17490 1.18568 905 1.86115 900 57.5
33.0 0.64940 75931 97510 1.53986 49638 14583 1.19236 329 1.83607 846 57. 0
33.5 0.66188 55611 95691 1.51083 51936 14901 1.19920 494 1.81180 103 56.5
34.0 0.67450 85168 42426 1.48256 09685 12740 1.20621 795 1.78829 165 56.0
34.5 0.68728 09586 01613 1.45500 90286 72445 1.21340 641 1.76551 728 55.5
35.0 0.70020 75382 09710 1.42814 80067 42114 1.22077 459 1.74344 680 55.0
35.5 0.71329 30678 97005 1.40194 82944 76336 1.22832 691 1.72205 082 54.5
36.0 0.72654 25280 05361 1.37638 19204 71173 1.23606 798 1.70130 162 54.0
36.5 0.73996 10750 ,28487 1.35142 24379 45808 1.24400 257 1.68117 299 53.5
37.0 0.75355 40501 02794 1.32704 48216 20410 1.25213 566 1.66164 014 53.0
37.5 0.76732 69879 78960 1.30322 53728 41206 1.26047 241 1.64267 963 52.5
38.0 0.78128 56265 06717 1.27994 16321 93079 1.26901 822 1.62426 925 52.0
38.5 0.79543 59166 67828 1.25717 22989 18954 1.27777 866 1.60638 793 51.5
39.0 0.80978 40331 95007 1.23489 71565 35051 1.28675 957 1.58901 573 51. 0
39.5 0.82433 63858 17495 1.21309 70040 92932 1.29596 700 1.57213 369 50.5
40.0 0.83909 96311 77280 1.19175 35925 94210 1.30540 729 1.55572 383 50.0
40.5 0.85408 06854 63466 1.17084 95661 12539 1.31508 700 1.53976 904 49.5
41.0 0.86928 67378 16226 1.15036 84072 21009 1.32501 299 1.52425 309 49.0
41.5 0.88472 52645 55944 1.13029 43863 61753 1.33519 242 1.50916 050 48.5
42.0 0.90040 40442 97840 1.11061 25148 29193 1.34563 273 1.49447 655 48.0
42.5 0.91633 11740 17423 1.09130 85010 69271 1.35634 170 1.48018 723 47.5
43. 0 0.93251 50861 37661 1.07236 87100 24682 1.36732 746 1.46627 919 47.0
43.5 0.94896 45667 14880 1.05378 01252 80962 1.37859 847 1.45273 967 46.5
44.0 0.96568 87748 07074 1.03553 03137 90569 1.39016 359 1.43955 654 46. 0
44.5 0.98269 72631 15690 1.01760 73929 72125 1.40203 206 1.42671 819 45.5
45.0 1.00000 00000 00000 1.00000 o~ooy 00000 1.41Q&356 1.414&356 45.0
90°-e cot tJ e
[ 1
C-i)4 [‘-4’31
L -1 [ 1
C-f!4
[ 1
c-:)3
200 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[(J4jl
(-62
[ ii3 1 [(-i$3 1 1
ELEMENTARY TRANSCENDENTAL FUNCTIONS 201
CIRCULAR FUNCTIONS FOR THE ARGUMENT f x Table 4.12
X csc;x 1-X
0.00 1 00000 00000 00000 00000 1.00
0.01 63.65674 1162; 71580 99500 1'00012 33827 39761 81169 63 66459 53060:0564 58546 0.99
0.02 31.82051 59537 73958 03934 1:00049 36832 37144 42400 31:83622 52090 97622 95566 0.98
0.03 21.20494 87896 88751 52283 1.00111 13587 85243 76109 21.22851 50958 16816 17580 0.97
0.04 15.89454 48438 65303 44576 1.00197 71730 71142 10978 15.92597 11099 08654 59358 0.96
0.05 12.70620 47361 74704 64602 1.00309 21984 82825 50283 12.74549 48431 82374 28619 0.95
0.06 10.57889 49934 05635 52417 1.00445 78193 57019 51480 10.62605 37962 83115 99865 0.94
0.07 9.05788 66862 38928 19329 1.00607 57361 86291 90575 9.11292 00161 49841 72675 0.93
0.08 7.91581 50883 05826 84427 1.00794 79708 09297 28943 7.97872 97555 59476 60149 0.92
0.09 7.02636 62290 41380 19848 1.01007 68726 13784 19104 7.09717 00264 69225 38129 0.91
0.10 6.31375 15146 75043 09898 1.01246 51257 88002 93136 6.39245 32214 99661 54704 0.90
0.11 5.72974 16467 24314 86192 1.01511 57576 62501 87437 5.81635 10329 24944 03199 0.89
0.12 5.24218 35811 13176 73758 1.01803 21481 91042 38259 5.33671 14122 92458 78659 0.88
0.13 4.82881 73521 92759 97818 1.02121 80406 26567 47910 4.93127 53949 49859 96253 0.87
0.14 4.47374 28292 11554 62415 1.02467 75534 55900 33566 4.58414 38570 27373 56913 0.86
0.15 4.16529 97700 90417 20387 1.02841 51936 65208 54585 4.28365 75697 31185 03924 0. 85
0.16 3789474 28549 29859 33474 1.03243 58734 17339 88710 4.02107 22333 75967 50952 0.84
0.17 3.65538 43546 52259 73004 1.03674 49162 32016 53065 3.78970 11465 59780 81919 0. 83
0.18 3:44202 25766 69218 62809 1.04134 80947 70681 14007 3.58434 36523 72161 57038 0.82
0.19 3.25055 08012 99836 37634 1.04625 16303 39647 78848 3.40089 40753 61802 31848 0. 81
0.20 3.07768 35371 75253 40257 1.05146 22242 38267 21205 3.23606 79774 99789 69641 0. 80
0.21 2.92076 09892 98816 40048 1.05698 70790 93232 61183 3.08720 66268 08416 38088 0.79
0.22 2.77760 68539 14974 88865 1.06283 39243 36113 96396 2.95213 47928 09339 97327 0.78
0.23 2.64642 32102 86631 86514 1.06901 10439 98926 01199 2.82905 56388 91501 64260 0.77
0.24 2.52571 16894 47304 99451 1.07552 73070 22247 78234 2.71647 18916 65871 74307 0.76
0.25 2.41421 35623 73095 04880 1.08239 22002 92393 96880 2.61312 59297 52753 05571 0.75
0.26 2.31086 36538 82410 63708 1.08961 58646 48705 30888 2.51795 36983 10349 34110 0.74
0.27 2.21475 44978 13361 51875 1.09720 91341 29537 26252 2.43004 88648 55296 52041 0.73
0.28 2.12510 81731 57202 76115 1.10518 35787 56399 59380 2.34863 46560 54351 86300 0.72
0..29 2.04125 39671 21703 26026 1.11355 15511 90413 37268 2.27304 15214 61957 72361 0.71
0.30 1.96261 05055 05150 58230 1.12232 62376 34360 80715 2.20268 92645 85266 62156 0.70
0.31 1.88867 13416 31067 67620 1.13152 17133 97749 42882 2.13707 26325 27611 85837 0.69
0.32 1.81899 32472 81066 27571 1.14115 30035 92241 17245 2.07574 96076 48793 05903 0.68
0.33 1.75318 66324 72237 08332 1.15123 61494 81376 51287 2.01833 18280 89559 43676 0. 67
0.34 1.69090 76557 85011 24674 1.16178 82810 72765 98515 1.96447 66988 67248 48330 0.66
0.35 1.63185 16871 28789 61767 1.17282 76966 14008 94955 1.91388 08554 30942 72280 0.65
0.36 1.57574 78599 68651 08688 1.18437 39497 36918 17500 1.86627 47167 00567 54120 0.64
0.37 1.52235 45068 96131 24085 1.19644 79450 89806 17366 1.82141 79214 74081 38479 0.63
0.38 1.47145 53158 19969 04283 1.20907 20434 06541 15436 1.77909 54854 79867 33350 0.62
0.39 1.42285 60774 31870 59031 1.22227 01770 86068 14117 1.73911 45497 30640 74960 0.61
0.40 1.37638 19204 71173 53820 1.23606 79774 99789 69641 1.70130 16167 04079 86436 0.60
0.41 1.33187 49515 02597 59439 1.25049 29154 09784 85573 1.66550 01910 65749 08074 0.59
0.42 1.28919 22317 85066 67042 1.26557 44560 72090 15648 1.63156 87575 13749 73007 0.58
0.43 1.24820 40363 53049 43751 1.28134 42308 20677 31999 1.59937 90408 68062 88301 0.57
0.44 1.20879 23504 09609 13115 1.29783 62271 84727 12712 1.56881 45035 05365 75750 0.56
0.45 1.17084 95661 12539 22520 1.31508 69998 90784 80424 1.53976 90432 22366 30748 0.55
0.46 1.13427 73492 55405 46422 1.33313 59054 50172 40410 1.51214 58610 31226 40092 0.54
0.47 1.09898 56505 36301 56382 1.35202 53634 40027 12805 1.48585 64735 81717 76608 0.53
0.48 1.06489 18403 24791 86700 1.37180 11480 64918 28453 1.46081 98491 22513 12750 0.52
0.49 1.03191 99492 80495 57182 1.39251 27141 49012 49662 1.43696 16493 57094 20394 0.51
0.50 1.00000 00000 00000 00000 1.41421 35623 73095 04880 1.41421 35623 73095 04880 0.50
l-x tan;x cscTx
2 SW;X X
202 ELEMENTARY TRANSCENDENTAL FUNCTIONS
s=12 s=14
0.50000 00000 0.86602 54038 0.43388 37391 0.90096 88679
0.86602 54038 0.50000 00000 0.78183 14825 0;62348 98019
1.00000 00000 +o.ooooo 00000 0.97492 79122 +0.22252 09340
0.86602 54038 -0.50000 00000 0.97492 79122 -0.22252 09340
0.50000 00000 -0.86602 54038 0.78183 14825 -0.62348 98019
0.00000 00000 -1.00000 00000 0.43388 37391 -0.90096 88679
0.00000 00000 -1.00000 00000
s=18 s=20
0.34202 01433 0.93969 26208 0 32469 94692? 94581 72417 0.30901 69944 0.95105 65163
0.64278 76097 0.76604 44431 0:61421 27127 0:78914 05094 0.58778 52523 0.80901 69944
0.86602 54038 0.50000 00000 0.83716 64782 0.54694 81581 0.80901 69944 0.50778 52523
0.98480 77530 +0.17364 81777 0.96940 02659 +0.24548 54872 0.95105 65163 0.30901 69944
0.98480 77530 -0.17364 81777 0.99658 44930 -0.08257 93455 1.00000 00000 +o.ooooo 00000
0.86602 54038 -0.50000 00000 0.91577 33266 -0.40169 54247 0.95105 65163 -0.30901 69944
0.64278 76097 -0.76604 44431 0.73572 39107 -0.67728 15716 0.80901 69944 -0.58778 52523
0.34202 01433 -0.93969 26208 0.47594 73930 -0.87947 37512 0.58778 52523 -0.80901 69944
0.00000 00000 -1.00000 00000 0.16459 45903 -0.98636 13034 0.30901 69944 -0.95105 65163
0.00000 00000 -1.00000 00000
=24 8 b-=25
0.25881 9045; 0.96592 58263 0.24868 98872 0.96858 31611
0.50000 00000 0.86602 54038 0.48175 36741 0.87630 66801
0.70710 67812 0.70710 67812 0.68454 71059 0.72896 86274
0.86602 54038 0.50000 00000 0.84432 79255 0.53582 67950
0.96592 58263 0.25881 90451 0.95105 65163 0.30901 69944
1.00000 00000 + 0.00000 00000 0.99802 67284 +0.06279 05196
0.96592 58263 -0.25881 90451 ii98228 72507 -0.18738 13146
0.86602 54038 -0.50000 00000 0.90482 70525 -0.42577 92916
0.70710 67812 -0.70710 67812 0.77051 32428 -0.63742 39898
0.50000 00000 -0.86602 54038 0.58178 52523 -0.80901 69944
0.25881 90451 -0.96592 58263 0.36812 45527 -0.92977 64859
0.00000 00000 -1.00000 00000 0.12533 32336 -0.99211 47013
ELEMENTARY TRANSCENDENTAL FWCTIONS 203
INVERSE CIRCULAR SINES AND TANGENTS Table 4.14
[ 1
(9’16
For use and extension of the table see Examples 21-25. For other inverse functions see 4.4 and
4.3.45. ;=1.57079 63267 95
Compilation of arcsin :c from National Bureau of Standards, Table of arcsin z. Columbia Univ.
Press, New York, N.Y., 1945 (with permission).
204 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[c-y1 [c-y1
0.450 0.46676 53390 47 0.42285 39261 33 0.500 0.52359 87755 98 0.46364 76090 01
1C-J)8
1 [c-y1 ;=1.57079
- 6326795
208 ELEMENTARY TRANSCENDENTAL FUNCTIONS
0.500 0.52359 87755 98 0.46364 76090 01 0.550 0.58236 42378 69 0.50284 32109 28
0.501 0.52475 38615 51 0.46444 72889 58 0.551 0.58356 20792 89 0.50361 06410 37
0.502 0.52590 97203 91 ' 0.46524 63286 62 0.552 0.58476 08688 33 0.50437 74226 73
0.503 0.52706 63552 20 0.46604 47278 61 0.553 0.58596 06104 84 0.50514 35557 57
0.504 0.52822 37691 54 0.46684 24863 09 0.554 0.58716 13082 43 0.50590 90402 12
0.505 0.52938 19653 22 0.46763 96037 63 0.555 0.58836 29661 37 0.50667 38759 68
0.506 0.53054 09468 69 0.46843 60799 83 0.556 0.58956 55882 10 0.50743 80629 53
0.507 0.53170 07169 56 0.46923 19147 34 0.557 0.59076 91785 32 0.50820 16011 02
0.508 0.53286 12787 56 0.47002 71077 82 0.558 0.59197 37411 92 0.50896 44903 52
0.509 0.53402 26354 61 0.47082 16589 00 0.559 0.59317 92803 04 0.50972 67306 43
0.510 0.53518 47902 76 0.47161 55678 62 0.560 0.59438 58000 01 0.51048 83219 17
0.511 0.53634 77464 20 0.47240 88344 48 0.561 0.59559 33044 41 0.51124 92641 21
0.512 0.53751 15071 30 0.47320 14584 38 0.562 0.59680 17978 05 0.51200 95572 04
0.513 0.53867 60756 57 0.47399 34396 20 0.563 0.59801 12842 95 0.51276 92011 19
0.514 0.53984 14552 69 0.47478 47777 82 0.564 0.59922 17681 37 0.51352 81958 22
0.515 0.54100 76492 49 0.47557 54727 17 0.565 0.60043 32535 81 0.51428 65412 69
0.516 0.54217 46608 96 0.47636 55242 22 0.566 0.60164 57448 99 0.51504 42374 25
0.517 0.54334 24935 25 0.47715 49320 97 0.567 0.60285 92463 89 0.51580 12842 52
0.518 0.54451 11504 67 0.47794 36961 45 0.568 0.60407 37623 71 0.51655 76817 18
0.519 0.54568 06350 69 0.47873 18161 73 0.569 0.60528 92971 89 0.51731 34297 96
0.520 0.54685 09506 96 0.47951 92919 93 0.570 0.60650 58552 13 0.51806 85284 57
0.521 0.54802 21007 28 0.48030 61234 17 0.571 0.60772 34408 36 0.51882 29776 79
0.522 0.54919 40885 61 0.48109 23102 64 0.572 0.60894 20584 75 0.51957 67774 41
0.523 0.55036 69176 11 0.48187 78523 54 0.573 0.61016 17125 74 0.52032 99277 27
0.524 0.55154 05913 07 0.48266 27495 12 0.574 0.61138 24076 01 0.52108 24285 22
0.525 0.55271 51130 97 0.48344 70015 67 0.575 0.61260 41480 49 0.52183 42798 14
0.526 0.55389 04864 46 0.48423 06083 50 0.576 0.61382 69384 37 0.52258 54815 96
0.527 0.55506 67148 37 0.48501 35696 94 0.577 0.61505 07833 09 0.52333 60338 62
0.528 0.55624 38017 69 0.48579 58854 40 0.578 0.61627 56872 37 0.52408 59366 09
0.529 0.55742 17507 59 0.48657 75554 29 0.579 0.61750 16548 17 0.52483 51898 38
0.530 0.55860 05653 43 0.48735 85795 05 0.580 0.61872 86906 72 0.52558 37935 52
0.531 0.55978 02490 72 0.48813 89575 18 0.581 0.61995 67994 52 0.52633 17477 57
0.532 0.56096 08055 18 0.48891 86893 19 0.582 0.62118 59858 34 0.52707 90524 63
0.533 0.56214 22382 69 0.48969 77747 65 0.583 0.62241 62545 21 0.52782 57076 82
0.534 0.56332 45509 33 0.49047 62137 12 0.584 0.62364 76102 44 0.52857 17134 28
0.535 0.56450 77471 34 0.49125 40060 25 0.585 0.62488 00577 61 0.52931 70697 19
0.536 0.56569 18305 17 0.49203 11515 68 0.586 0.62611 36018 60 0.53006 17765 76
0.537 0.56687 68047 44 0.49280 76502 10 0.587 0.62734 82473 54 0.53080 58340 23
0.538 0.56806 26734 97 0.49358 35018 23 0.588 0.62858 39990 87 0.53154 92420 86
0.539 0.56924 94404 76 0.49435 87062 83 0.589 0.62982 08619 28 0.53229 20007 93
0.540 0.57043 71094 00 0.49513 32634 68 0.590 0.63105 88407 78 0.53303 41101 77
0.541 0.57162 56840 08 0.49590 71732 62 0.591 0.63229 79405 66 0.53377 55702 73
0.542 0.57281 51680 58 0.49668 04355 48 0.592 0.63353 81bb2 50 0.53451 63811 18
0.543 0.57400 55653 28 0.49745 30502 17 0.593 0.63477 95228 17 0.53525 65427 53
0.544 0.57519 68796 15 0.49822 50171 59 0.594 0.63602 20152 84 0.53599 60552 20
0.545 0.57638 91147 36 0.49899 63362 71 0.595 0.63726 56487 00 0.53673 49185 66
0.546 0.57758 22745 29 0.49976 70074 50 0.596 0.63851 04281 42 0.53747 31328 39
0.547 0.57877 63628 51 0.50053 70305 98 0.597 0.63975 63587 17 0.53821 06980 90
0.548 0.57997 13835 79 0.50130 64056 22 0.598 0.64100 34455 66 0.53894 76143 74
0.549 0.58116 73406 12 0.50207 51324 28 0.599 0.64225 16938 57 0.53968 38817 48
[IC-l)1 c(-y 1 f=
0.550 0.58236 42378 69 0.50284 32109 28 0.600 0.64350 11087 93 0.54041 95002 71
1 [(-;I2 1 [c-y1
1.570796326795
ELEMENTARY TRANSCENDENTAL FUNCTIONS 209
INVERSE CIRCULAR SINES AND TANGENTS Table 4.14
[ (-;I2
1 (-JP
[ I- ;=1.57079 6326795 [ 5 1 [c-y3
1
210 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[(-[)7
1 [ 4 1 .[ I,-6)1
6 1 1C-l)7 1
;=1.57079 6326795
212 ELEMENTARY TRANSCENDENTAL FUNCTIONS
[ C-j)3
1 [(-[)3
1 [C-i’2
1
216 ELEMENTARY TRANSCENDENTAL FUNCTIONS
Table 4.15 HYPERBOLIC FUNCTIONS
cc-y1 c 1
c-p
218 ELEMENTARY TRANSCENDENTAL FUNCTIONS
0.05 1.17008 87875 0.85463 59992 0.15772 63942 1.01236 23933 0.15580 03292
0.06 1.20743 17210 0.82820 41813 0.18961 37699 1.01781 79512 0.18629 43856
0. 07 1.24596 64399 0.80258 98355 0.22168 83022 1.02427 81377 0.21643 36952
0.08 1.28573 09795 0.77776 76792 0.25398 16502 1.03174 93294 0.24616 60434
0. 09 1.32676 45892 0.75371 32120 0.28652 56886 1.04023 89006 0.27544 21974
0.10 1.36910 77706 0.73040 26910 0.31935 25398 1.04975 52308 0.30421 61929
0.11 1.41280 23184 0.70781 31080 0.35249 46052 1.06030 77132 0.33244 55730
0.12 1.45789 13610 0.68592 21659 0.38598 45975 1.07190 67634 0.36009 15776
0.13 1.50441 94029 0.66470 82576 0.41985 55727 2.08456 38303 0.38711 92833
0.14 1.55243 23694 0.64415 04440 0.45414 09627 1.09829 14067 0.41349 76928
0.15 1.60197 76513 0.62422 84336 0.48887 46088 1.11310 30425 0.43919 97777
0.16 1.65310 41518 0.60492 25628 0.52409 07945 1.12901 33573 0.46420 24748
0.17 1.70586 23348 0.58621 37756 0.55982 42796 1.14603 80552 0.48848 66406
0.18 1.76030 42750 0.56808 36059 0.59611 03346 1.16419 39405 0.51203 69673
0.19 1.81648 37088 0.55051 41583 0.63298 47753 1.18349 89335 0.53484 18637
0.20 1.87445 60876 0.53348 80911 0.67048 39982 1.20397 20893 0.55689 33069
0.21 1.93427 86325 0.51698 85988 0.70864 50169 li22563 36157 0.57818 66683
0.22 1.99601 03910 0.50099 93958 0.74750 54976 1.24850 48934 0.59872 05188
0.23 2.05971 22948 0.48550 47001 0.78710 37973 1.27260 84975 0.61849 64181
0.24 2.12544 72203 0.47048 92177 0.82747 90013 1.29796 82190 0.63751 86920
0.25 2.19328 00507 0.45593 81278 0.86867 09615 1.32460 90893 0.65579 42026
0.26 2.26327 77398 0.44183 70677 0.91072 03361 1.35255 74038 0.67333 21140
0. 27 2.33550 93782 0.42817 21192 0.95366 86295 1.38184 07487 0.69014 36583
0.28 2.41004 62616 0.41492 97945 0.99755 82336 1.41248 80280 0.70624 19035
0.29 2.48696 19609 0.40209 70227 1.04243 24691 1.44452 94918 0.72164 15276
0.30 2.56633 23952 0.38966 11374 1.08833 56289 1.47799 67663 0.73635 85995
0. 31 2.64823 59064 0.37760 98638 1.13531 30213 1.51292 28851 0.75041 03695
0. 32 2.73275 33366 0.36593 13069 1.18341 10148 1.54934 23218 0.76381 50706
0.33 2.81996 81081 0.35461 39395 1.23267 70843 1.58729 10238 0.77659 17313
0.34 2.90996 63054 0.34364 65907 1.28315 98573 1.62680 64481 0.78876 00021
0.35 3.00283 67606 0.33301 84355 1.33490 91626 1.66792 75980 0.80033 99933
0.36 3.09867 11407 0.32271 89833 1.38797 60787 1.71069 50620 0.81135 21279
0. 37 3.19756 40381 0.31273 80681 1.44241 29850 1.75515 10531 0.82181 70068
0.38 3.29961 30643 0.30306 58385 1.49827 36129 1.80133 94514 0.83175 52873
0. 39 3.40491 89460 0.29369 27474 1.55561 30993 1.84930 58467 0.84118 75743
0.40 3.51358 56243 0.28460 95433 1.61448 80405 1.89909 75838 0.85013 43239
0.41 3.62572 03579 0.27580 72607 1.67495 65486 1.95076 38093 0.85861 57589
0.42 3.74143 38283 0.26727 72113 1.73707 83085 2.00435 55198 0.86665 17947
0.43 3.86084 02496 0.25901 09757 1.80091 46370 2.05992 56127 0.87426 19762
0. 44 3.98405 74810 0.25100 03946 1.86652 85432 2.11752 89378 0.88146 54241
0. 45 4.11120 71429 0.24323 75614 1.93398 47907 2.17722 23522 0.88828 07899
0.46 4.24241 47373 0.23571 48138 2.00334 99617 2.23906 47756 0.89472 62194
0. 47 4.37780 97717 0.22842 47266 2.07469 25226 2.30311 72491 0.90081 93236
0.48 4.51752 58864 0.22136 01040 2.14808 28912 2.36944 29952 0.90657 71557
0.49 4.66170 09873 0.21451 39731 2.22359 35071 2.43810 74802 0.91201 61950
0.50 4.81047
c
(-;I6
73810
1
0.20787 95764
r( -p) 11
L 0 J
2.30129
[ 1
(d’3
89023 2.50917 84787 0.91715
[ c-y1 23357
Compiled from British Association for the Advancement of Science, Mathematical Tables, vol. 1. Circular and
hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian
probability functions, 3d ed. Cambridge Univ. Press, Cambridge, England, 1951 (with permission). Known
errors have been corrected.
220 ELEMENTARY TRANSCENDENTAL FUNCTIONS
t-1 arcsinh z-111 z arccosh z-ln I <Zi 2-I arcsinh z-111 z arccosh z-ln z
0.50 0.75048 82946 0.62381 07164 0.70841 81861 0.67714 27078
0.49 0.74839 16011 0.62685 90940 z 0. 24
0.25 0.70724 57326 0.67842 57947
0.48 0.74632 48341 0.62981 77884 0.70611 72820 0.67965 18411
0.47 0.74428 85962 0.63268 90778 t 0.23
0.22 0.70503 32895 0.68082 14660
0.46 0.74228 34908 0.63547 51194 2 0.21 0.70399 41963 0.68193 52541
0.45 0.74031 01215 0.63817 79566 : 0.19
0. 20 0.70300 04288 0.68299 37571
0.44 0.73836 90921 0.64079 95268 0.70205 23983 0.68399 74947
0.43 0.73646 10057 0.64334 16670 t 0.18
0.17 0.70115 05002 0.68494 69555
0.42 0.73458 64641 0.64580 61207 0.70029 51134 0.68584 25981
0.41 0.73274 60676 0.64819 45429 2 0.16 0.69948 66000 0.68668 48518
0.40 0.73094 04145 0.65050 85051 3 0.15 0.69872 53043 0.68747 41175
0.39 0.72917 01001 0.65274 95004 3; 14 0.69801 15527 0.68821 07683
0. 38 0.72743 57167 0.65491 89477 : 0.13 0.69734 56533 0.68889 51504
0.37 0.72573 78524 0.65701 81952 0.12 0.69672 78946 0.68952 75836
0. 36 0.72407 70912 0.65904 85249 : 0.11 0.69615 85462 0.69010 83616
0.35 0.72245 40117 0.66101 11555 3 0.10 0.69563 78573 0.69063 77531
0.34 0.72086 91873 0.66290 72458 0.09 0.69516 60572 0.69111 60018
0.33 0.71932 31846 0.66473 78974 3 0.08 0.69474 33542 0.69154 33269
0.32 0.71781 65636 0.66650 41577 0.07 0.69436 99357 0.69191 99235
0.31 0.71634 98766 0.66820 70226 : 0. 06 0.69404 59680 0.69224 59631
0.30 0.71492 36678 0.66984 74382 0.05 0.69377 15954 0.69252 15938
0.29 0.71353 84725 0.67142 63038 : 0.04 0.69354 69408 0.69274 69403
0.28 0.71219 48165 0.67294 44732 0.03 0.69337 21047 0.69292 21046
0.27 0.71089 32154 0.67440 27575 :: 0.02 0.69324 71656 0.69304 71656
0.26 0.70963 41742 0.67580 19258 4 0. 01 0.69317 21796 0.69312 21796
0.25 0.70841 81861 0.67714 27078 4 0. 00 0.69314 71806 0.69314 71806
[
C-56)5
3 [
C-65) 1
I
0>=nearest integer to .I’.
[ 1
C-56)6 *
[ 1
t-y
: 4.73004
7.85320 46
07
i 10.99560
14.13716 78
55
5 17.27875 96
For ~25, .J,,=; [2~+l]r
x-1 Xl x3 <A>
-1.00 2.02876 4.932318 7.97867 ll'r$554 14 i:744 17.37638 20 4?917 23.6?428 26 7:092
-0.95 2.01194 4.90375 7.97258 11:08110 14:20395 17.33351 20:46673 23.60217 26:73905 1:
-0.90 1.99465 4.89425 7.96648 11.07665 14.20046 17.33064 20.46430 23.60006 26.73718 -1
-0.85 1.97687 4.88468 7.96036 11.07219 14.19697 17.32777 20.46187 23.59795 26.73532
-0.80 1.95857 4.87504 7.95422 11.06773 14.19347 17.32490 20.45943 23.59584 26.73345 1:
-0.75 1.93974 4.86534 7.94807 11.06326 14.18997 17.32203 20.45700 23.59372 26.73159
-0.70 1.92035 4.85557 7.94189 11.05879 14.18647 17.31915 20.45456 23.59161 26.72972
-0.65 1.90036 4.84573 7.93571 11.05431 14.18296 17.31628 20.45212 23.58949 26.72785
-0.60 1.87976 4.83583 7.92950 11.04982 14.17946 17.31340 20.44968 23.58738 26.72598
-0.55 1.85852 4.82587 7.92329 11.04533 14.17594 17.31052 20.44724 23.58526 26.72411
-0.50 1.83660 4.81584 7.91705 11.04083 14.17243 17.30764 20.44480 23.58314 26.72225
-0.45 1.81396 4.80575 7.91080 11.03633 14.16892 17.30476 20.44236 23.58102 26.72038
-0.40 1.79058 4.79561 7.90454 11.03182 14.16540 17.30187 20.43992 23.57891 26.71851
-0.35 1.76641 4.78540 7.89827 11.02730 14.16188 17.29899 20.43748 23.57679 26.71664
-0.30 1.74140 4.77513 7.89198 11.02278 14.15835 17.29610 20.43503 23.57467 26.71477
-0.25 1.71551 4.76481 7.88567 11.01826 14.15483 17.29321 20.43259 23.57255 26.71290
-0.20 1.68868 4.75443 7.87936 11.01373 14.15130 17.29033 20.43014 23.57043 26.71102
-0.15 1.66087 4.74400 7.87303 11.00920 14.14777 17.28744 20.42769 23.56831 26.70915
-0.10 1.63199 4.73351 7.86669 11.00466 14.14424 17.28454 20.42525 23.56619 26.70728
-0.05 1.60200 4.72298 7.86034 11.00012 14.14070 17.28165 20.42280 23.56407 26.70541
0.00 1.57080 4.71239 7.85398 10.99557 14.13717 17.27875 20.42035 23.56194 26.70354
0.05 1.53830 4.70176 7.84761 10.99102 14.13363 17.27586 20.41790 23.55982 26.70166
0.10 1.50442 4.69108 7.84123 10.98647 14.13009 17.27297 20.41545 23.55770 26.69979
0.15 1.46904 4.68035 7.83484 10.98192 14.12655 17.27007 20.41300 23.55558 26.69792
0.20 1.43203 4.66958 7.82844 10.97736 14.12301 17.26718 20.41055 23.55345 26.69604
0.25 1.39325 4.65878 7.82203 10.97279 14.11946 17.26428 20.40810 23.55133 26.69417
0.30 1.35252 4.64793 7.81562 10.96823 14.11592 17.26138 20.40565 23.54921 26.69230
0.35 1.30965 4.63705 7.80919 10.96366 14.11237 17.25848 20.40320 23.54708 26.69042
0.40 1.26440 4.62614 7.80276 10.95909 14.10882 17.25558 20.40075 23.54496 26.68855
0.45 1.21649 4.61519 7.79633 10.95452 14.10527 17.25268 20.39829 23.54283 26.68668
0.50 1.16556 4.60422 7.78988 10.94994 14.10172 17.24978 20.39584 23.54071 26.68480
0.55 1.11118 4.59321 7.78344 10.94537 14.09817 17.24688 20.39339 23.53858 26.68293
0.60 1.05279 4.58219 7.77698 10.94079 14.09462 17.24398 20.39094 23.53646 26.68105
0.65 0.98966 4.57114 7.77053 10.93621 14.09107 17.24108 20.38848 23.53433 26.67918
0.70 0.92079 4.56007 7.76407 10.93163 14.08752 17.23817 20.38603 23.53221 26.67730
0.75 0.84473 4.54899 7.75760 10.92704 14.08396 17.23527 20.38357 23.53008 26.67543
0.80 0.75931 4.53789 7.75114 10.92246 14.08041 17.23237 20.38112 23.52796 26.67355
0.85 0.66086 4.52678 7.74467 10.91788 14.07686 17.22946 20.37867 23.52583 26.67168
0.90 0.54228 4.51566 7.73820 10.91329 14.07330 17.22656 20.37621 23.52370 26.66980 :
0.95 0.38537 4.50454 7.73172 10.90871 14.06975 17.22366 20.37376 23.52158 26.66793 1
1.00 0.00000 4.49341 7.72525 10.90412 14.06619 17.22075 20.37130 23.51945 26.66605
For h-0, seejl. Sof Table 10.6. <x>=nearest integer to X.
ELEMENTARY TRANSCENDENTAL FUNCTIONS 225
ROOTS x,, OF cot xn =Xx,, Table 4.20
A x1 22 53 24 25 % x7 29
0.00 1.57080 4.71239 7.85398 10.99557 14,13717 17.27876 20.42035 23.5*6194 26 70354
0.05 1.49613 4.49148 7.49541 10.51167 13.54198 16.58639 19.64394 22.71311 25:79232
0.10 1.42887 4.30580 7.22811 10.20026 13.21418 16.25936 19.32703 22.41085 25.50638
0.15 1.36835 4.15504 7.04126 10.01222 13.03901 16.10053 19.18401 22.28187 25.38952
0.20 1.31384 4.03357 6.90960 9.89275 12.93522 16.01066 19.10552 22.21256 25.32765
0.25 1.26459 3.93516 6.81401 9.81188 12.86775 15.95363 19.05645 22.16965 25.28961
0.30 1.21995 3.85460 6.74233 9.75407 12.82073 15.91443 19.02302 22.14058 25.26392
0.35 1.17933 3.78784 6.68698 9.71092 12.78621 15.88591 18.99882 22.11960 25.24544
0.40 1.14223 3.73184 6.64312 9.67758 12.75985 15.86426 18.98052 22.10377 25.23150
0.45 1.10820 3.68433 6.60761 9.65109 12.73907 15.84728 18.96619 22.09140 25.22062
0.50 1.07687 3.64360 6.57833 9.62956 12.72230 15.83361 18.95468 22.08147 25.21190
0.55 1.04794 3.60834 6.55380 9.61173 12.70847 15.82237 18.94523 22.07333 25.20475
0.60 1.02111 3.57756 6.53297 9.59673 12.69689 15.81297 18.93734 22.06653 25.19878
0.65 0.99617 3.55048 6.51508 9.58394 12.68704 15.80500 18.93065 22.06077 25.19373
0.70 0.97291 3.52649 6.49954 9.57292 12.67857 15.79814 18.92490 22.05583 25.18939
0.75 0.95116 3.50509 6.48593 9.56331 12.67121 15.79219 18.91991 22.05154 25.18563
0.80 0.93076 3.48590 6.47392 9.55486 12.66475 15.78698 18.91554 22.04778 25.18234
0.85 0.91158 3.46859 6.46324 9.54738 12.65904 15.78237 18.91168 22.04447 25.17943
0.90 0.89352 3.45292 6.45368 9.54072 1?.65395 15.77827 18.90825 22.04151 25.17684
0.95 0.57647 3.43865 6.44508 9.53473 12.64939 15.77459 18.90518 22.03887 25.17453
1.00 0.86033 3.42562 6.43730 9.52933 12.64529 15.77128 18.90241 22.03650 25.17245
A-’ Xl 22 53 x4 X5 X6 x7 3% 59 <A>
1.00 0.86033 3.42562 6.43730 9.52933 12.64529 15.77128 18.90241 22.03650 25.17245
0.95 0.84426 3.41306 6.42987 9.52419 12.64138 15.76814 18.89978 22.03424 25.17047 :
0.90 0.82740 3.40034 6.42241 9.51904 12.63747 15.76499 18.89715 22.03197 25.16848
0.85 0.80968 3.38744 6.41492 9.51388 12.63355 15.76184 18.89451 22.02971 25.16650 :
0.80 0.79103 3.37438 6.40740 9.50871 12.62963 15.75868 18.89188 22.02745 25.16452 1
0.75 0.77136 3.36113 6.39984 9.50353 12.62570 15.75553 18.88924 22.02519 25.16254
0.70 0.75056 3.34772 6.39226 9.49834 12.62177 15.75237 18.88660 22.02292 25.16055 :
0.65 0.72851 3.33413 6.38464 9.49314 12.61784 15.74921 18.88396 22.02066 25.15857
0.60 0.70507 3.32037 6.37700 9.48793 12.61390 15.74605 18.88132 22.01839 25.15659 H
0.55 0.68006 3.30643 6.36932 9.48271 12.60996 15.74288 18.87868 22.01612 25.15460 2
0.50 0.65327 3.29231 6.36162 9.47749 12.60601 15.73972 18.87604 22.01386 25.15262 2
0.45 0.62444 3.27802 6.35389 9.47225 12.60206 15.73655 18.87339 22.01159 25.15063 2
0.40 0.59324 3.26355 6.34613 9.46700 12.59811 15.73338 18.87075 22.00932 25.14864
0.35 0.55922 3.24891 6.33835 9.46175 12.59415 15.73021 18.86810 22.00705 25.14666 z
0.30 0.52179 3.23409 6.33054 9.45649 12.59019 15.72704 18.86546 22.00478 25.14467 3
0.25 0.48009 3.21910 6.32270 9.45122 12.58623 15.72386 18.86281 22.00251 25.14268 4
0.20 0.43284 3.20393 6.31485 9.44595 12.58226 15.72068 18.86016 22.00024 25.14070
0.15 0.37788 3.18860 6.30696 9.44067 12.57829 15.71751 18.85751 21.99797 25.13871 G
0.10 0.31105 3.17310 6.29906 9.43538 12.57432 15.71433 18.85486 21.99569 25.13672
0.05 0.22176 3.15743 6.29113 9.43008 12.57035 15.71114 18.85221 21.99342 25.13473 :oo
0.00 0.00000 3.14159 6.28319 9.42478 12.56637 15.70796 18.84956 21.99115 25.13274 Co
* [c-y] ['-;'l] [y;"] ~[(-y'! [(-p] ['-y] ['-;"] [(-p]
For h-l > .20, the maximum error in linear interpolation is (- 4)7; five-point interpolation gives 5D.
*see page n.
5. Exponential Integral and Related Functions
WALTER GAUTSCHI l AND WILLIAM I?. CAHILL 2
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 228
5.1. Exponential Integral . . . . . . . . . . . . . . . . . . 228
5.2. Sine and Cosine Integrals . . . . . . . . . . . . . . . . 231
References . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Table 5.1. Sine, Cosine and Exponential Integrals (O<sl 10) . . . . 238
x-‘Si(x), x-2[Ci(x)-ln x-r]
cc-‘[Ei(x)-ln x-r], x-‘[E,(x)+ln x+-r], x=0(.01).5, 10s
Si(x), Ci(x), 10D; Ei(x), El(x), 9D; x=.5(.01)2
Si(x), Ci(x), 10D; xP Ei(x), xez El(x), 9D; x=2(.1)10
Table 5.2. Sine, Cosine and Exponential Integrals for Large Arguments
(101x1 a) . . . . . . . . . . . . . . . . . . . . . . . . . 243
qf(x), 9D; x2g(x), 7D; ze-“Ei(x), 8D; xezEl(s), 10D
f(x)=-si(x) cos x+Ci(x) sin 2, g(x)=-si(x) sin x-C;(x) co9 x
x-‘=.l(-.005)0
Table 5.3. Sine and Cosine Integrals for Arguments TIZ (0 Ix 210) . . . 244
Si(rx), Gin(m), x=0(.1)10, 7D
Table 5.5. Exponential Integrals E,(x) for Large Arg,urnents (2 Ix _< 03) a 248
(x+n)e”E,(x), n=2, 3, 4, 10, 20, x-‘=.5(-.05).1(-.01>0, 5D
227
5. Exponential Integral and Related Functions
Mathematical Properties
5.1.9
(larg zl<?r>
&(z) =n!2-n-1(e” [l-z+&. . . +(-l)“$]
f” e-’ r” edt
5.1.2 Ei(x)=--J1 t dt=T- t (xx9
2 m 22
--e-z (l+~+~+ . . . +$I .
5.1.3 (x>l)
5.1.4
E,(z)=jy e; at (n=O, 1,2, . . .; ~z>O)
(D
5.1.5
%(4’
s 1
t*e-z*dt ‘(n=O, 1,2, . . .; 92>0)
Interrelations
5.1.7 i
E1(-xfiO)=-Ei(x):)ii?r,
-Ei(x)=~[~l(-x+iO)+E1(-x--iO)] (x>O)
5.1.18
J%(4<J%1wL+1(4 (x>O;n=1,2,3,. . .)
5.1.19
5.1.20
<e%(x)<h (I+:) (x>O)
n:l
-151 \’ 5.1.21
5.1.22
5.1.10 Ei(z)=r+lnz+FI n$ (XX)
a . E,(z)=e-*
(
--$$-n$-&. . .) (b-g 4-G)
Special Values
5.1.12
5.1.23 am =A (n>l)
Derivatives 5.1.36
le --Otsin bt
5 . 126 --
dE”‘z)---E (n==1,2,3,. . .) dt=arctan k +Y&(a+ib)
. 12- 1 (z) t
dz S0
(a>O, b real)
5.1.27
5.1.37
S t bt)
~n[e”El(z)l=&l [e”E,(z)l 1 e”‘(l-cos
+Ei(a)
0
+(-W-l)! (+l 2 3
, , ,-**
)
2” +WE,(-a+ib) (a>O,b real)
5.1.38
Definite and Indefinite Integrals
1 e-“‘(l--OS
t bt)
(For more
[5.3], [5.6],
extensive
[5.11], [5.12],
tables of integrals see
[5.13]. For integrals
S 0
dt=$ In (1+$)-&(a)
z e’-1
5.1.40 - dt=Ei (x)-In x--y (XX)
5.1.29 S0 t
5.1.41
S op & dt=eciQbE,(--iab) (a>% b>O)
S
G2 dx=$ [e-“E,(-a--ix)-ee”E,(u--ix)]
5.1.30
+const.
S- t+ib
o t2+b2 eI”ldt=e-““(-Ei(ub)+ir)
5.1.43
+const.
(a>% b>O)
a& dx=-; 9(ef”EI(-x+ia))+const. (a>O)
5.1.32
Sm0
e-“‘-emb’
t
dt=h b
a
S
5.1.44
5.1.33
S 0
mE:(t)dt=2 ln 2 S s2 dx=-L2(ef”EI(-x+ia))+const. (a>O)
5.1.34
m 5.1.45 E,(z)=P-'r(l-n,
2)
S 0
e+E,(t)dt= 5.1.46
5.1.47
q&)=2-*--lIyTL+1,
j3,(z)=z-“-‘[r(n+l, -z)-Iyn+1,
2)
z)]
q[ln (1 +a) +g: q] (a>-1) Relation to Spherical Bessel Functions (see 10.2)
(a>% b>O)
EXPONENTIAL INTEGRAL AND RELATED PUNG!CIONS 231
Number-Theoretic Significance of li (z) ao= - .57721 566 a3= .05519 968
al= .999!39 193 a4= - .00976 004
(Assuming Riemann’s hypothesis that all non-
az= - .249!Jl 055 as= .00107 857
real zeros of t(z) have a real part of 3)
5.1.54 l<z<=J
5.1.50 li (x)-7r(x)=O(@ In 2) (x+=>
r(x) is the number of primes less than or equal xe=J%c4=x2+b x2+a:x+T+6
x+b (2)
to 2.
Y I&> 1<5x 10-6
a,=2.334733 b1=3.330657
a2= 250621 bz= 1.681534
5.1.55 lOSx<w
l&)l<lO-’
a1=4.03640 b,=5.03637
%=1.15198 b2=4.19160
5.1.56 lix<w
I I
0Y I I I ~e(x>1<2Xlo-*
200 400 600 000 1000 L-X
En(z)m e+il-n+n(n+l)
2 2 7- n(n+l)(n+2)+
z3 j 5.2. Sine and Cosine Integrals
*-*
Definitions
(larg 4 <%d
5.1.52 5.2.2 ta
5.2.4 7
Shi(z)=
S 0
-
t
dt
Auxiliary Functions
5.2.17 Shi(z)=~o (2n+$L+1)!
5.2.6 f(z)=Ci(z) sin z--i(z) co9 2
5.2.7 g(z)= -Ci(z) cos z-si(z) sin 2 5.2.18 Chi(z)=r+ln dgl &,
Sine and Cosine Integrals in Terms of Auxiliary Symmetry Relations
Functions
5.2.19 Si(--z)=-Si(z), Si(Z)=si(z)
5.2.8 Si(z)=i--f(z) co9 z-g(z) sin 2
5.2.20
5.2.9 Ci(z)=f(z) sin z-g(z) co9 2 Ci(-z)=Ci(z)--.i7r (O<w z<d
Integral Representations C;(Z)=-)
I
Relation to Exponential Integral
5.2.10 si(z)=- e-2 cont cos (2 sin t)dt 5.2.21
Si(z)=& [E,(k)-EI(--iz)l+~ (kg zl<t$
5.2.11 Ci(z) +&(z)=l’ e-’ coB’ sin (z sin t)dt
5.2.13
so-cos
g(z)= t+zt (jt= (&>O)5.2.23
Ci(z)=-k [E,(iz)+IG(--iz)] (larg 4-C:)
Value at Infinity
Integrals
FIGURE 5.6. y=Si(z) and y=Ci(z) 5.2.28 a e-Wi (t)dt= -& In (l+u*) (.@a>O)*
s Cl
Series Expansions
5.2.14 5.2.!29
S m e-=‘si (t)dt=--1
a
arctan a (9?a>o)
m
0
s’(1-e-a’)
‘OS
bt&=i
ln(I+$)
+Ci
(b) a,=7.547478 b1=12.723684 *
a~= 1.564072 bz= 15.723606 *
0 t
5.2.38 l<z<m
+ %?I$ (a+ ib) (a real, b>O)
Asymptotic Expansions
5.2.34 le(z)1<5Xlo-’
(larg zl<r) al= 38.027264 b1= 40.021433
,(z)+(l-$+$-$+. . .)
&-265.187033 bz=322.624911
5.2.35 a3=335A77320 b3=570.236280
a,= 38.102495 b,=157.105423
5.2.39 l<s<=J
Rational Approximations *
5.2.36 152<a
(&):<3Xlo-’
Numerical Methods
5.3. Use and Extension of the Tables Example 3. Compute Si (20) to 5D.
Since l/20=.05 from Table 5.2 we find
Example 1. Compute Ci (.25) to 5D.
j(20) = .049757, g(20) = .002464. From Table 4.8,
From Tables 5.1 and 4.2 we have
sin 20 = .912945, cos 20 = .408082. Using 5.2.8
Ci (.25) - M.25) -Y= _ .24g350
(.25)* I
Si(20) =;-j(20) co9 20-g(20) sin 20
Ci (.25)=(.25)2(-.249350)+(-l.38629) =1.570796-.022555=1.54824.
+.577216= -.82466.
Example 4. Compute E,(z), n=l(l)N, to 5S
Example 2. Compute Ei (8) to 5s.
for z= 1.275, N= 10.
From Table 5.1 we have ze-‘Ei (z) =1.18185 for
If z is less than about five, the recurrence
s=8. From Table 4.4, e8=2.98096X103. Thus
relation 5.1.14 can be used in increasing order of n
Ei (8) =440.38.
without serious loss of accuracy.
By quadratic interpolation in Table 5.1 we get
*see page II.
8 From C. Hastings, Jr., Approximations for digital & (1.275) = .1408099, and from Table 4.4, e-l.*”
computers, Princeton Univ. Press, Princeton, N.J., 1955 =. 2794310. The recurrence formula 5.1.14 then
(with permission). yields
234 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
1: 0.lw) n5 a”(l)
- .324297 f(z)==.063261 -.105354i
e-‘==.031510 -.022075i
10 .280560 4 552373 El(z)= --.000332 -.004716i
9 - .2oE?z 3 - .449507
8 .319908 2 .878885 Repeating the calculation with zo=3+6i and
7 -.253812 1 - .735759 Az=.2578+.8943,i we get the same result.
6 .40465 0 2.350402 An alternative ,procedure is to perform bivariate
interpolation in t.he real and imaginary parts of
The functions ,&(s) and &(x) can be obtained ze“El (2).
from Table 10.8 using 5.1.48, 5.1.49.
Example 10. Compute E,(z) for z=-4.2
Example 9. Compute E,(z) for z=3.2578 + 12.7i.
+6.8943i. Using the formula at the bottom of Table 5.6
From Table 5.6 we have for z,,=z0+iy,=3+7i
.711093
e”E,(z) =
z&~El (zo) = .934958+ .095598i, -3.784225+-12.7i
.278518 .010389
e”oE,(zo)=.059898-.107895i. -!- -1.90572+12.7i+2.0900+12.7i
From Taylor’s formula with f(z) =ezEI (z) we have - .0184106- .0736698i
E,(z)z--1.87133-4.7054Oi.
+jq (A,@+. . .
References
Texts
[5.1] F. J. Corbat6, On the computation of auxiliary [5.6] W. Grijbner and N. Hofreiter, Integraltafel
functions for two-center integrals by means of a (Springer-Verlag, Wien and Innsbruck, Austria,
high-speed computer, J. Chem. Phys. 24,452-453 1949-50).
(1956). [5.7] C. Hastings, *Jr., Approximations for digital com-
[5.2] A. Erdelyi et al., Higher transcendental functions, puters (Princeton Univ. Press, Princeton, N.J.,
vol. 2 (McGraw-Hill Book Co., Inc., New York, 1955).
N.Y., 1953). [5.8] E. Hopf, Mathematical problems of radiative
equilibrium, Cambridge Tracts in Mathematics
[5.3] A. Erdelyi et al., Tables of integral transforms, ~01s.
and Mathematical Physics, No. 31 (Cambridge
1, 2 (McGraw-Hill Book Co., Inc., New York, Univ. Press, Cambridge, England, 1934).
N.Y., 1954).
[5.9] V. Kourganofl’, Basic methods in transfer problems
[5.4] W. Gautschi, Some elementary inequalities relating (Oxford Un:v. Press, London, England, 1952).
to the gamma and incomplete gamma function, [5.10] F. Lijsch and F. Schoblik, Die Fakultiit und ver-
J. Math. Phys. 38, 77-81 (1959). wandte Funktionen (B. G. Teubner, Leipzig,
[5.5] W. Gautschi, Recursive computation of certain Germany, 1951).
integrals, J. Assoc. Comput. Mach. 8, 21-40 [5.11] N. Nielsen, Theorie des Integrallogarithmus (B. G.
(1961). Teubner, Leipzig, Germany, 1906).
236 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
[ C-j)5
1 ll 1
c-5)2
5
C--5)1
I: 15 [(-;I91
EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS 241
SINE, COSINE AND EXPONENTIAL INTEGRALS Table 5.1
Si(X) Ci(x) Ei(x’l
l.GO 1.32468
1.51 1.33131
35312
36664
0.47035
0.47079
63172
32232
3.30128
3.33121
5449
3449 0.09854
El(x)
0.10001 9582
4365
1.52 1.33790 40489 0.47116 13608 3.36124 2701 0.09709 3466
1.53 1.34445 45453 0.47146 15952 3.39137 4858 0.09566 6424
1.54 1.35096 50245 0.47169 47815 3.42161 1576 0.09426 2786
1.60541 29768
[C-j)5 cc-y1
2.00 0.42298 08288 4.95423 4356 0.04890 0511
1 c 1(592
c 1 (-!I3
242 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
Table 5.2
[ 1
(-s3)5
[ 1
(-s3)6
[I(-P4
1
EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS 247
EXPONENTIAL INTEGRALS E,(z) Table 5.4,
.I: 82 (.r) &::I(J) JY4 (A) ~,~(.r) -f320 (.I.)
1.00 0.14849 55 0.10969 20 0.08606 25 0.03639 40 0.01834 60
1. 01 0.14631 99 0.10821 79 0.08497 30 0.03599 29 0.01815 39
1.02 0.14418 04 0.10676 54 0.08389 81 0.03559 63 0.01796 39
1. 03 0.14207 63 0.10533 42 0.08283 76 0.03520 41 0.01777 59
1.04 0.14000 68 0.10392 38 0.08179 13 0.03481 63 0.01758 98
1.05 0.13797 13 0.10253 39 0.08075 90 0.03443 28 0.01740 57
1. 06 0.13596 91 0.10116 43 0.07974 06 0.03405 35 0.01722 35
1. 07 0.13399 96 0.09981 45 0.07873 57 0.03367 85 0.01704 33
1. 08 0.13206 22 0.09848 42 0.07774 42 0.03330 77 0.01686 49
1.09 0.13015 62 0.09717 31 0.07676 59 0.03294 10 0.01668 84
1.10 0.12828 11 0.09588 09 0.07580 07 0.03257 84 0.01651 37
1. 11 0.12643 62 0.09460 74 0.07484 83 0.03221 98 0.01634 09
1.12 0.12462 10 0.09335 21 0.07390 85 0.03186 52 0.01616 99
1.13 0.12283 50 0.09211 49 0.07298 12 0.03151 45 0.01600 07
1. 14 0.12107 75 0.09089 53 0.07206 61 0.03116 78 0.01583 33
1.15 0.11934 81 0.08969 32 0.07116 32 0.03082 49 0.01566 76
1. 16 0.11764 62 0.08850 83 0.07027 22 0.03048 58 0.01550 37
1.17 0.11597 14 0.08734 02 0.06939 30 0.03015 05 0.01534 14
1. 18 0.11432 31 0.08618 88 0.06852 53 0.02981 89 0.01518 09
1.19 0.11270 08 0.08505 37 0.06766 91 0.02949 10 0.01502 21
1.20 0.11110 41 0.08393 47 0.06682 42 0.02916 68 0.01486 49
1. 21 0.10953 25 0.08283 15 0.06599 04 0.02884 61 0.01470 94
1.22 0.10798 55 0.08174 39 0.06516 75 0.02852 90 0.01455 55
1.23 0.10646 27 0.08067 17 0.06435 55 0.02821 55 0.01440 32
1.24 0.10496 37 0.07961 46 0.06355 40 0.02790 54 0.01425 26
1.25 0.10348 81 0.07857 23 0.06276 31 0.02759 88 0.01410 35
1.26 0.10203 53 0.07754 47 0.06198 25 0.02729 55 0.01395 59
1.27 0.10060 51 0.07653 16 0.06121 22 0.02699 57 0.01381 00
1.28 0.09919 70 0.07553 26 0.06045 19 0.02669 91 0.01366 55
1.29 0.09781 06 0.07454 76 0.05970 15 0.02640 59 0.01352 26
1.30 0.09644 55 0.07357 63 0.05896 09 0.02611 59 0.01338 11
1. 31 0.09510 15 0.07261 86 0.05822 99 0.02582 91 0.01324 12
1.32 0.09377 80 0.07167 42 0.05750 85 0.02554 55 0.01310 27
1.33 0.09247 47 0.07074 29 0.05679 64 0.02526 51 0.01296 57
1.34 0.09119 13 0.06982 46 0.05609 36 0.02498 78 0.01283 01
1.35 0.08992 75 0.06891 91 0.05539 98 0.02471 35 0.01269 59
1. 36 0.08868 29 0.06802 60 0.05471 51 0.02444 23 0.01256 31
1. 37 0.08745 71 0.06714 53 0.05403 93 0.02417 41 0.01243 17
1. 38 0.08624 99 0.06627 68 0.05337 22 0.02390 88 0.01230 17
1. 39 0.08506 10 0.06542 03 0.05271 37 0.02364 65 0.01217 31
1. 40 0.08388 99 0.06457 55 0.05206 37 0.02338 72 0.01204 58
1.41 0.08273 65 0.06374 24 0.05142 22 0.02313 06 0.01191 98
1.42 0.08160 04 0.06292 07 0.05078 89 0.02287 70 0.01179 52
1.43 0.08048 13 0.06211 04 0.05016 37 0.02262 61 0.01167 19
1.44 0.07937 89 0.06131 11 0.04954 66 0.02237 80 0.01154 99
1.45 0.07829 30 0.06052 27 0.04893 74 0.02213 27 0.01142 91
1.46 0.07722 33 0.05974 52 0.04833 61 0.02189 01 0.01130 96
1.47 0.07616 94 0.05897 82 0.04774 25 0.02165 01 0.01119 14
1. 48 0.07513 13 0.05822 17 0.04715 65 0.02141 28 0.01107 44
1.49 0.07410 85 0.05747 55 0.04657 80 0.02117 82 0.01095 86
1.50 0.07310 08 0.05673 95 0.04600 70 0.02094 61 0.01084 40
1.51 0.07210 80 0.05601 35 0.04544 32 0.02071 67 0;01073 07
1.52 0.07112 98 0.05529 73 0.04488 67 0.02048 97 0.01061 85
1.53 0.07016 60 0.05459 08 0.04433 72 0.02026 53 0.01050 75
1.54 0.06921 64 0.05389 39 0.04379 48 0.02004 33 0.01039 77
1.55 0.06828 07 0.05320 64 0.04325 93 0.01982 38 0.01028 90
1.56 0.06735 87 0.05252 83 0.04273 07 0.01960 67 0.01018 15
1. 57 0.06645 02 0.05185 92 0.04220 87 0.01939 21 0.01007 50
1.58 0.06555 49 0.05119 92 0.04169 35 0.01917 98 0.00996 97
1.59 0.06467 26 0.05054 81 0.04118 47 0.01896 98 0.00986 56
1.60 0.06380 32 0.04990 57 0.04068 25 0.01876 22 0.00976 24
[ 1 [ 1
C-3615 C-i)3
[
c-y
I [
(-;I6
I [ 1
C-l)3
248 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
1.021896 0.036759 1.020942 0.038361 1.019824 0.039950 1.018533 0.041505 1.017066 0.043001
-8 -7 -6 -5
1.152759 0.003489 1.181848 0.008431 1.222408 0.020053 1.278884 0.046723 1.353831 0.105839
1.146232 0.026376 1.169677 0.038841 1.199049 0.060219 1.233798 0.097331 1.268723 0.160826
1.134679 0.044579 1.151385 0.060814 1.169639 0.085335 1.186778 0.122162 1.196351 0.175646
1.120694 0.057595 1.131255 0.074701 1.140733 0.098259 1.146266 0.130005 1.142853 0.170672
1.106249 0.065948 1.111968 0.082156 1.115404 0.102861 1.114273 0.128440 1.105376 0.158134
1.092564 0.070592 1.094818 0.085055 1.094475 0.102411 1.089952 0.122397 1.079407 0.143879
1.080246 0.072520 1.080188 0.084987 1.077672 0.099188 1.071684 0.114638 1.061236 0.130280
1.069494 0.072580 1.067987 0.083120 1.064339 0.094618 1.057935 0.106568 1.048279 0.118116
1.060276 0.071425 1.057920 0.080250 1.053778 0.089537 1.047493 0.098840 1.038838 0.107508
1.052450 0.069523 1.049645 0.076885 1.045382 0.084405 1.039464 0.091717 1.031806 0.098337
1.045832 0.067197 1.042834 0.073340 1.038659 0.079462 1.033205 0.085271 1.026459 0.090413
:i 1.040241 0.064664 1.037210 0.069803 1.033231 0.074821 1.028260 0.079488 1.022317 0.083544
1.035508 0.062063 1.032539 0.066381 1.028808 0.070524 1.024300 0.074315 1.019052 0.077561
:: 1.031490 0.059482 1.028638 0.063128 1.025171 0.066576 1.021090 0.069688 1.016439 0.072320
14 1.028065 0.056975 1.025359 0.060070 1.022152 0.062962 1.018458 0.065542 1.014319 0.067702
1.025132 0.054573 1.022583 0.057215 1.019626 0.059658 1.016277 0.061817 1.012577 0.063610
:'b 1.022608 0.052291 1.020219 0.054559 1.017494 0.056638 1.014452 0.058460 1.011130 0.059962
1.020426 0.050135 1.018192 0.052094 1.015681 0.053874 1.012912 0.055424 1.009915 0.056694
:; 1.018530 0.048106 1.016444 0.049806 1.014129 0.051341 1.011600 0.052670 1.008887 0.053752
19 1.016874 0.046201 1.014929 0.047684 1.012790 0.049015 1.010476 0.050161 1.008009 0.051092
20 1.015422 0.044413 1.013607 0.045714 1.011629 0.046875 1.009505 0.047870 1.007254 0.048675
For [21>4, linear interpolation will yield about four decimals, eight-point interpolation will
yield about six decimals.
See Example39 -10.
250 EXPONENTIAL INTEGRAL AND RELATED FUNCTIONS
Y\X 1 2 3 4 5
0 0.596347 0.000000 0.722657 0.000000 0.786251 0.000000 0.825383 0.000000 0.852111 0.000000
: 0.777514
0.673321 0.147864
0.1865.70 0.747012
0.796965 0.075661
0.118228 0.797036 0.045686 0.831126 0.030619 0.855544 0.021985
0.823055 0.078753 0.846097 0.055494 0.864880 0.040999
0.853176 0.096659 0.865521 0.072180 0.877860 0.055341
: 0.847468
0.891460 0.165207
0.181226 0.844361
0.881036 0.131686
0.132252 0.880584 0.103403 0.885308 0.081408 0.892143 0.064825
0.919826 0.148271 0.907873 0.125136 0.903152 0.103577 0.903231 0.085187 0.906058 0.070209
2 0.938827 0.132986 0.927384 0.116656 0.921.006 0.100357 0.918527 0.085460 0.918708 0.072544
0.952032 0.119807 0.941722 0.107990 0.934958 0.095598 0.931209 0.083666 0.929765 0.072792
i 0.961512 0.108589 0.952435 0.099830 0.945868 0.090303 0.941594 0.080755 0.939221 0.071700
9 0.968512 0.099045 0.960582 0.092408 0.954457 0.084986 0.950072 0.077313 0.947219 0.069799
10 0.973810 0.090888 0.966885 0.085758 0.961283 0.079898 0.957007 0.073688 0.953955 0.067447
0.977904 0.083871 0.971842 0.079836 0.966766 0.075147 0.962708 0.070080 0.959626 0.064878
:: 0.981127 0.077790 0.975799 0.074567 0.971216 0.070769 0.967423 0.066599 0.964412 0.062242
13 0.983706 0.072484 0.979000 0.069873 0.974865 0.066762 0.971351 0.063300 0.968464 0.059630
14 0.985799 0.067822 0.981621 0.065679 0.977888 0.063104 0.974646 0.060206 0.971911 0.057096
15 0.987519 0.063698 0.983791 0.061921 0.980414 0.059767 0.977430 0.057322 0.974858 0.054671
0.985606 0.058539 0.982544 0.056723 0.979799 0.054644 0.977391 0.052371
:; 0.988949
0.990149 0.060029
0.056745 0.987138 0.055485 0.984353 0.053941 0.981827 0.052162 0.979579 0.050200
18 0.991167 0.053792 0.988442 0.052717 0.985902 0.051394 0.983574 0.049861 0.981478 0.048160
19 0.992036 0.051122 0.989561 0.050199 0.987237 0.049057 0.985089 0.047728 0.983135 0.046245
20 0.992784 0.048699 0.990527 0.047900 0.988395 0.046909 0.986410 0.045749 0.984587 0.044449
Y\X 6 7 8 9 10
0 0.871606 0.000000 0.886488 0.000000 0.898237 0.000000 0.907758 0.000000 0.915633 0.000000
0.873827 0.016570 0.888009 0.012947 0.899327 0.010401 0.908565 0.008543 0.916249 0.007143
: 0.880023 0.031454 0.892327 0.024866 0.902453 0.020140 0.910901 0.016639 0.918040 0.013975
3 0.889029 0.043517 0.898793 0.034995 0.907236 0.028693 0.914531 0.023921 0.920856 0.020230
4 0.899484 0.052380 0.906591 0.042967 0.913167 0.035755 0.919127 0.030145 0.924479 0.025717
5 0.910242 0.058259 0.914952 0.048780 0.919729 0.041242 0.924336 0.035208 0.928664 0.030334
0.920534 0.061676 0.923283 0.052667 0.926481 0.045242 0.929836 0.039123 0.933175 0.034063
; 0.929945 0.063220 0.931193 0.054971 0.933096 0.047942 0.935365 0.041986 0.937807 0.036944
0.938313 0.063425 0.938469 0.056047 0.939359 0.049570 0.940731 0.043936 0.942398 0.039060
i 0.945629 0.062714 0.945023 0.056211 0.945154 0.050349 0.945812 0.045128 0.946833 0.040514
10 0.951965 0.061408 0.950850 0.055725 0.950427 0.050481 0.950535 0.045711 0.951035 0.041413
0.957427 0.059735 0.955987 0.054790 0.955176 0.050135 0.954870 0.045818 0.954959 0.041861
:: 0.962128 0.057855 0.960495 0.053560 0.959421 0.049444 0.958814 0.045563 0.958586 0.041948
13 0.966178 0.055877 0.964444 0.052146 0.963201 0.048514 0.962379 0.045038 0.961913 0.041755
14 0.969673 0.053874 0.967903 0.050627 0.966559 0.047425 0.965591 0.044319 0.964949 0.041347
15 0.972699 0.051894 0.970935 0.049062 0.969539 0.046236 0.968477 0.043463 0.967710 0.040780
0.975326 0.049966 0.973551 0.047489 0.972185 0.044992 0.971067 0.042516 0.970214 0.040095
:7" 0.977617 0.048109 0.975940 0.045935 0.974538 0.043724 0.973393 0.041512 0.972484 0.039329
0.979622 0.046332 0.978009 0.044419 0.976632 0.042456 0.975481 0.040477 0.974540 0.038508
:9" 0.981384 0.044641 0.979839 0.042951 0.978500 0.041205 0.977357 0.039431 0.976402 0.037653
20 0.982938 0.043036 0.981465 0.041538 0.980169 0.039980 0.979047 0.038388 0.978090 0.036781
* If ~~10 or y>lO then (see [5.15])
eZEl(z)x 0.711093 +------+--------+e,161<3~10-6.
0.278518 0.010389
2+0.415775 z+2.29428 z+6.2900
0.932672 0.026361 0.936356 0.023091 0.939729 0.020373 0.942816 0.018095 0.945640 0.016169
0.936400 0.029857 0.939462 0.026339 0.942338 0.023378 0.945024 0.020867 0.947522 0.018725
0.940297 0.032670 0.942757 0.029036 0.945140 0.025934 0.947419 0.023273 0.949582 0.020980
0.944229 0.034847 0.946132 0.031205 0.948047 0.028052 0.949933 0.025315 0.951765 0.022931
0.948093 0.036453 0.949500 0.032887 0.950985 0.029756 0.952502 0.027004 0.954018 0.024582
10 0.951816 0.037566 0.952792 0.034134 0.953895 0.031081 0.955075 0.028365 0.956296 0.025949
11 0.955347 0.038261 0.955958 0.035004 0.956729 0.032068 0.957610 0.029426 0.958563 0.027052
12 0.958659 0.038612 0.958968 0.035552 0.959454 0.032761 0.960073 0.030221 0.960787 0.027915
13 0.961739 0.038684 0.961800 0.035833 0.962049 0.033201 0.962443 0.030781 0.962947 0.028564
14 0.964583 0.038534 0.964447 0.035893 0.964499 0.033428 0.964702 0.031140 0.965026 0.029024
0.967199 0.038211 0.966907 0.035775 0.966799 0.033479 0.966843 0.031327 0.967011 0.029320
El 0.969597 0.037756 0.969184 0.035515 0.968947 0.033384 0.968860 0.031370 0.968897 0.029476
17 0.971789 0.037200 0.971285 0.035144 0.970946 0.033172 0.970752 0.031293 0.970680 0.029512
18 0.973792 0.036572 0.973220 0.034687 0.972802 0.032865 0.972521 0.031117 0.972359 0.029448
19 0.975621 0.035893 0.974999 0.034166 0.974521 0.032485 0.974172 0.030862 0.973936 0.029301
20 0.977290 0.035179 0.976634 0.033597 0.976112 0.032049 0.975709 0.030542 0.975414 0.029086
y\x 16 17 18 19 20
0 0.944130 0.000000 0.947100 0.000000 0.949769 0.000000 0.952181 0.000000 0.954371 0.000000
0.944306 0.003128 0.947250 0.002804 0.949897 0.002527 0.952291 0.002290 0.954467 0.002085
: 0.944829 0.006196 0.947693 0.005560 0.950277 0.005016 0.952619 0.004549 0.954752 0.004144
0.945678 0.009150 0.948416 0.008223 0.950898 0.007430 0.953156 0.006745 0.955219 0.006151
4' 0.946824 0.011940 0.949395 0.010754 0.951741 0.009735 0.953887 0.008853 0.955856 0.008084
0.948226 0.014529 0.950600 0.013121 0.952782 0.011904 0.954793 0.010847 0.956650 0.009922
2 0.949842 0.016886 0.951995 0.015296 0.953995 0.013916 0.955853 0.012709 0.957581 0.011649
7 0.951624 0.018994 0.953545 0.017265 0.955349 0.015753 0.957043 0.014425 0.958631 0.013253
0.953527 0.020847 0.955212 0.019019 0.956815 0.017409 0.958337 0.015986 0.959779 0.014723
9" 0.955509 0.022445 0.956960 0.020555 0.958363 0.018878 0.959712 0.017387 0.961004 0.016056
0.957530 0.023797 0.958758 0.021878 0.959966 0.020163 0.961144 0.018628 0.962288 0.017250
:1" 0.959559 0.024917 0.960576 0.022998 0.961598 0.021270 0.962612 0.019712 0.963611 0.018305
0.961568 0.025823 0.962391 0.023927 0.963238 0.022207 0.964097 0.020645 0.964956 0.019227
:: 0.963534 0.026534 0.964181 0.024679 0.964868 0.022984 0.965582 0.021436 0.966310 0.020021
14 0.965443 0.027070 0.965931 0.025271 0.966472 0.023616 0.967052 0.022094 0.967658 0.020694
20 0.975215 0.027685 0.975099 0.026343 0.975057 0.025062 0.975079 0.023842 0.975155 0.022684
El(z)+ln z
Y\" -2.0 -1.5 -1.0 -0.5 0
-2.895820 0.000000 -1.895118 0.000000 -1.147367 0.000000 -0.577216 0.000000
FE -4.219228
-4.261087 0.636779
0.000000 -2.867070 0.462804 -1.875155 0.342700 -1.133341 0.258840 -0.567232 0.199556
-2.781497 0.917127 -1.815717 0.679691 -1.091560 0.513806 -0.537482 0.396461
2 -4.094686
-3.890531 1.260867
1.859922 -2.641121 1.354712 -1.718135 1.005410 -1.022911 0.761122 -0.488555 0.588128
0:8 -3.611783 2.422284 -2.449241 1.767748 -1.584591 1.314586 -0.928842 0.997200 -0.421423 0.772095
1.0 -3.265262 2.937296 -2.210344 2.149077 -1.418052 1.602372 -0.811327 1.218731 -0.337404 0.946083
PHILIP J. DAVIS 1
Contents
Page
Mathematical Properties. . . . . . . . . . . . . . . . . . . . 255
6.1. Gamma Function. . . . . . . . . . . . . . . . . . . . 255
6.2. Beta Function . . . . . . . . . . . . . . . . . . . . . 258
6.3. Psi (Digamma) Function. . . . . . . . . . . . . . . . . 258
6.4. Polygamma Functions. . . . . . . . . . . . . . . . . . 260
6.5. Incomplete Gamma Function. . . . . . . . . . . . . . . 260
6.6. Incomplete Beta Function. . . . . . . . . . . . . . . . 263
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 263
6.7. Use and Extension of the Tables. . . . . . . . . . . . . 263
6.8. Summation of Rational Series by Means of Polygamma Func-
tions. . . . . . . . . . . . . . . . . . . . . . . . . 264
References. . . . . . . . . . . . . . . . . . . . . . . . . . . 265
Table 6.1. Gamma, Digamma and Trigamma Functions (1 Is_< 2) . . 267
r(x), In I’(z), #(z), +‘(x)), x=1(.005)2, 10D
Table 6.2. Tetragamma and Pentagamma Functions (15 ~12) . . . 271
$“(s), $b3’(x), x=1(.01)2, 10D
Table 6.3. Gamma and Digamma Functions for Integer and Half-
Integer Values (1 In< 101) . . . . . . . . . . . . . . . . . . 272
uw, 9s n!/[(27r)%P++]e?, 8D
r(n+3), 8s In n+(n), 8D
n=l(l.>lOl
Table 6.4. Logarithms of the Gamma Function (1 In < 101). . . . . . 274
klo r(n+$>, 8s
n=l(l)lOl
253
254 GAMMA FUNCTION AND RELATED FUNCTIONS
Page
Table 6.5. Auxiliary Functions for Gamma and Digamma Func-
tions (66I:ZIm) . . . . . . . . . . . . . . . . . . . . . . . 276
z-l=.015 (-.OOl)O, 8D
-3
Recurrence Formulas
i.1.29 r(i~)r(-iy)=lr(iy)[2=ysi~ny
6.1.15 r(z+1)=zr(z)=z!=z(z-l)!
k
6.1.20 r(nz)=(2rr)t"-"'~~'-l~~r(~+~) '1 l.oooooc~oooo 000000
a
2 0.57721 56649 015329 I- *-,
Binomial Coefficient 3 -0.65587 80715 202538 \ I
’ s--z uz+l) 4 -0.04200 26350 340952'
6.1.21
0?V w!(z--w)! r(w+i)r+-w+i) 5
6
0.16653 86113
-0.04219 77345
822915'5
555443Lit
Pochhammer’s Symbol 7 -0.00962 19715 278770 '
6.1.22 8 0.00721 89432 466630 -
(z>c= 1, 9 -0.00116 51675 918591
10 -0.00021 52416 741149
(z).=z(z+l)(z+2) . . . @+?I-l)=W 0.00012 80502 823882
11
12 -0.00002 01348 547807 0
Gamma Function in the Complex Plane -0.00000 12504 934831 1
13
14 0.00000 11330 272320 2,
6.1.23 r(z)=r(2); In r(B)=ln rl.2)
15 -0.00000 02056 338417 3
16 0.00000 00061 160950 '-f
6.1.24 arg r(z+i)=arg r(z)+arhnf 17 0.00000 00050 020075 3
18 -0.00000 00011 812746 b
19 0.00000 00001 0434271
20 0.00000 00000 0778234
21 -0.00000 00000 036968 A
6.1.26
22 0.00000 00000 005100 I'
6.1.27 23 -0.00000 00000 000206k,
24 -0.00000 00000 000054b
mgr b+id =8m +n$o(--&-arctan&-) 25 0.00000 00000 000014 i;
26 0.00000 00000 000001 (5
(z+iy#0,-1,-2, .. .)
2 The coeffhxents ck are from H. T. Davis, Tables of
where ~W=rWr(2) higher mathematical functions, 2 vols., Principia Press,
Bloomington, Ind., 1933, 1935 (with permission); with
6.1.28 r(i+iy)=iy r(iy) corrections due to H. E. Salzer.
GAMMA FUNCTION AND RELATED FUNCTIONS 257
Polynomial Approximationsa Error Term for Asymptotic Expansion
r(x+1)=2!=1+~,z+b,22+ . . . +b,xa+,(x)
(&)(~3XlO-7 where
* Prom C. Hastings, Jr., Approximations for digital as Z+W along any curve joining z=O and z= m,
computers, Princeton Univ. Press, Princeton, N.J., 1955 providingzf --a, --a-l, . . . ; z# -b, -b-l,
(with permission). . . . .
258 GAMMA FUNCTION AND RELATED FUNCTIONS
Continued Fraction
6.1.48
=- a0 _-al a2 ---a3 a4 a5
(92 > 0)
z+ z+ z+ z+ z+ z+ . . .
\
i &:/’
1
1 1 53 695
I / &=---, al=-9 a2=-) a3=-,
12 30 210 371
22999 29944523 10953534lOO9
a4=ii%%’ “=19733142’ “= 48264275462
Wallis’ Formula 4
6.1.49
-4
-6
r
Some Definite Integrals
I n-1
In r(z)= o- (z-l) e-Qy-$J f (g z > 0) 6.3.2 : lw)= 7, It(n)=--+ p-l (n22>
S[
2+
x_. . ..’
=(z-&) In z-z+& In 2s
Fractional Values
(s?~>O)
6.3.3
6.2. Beta Function
6.2.1 9G) =-y-2ln 2=-1.96351 00260 21423 . . .
1
B(z,w) =
S 0
4
p-1 (l--t)“-’ dt=
s
om(l;;;z+w at 6.3.4
=2
S o (sin t)2E-1 (cos t)2w-1 dt
(92~>0,~‘>00)
#(n, +$)=--r--2ln2+2
(
l+i+... +A) -
(nil)
Duplication Formula
6.3.10
- -
s?~(iy)=sz~(-iy)=%?‘1L(l+iy)=5z’9(1--iy)
n 2% r (x?J
6.3.11 Y-$(iy)=+y-‘+; ?r coth ny __ _
>
6.3.12 Y+(i+&) =+T tanh 7~ $1.462 +o.sss
-0.504 -3.545
- 1.573 +2.302
6.3.13 j$(l+iy)=-+y++ rcoth my -2.611 -0.888
-3.635 $0.245
=Yn~l(n’+Y2~ -1 -4.653 -XL053
-5.667 +o. 009
Series Expansions -6.678 -0.001
- -
6.3.14 #(l+~)=--r+~~(-l)~{(n)~-~ (lzl<l)
. -- x0=1.46163 21449 68362 gi I+&
6.3.15 \3
I’(x,)= .88560 31944 10889
~(l+z)=+z-‘-+ cot ?fz-(l--22)-1+1--y
6.3.20 xd=-n+(ln n>-‘+o[(ln n>-‘1
-~~Ik(2n+~wn Ma Definite Integrals -
6.3.16 6.3.21
=lnz--- S tat
I[ 0
~~(l+iy)=l--Y-&
l2-
22 0 (t”+z”)(P-1)
+gl (-l)“+‘[r(2n+l)--lly2”
(I&E 4< ;)
(lYl<2)
6.3.22
=-r+y2 n$l n-‘(n2Sy2) -l
(--~<Y<~)
Asymptotic Formulas
6.3.18
(n+l)!
(n=1,2,3, . . . ) -Tl(n+2)z+@i,*{(n+3)9-...
. 1
Tifl<l)
6.4.10
+‘“‘(z)=(-l)n+Ln!~O (z+k)-“-1
)L(“)(2),(?%=0,1, . . .), is a single valued analytic
function over the cnt,irc complex plane save at, (z#O,-1,-2,. . .)
the points z=-m(m=0,1,2, . . . ) where it pos-
sesses poles of order (n + 1). Asymptotic Formulas
0 6.4.11
Integer Values
6.4.2'
#‘“‘(l)=(- l)“+‘n!{(n+ 1) (n=l,2,3, . .) (2 + 02 in 1arg 2 1<T)
6.4.3 6.4.12
$(m)(n+l)=(-lpm!
1
i+
+2m+’ -
- - * +nm+l 1 (z+- in 1arg 21-&j
. 6.4.13
Fractional Values
6.4.4 ' '-'
p”‘(*) = (- l)“+‘n!(2”+‘- l)l(n+ 1) (z--,-in I arg 2 I<*)
(12=1,2, . .
6.4.14
6.4.5 #‘(n++) =*nJ-4 PI (2k- 1) -*
Recurrence Formuh
(z+- in I arg 2 I<T)
I 2 3 4
-3 -2 -I 0
-4
6.5.6
6.5.18 G x7*(+,-x2)=
S 0
‘ef2dt
164
P> sume-v&
=r(pl+l) 0 6.5.20 r(a,ix)=e*rf” [C(z,a)--iiS(
=Np+l, dp+l)
Recurrence Formulas
6.5.7 C(x,u)=
Sz-ta-1 co5t dt Wu<l) 6.5.21 P(a+l, x>=P(u, x)-&
a,(d=sm
6.5.10
6.5.24
e-=‘t”dt=x-“-lr(1+7,x)
1 (?j$)mmo =-lmeT-ln x=-&(x)--In 2
6.5.11
6.5.25 ar (w)
-= ar(a2)
------1-=y-le-z
ax ax
Incomplete Gamma Function as a Confluent
Hypergeometric Function (see chapter 13) 6.5.26
r(a, x+Y)-r(a, 5)
m
6.5.36
,e-zxa-l cm
n=o
(a-1)(a-2).
_____
Xn
. . (4[l-e-Ye
?I
(y)J
S 0
e -“‘r(b,ct) ha
y
l-
&]
(lYl<l4) (c%%+c)>O,9b>-1)
6.5.37
Smp-lr(b,t)
trt=!%!.k!i
Continued Fraction
6.5.31
0 a
1 1-a 1 2-u 2 (9 (a+ b) >O, Bu>O>
r(u,x)=e-‘.P
( &1+icp 1+ s+“’ >
(x>O,bl<m> 6.6. Incomplete Beta Function
6.5.32
Asymptotic Expansions 6.6.1 B,(a,b)=
S 0
= ta-‘(l--2)b-‘dt
6.6.2 I,(a,b)=B,(u,b)lB(u,b)
r(a, Z)-p-le-r ,+!!$+(a-y-2)+. . .]
[ For statistical applications, see 26.5.
1--I,-,(b,a)
SUppOW R,(u,z)=u,+,(u,z)+ . . . is t,lie re-
mainder after R terms in this scrics. Then if a,2 Relation to Binomial Expansion
arc real, WC have for n>u-2
6.6.4 I,(a,n-a+l)=I$ ()$(1--p)‘-
!nn(w)! _<lu,+,b,z)l
Numerical Methods
6.7. Use and Extension of the Tables
The error of linear interpolation in the table of
Example Compute r(6.88) to 8s. Using
1. the function.fi is smaller tlian lo-’ in this region.
the rccurrcIicc relation 6.1.16 nntl Table 6.1 WC Hence, f,(56.38)=.92041 67 and In r(56.38)=
11nvr, 169.85497 42.
r(6.R8)=[(5.38)(4.:~8)(:~.R8)(2.38)(1..78)]r(1.38) Direct interpolation in Table 6.4 of log,, I’(n)
=232.43671. eliminates tlir necessity of employing logarithms.
However, the error of linear interpolation is .002 so
Example 2. Compute ln r(56.38), using Table that log,0 r(n) is obtained wit11 a relative error
6.4 and linear illtcrpolntion ill.f,. \\-c have of lo-“.
In I’(56.38) = (56.38---a) In (56.38) - (56.38)
Wee page 11.
+J2(56.38)
264 GAMMA FUNCTION AND RELATED FUNCTIONS
Example 3. Compute lt(6.38) to 8s. Using the 6.8. Summation of Rational Series by Means
recurrence relation 6.3.6 and Table 6.1. of Polygamma Functions
-+ In 27r= - .91894
(;+7i) In 2= 1.73287+ 4.85203i
In r(++Ji)=-3.31598+ 2.32553i Then, we may express 2 u, in terms of the
75-l
In r(Z+$i) = -2.66047+ 293869i constants appearing in this partial fraction expan-
In I’(3+7i)=-5.16252+10.11625i sion as follows
Then,
(2.5$7i) In (3+7i) =-3. 0857779+ 17.1263119i
-(3+7i)=-3. ooooooo- 7. oooooooi
+ In (2~r)= . 9189385
[12(3+7i)]-‘= .0043103- .0100575i
-[360(3+7i)3]-1= .0000059- .0000022i
---------_-__ Higher order repetitions in the denominator are
ln I’(3+7i)=-5. 16252 +lO. 11625i handled similarly. If the denominator contains
GAMMA FUNCTION AND RELATED FUNCTIONS 265
i 1 1
we have 72 ( nii-n--2i >*
i --i --i i
Hence, al=-t a2=--, as=-) ad=--’
6 6 12 12
cY1=i, az=---i, (Y3=2i, (yq=-2i,
and therefore
Example 10.
References
Texts Tables
[6.1] E. Artin, Einfiihrung in die Theorie der Gamma- [6.9] A. Abramov, Tables of In r(z) for complex argu-
funktion (Leipzig, Germany, 1931). ment. Translated from the Russian by D. G.
Fry (Pergamon Press, New York, N.Y., 1960).
[6.2] P. E. Bohmer, Differenzengleichungen und be- In r(z+iy), 2=0(.01)10, y=O(.O1)4, 6D.
stimmte Integrale, chs. 3, 4, 5 (K. F. Koehler,
Leipzig, Germany, 1939). [6.10] Ballistic Research Laboratory, A table of the facto-
rial numbers and their reciprocals from l! through
[6.3] G. Doetsch, Handbuoh der Laplace-Transforma-
lOOO! to 20 significant digits. Technical Note No.
tion, vol. II, pp. 52-61 (Birkhauser, Basel,
351, Aberdeen Proving Ground, Md., 1951.
Switzerland, 1955).
[6.4] A. Erdelyi et al., Higher transcendental functions, [6.1 I] British Association for the Advancement of Science,
vol. 1, ch. 1, ch. 2, sec. 5; vol. 2, ch. 9 (McGraw- Mathematical tables, vol. 1, 3d ed., pp. 40-59
Hill Book Co., Inc., New York, N.Y., 1953). (Cambridge Univ. Press, Cambridge, England,
1951). The gamma and polygamma functions.
[6.5] C. Hastings, Jr., Approximations for digital com-
puters (Princeton Univ. Press, Princeton, N.J., Also lf = loglo (t)!dt, z=o(.Ol)l, 10D.
s0
1955).
[6.12] H. T. Davis, Tables of the higher mathematical
[6.61-F. L6soh and F. Schoblik, Die. Fakultiit und ver- functions, 2 ~01s. (Principia Press, Bloomington,
wandte Funktionen (B. G. Teubner, Leipzig, Extensive, many place tables
Ind., 1933, 1935).
Germany, 1951). of the gamma and polygamma functions up to
[6.7] W. Sibagaki, Theory and applications of the gamma @o(x) and of their logarithms.
function (Iwanami Syoten, Tokyo, Japan, 1952). [6.13] F. J. Duarte, Nouvelles tables de log10 n! 2133 d&i-
16.81 E. T. Whittaker and G. N. Watson, A course of males depuis n= 1 jusqu’il n=3000 (Kundig,
modern analysis, ch. 12, 4th ed. (Cambridge Geneva, Switzerland; Index Generalis, Paris,
Univ. Press, Cambridge, England, 1952). France, 1927).
266 GAMMA FUNCTION AND RELATED FUNCTIONS
[6.14] National Bureau of Standards, Tables of nl and (6.181 E. S. Pearson, Table of the logarithms of the com-
P(n++) for the first thousand values of n, Ap- plete P-function, arguments 2 to 1200, Tracts for
plied Math. Series 16 (U.S. Government Printing Computers No. VIII (Cambridge Univ. Press,
Office, Washington, D.C., 1951). n!, lGS;P(n+&), Cambridge, England, 1922). Loglo P(p), p=2(.1)
85. 5(.2)70(1)1200, 10D.
[6.15] National Bureau of Standards, Table of Coulomb
wave functions, vol. I, pp. 114-135, Applied [6.19] J. Peters, Ten-place logarithm tables, vol. I, Ap-
Math. Series 17 (U.S. Government Printing
pendix, pp. 58-68 (Frederick Ungar Publ. Co.,
Office, Washington, D.C., 1952).
New York, N.Y., 1957). nl, n=1(1)60, exact;
&‘[F’(l+i~)/F(l +iq],rl=O(.OO5)2 (.01)6 (.02)10(.1) (n!)-I, n=1(1)43, 54D; Logto( n=1(1)1200,
2O(.2)60(.5)110,10D;a~gF(l+i~),~=0(.01)1(.02) 18D.
3 (.05)10(.2)20(.4)30(.5)85, 8D.
[6.20] J. P. Stanley and M. V. Wilkes, Table of the recip-
[6.16] National Bureau of Standards, Table of the gamma rocal of the gamma function for complex argu-
function for complex arguments, Applied Math.
ment (Univ. of Toronto Press, Toronto, Canada,
Series 34 (U.S. Government Printing Office,
Washington, D.C., 1954). 1950). x=-.5(.01).5, y=O(.Ol)l, 6D.
[C-56)4I [
(-i)4
I
log,, e=O.43429 44819
[c-y1 ” (-;)9
[ 1
*See page II.
GAMMA FUNCTION AND RELATED FUNCTIONS 269
GAMMA, DIGAMMA AND TRIGAMMA FUNCTIONS Table 6.1
[ 1
(El)3
[ 1
(-46)3
log,, e=0.43429
[ 1
44819
(-f)3
[ 1
(-56)4
270 GAMMA FUNCTION AND RELATED FUNCTIONS
II 3
c-y
[ 1
(512
log,o e=0.43429
[
c-y
44819
I
GAMMA FUNCTION AND RELATED FUNCTIONS 271
TETRAGAMMA AND PENTAGAMMA FUNCTIONS Tal,le 6.2
‘l-al ,Ic 6.3 GAMMA AND DICAMMA Fl’NC’lM)NS FOR lh’l’IS(;l3R /III) II.\I,F-I;\‘I’FXER Vi\I,c’E:s
( 0)1.00000 00000 0)1.00000 000 -1)8.86226 93 -0.57721 56649 1.08443 755 0.57721 566
1.00000 00000
i 0i 2.00000
6.00000 I- 1
0)11.00000
5.00000 000
1.66666 667 I 0
10)1.32934
13.32335
1.16317 10 1.25611 43351
04 +0.42278
28 0.92278 76684 1.02806 452
1.02100
1.04220 830 0.27036
712 0.17582 795
0.13017 669
285
( 1)2.40000 00000 (- 2)4.16666 667 ( 1)5.23427 78 1.50611 76684 1.01678 399 0.10332 024
(- 3)8.33333 333 ( 2)2.87885 28 1.70611 76684 1.01397 285 0.08564 180
3)1.87125 43 1.87278 43351 1.01196 776 0.07312 581
4)1.40344 07 2.01564 14780 1.01046 565 0.06380 006
2.14064 14780 1.00929 843 0.05658 310
(- 6)2.75573 192 I 2.25175 25891 1.00836 536 0.05083 250
11 (- 7)2.75573 192 ( 7)1.18994 23 2.35175 25891 1.00760 243 0.04614 268
3.99168 00000 I 8)1.36843 37 2.44266 16800 1.00696 700 0.04224 497
:3 9 1.71054 21 2.52599 50133 1.00642 958 0.03895 434
10 2.30923 18 2.60291 80902 1.00596 911 0.03613 924
:54 (-11)1.14707 456 11)3.34838 61 2.67434 66617 1.00557 019 0.03370 354
:76
I1312)1.30767
2.09227 89888
43680 I -13
-14 17.64716
4.77947 733
373 13 8.56349 74
(12)5.18999 85 2.80351
2.74101 33283 1.00491 124
1.00522 0.02970 002
343 0.03157 539
18
19 I 14 6.40237 42810
15i 3.55687 37057 -15)2.81145 725 I 15I 1.49861 21 2.86233 24133
2.91789 68577 1.00463 988
1.00439 519 0.02654
0.02803 490
657
20 (17)1.21645 10041 2.97052 39922 1.00417 501 0.02520 828
21 (-19)4.11031 762 (19)1.10827 98 3.02052 39922 1.00397 584 0.02399 845
22 5.1OYO942172 3.06814 30399 1.00379 480 0.02289 941
23 3.11359 75853 1.00362 953 0.02189 663
24 3.15707 58462 1.00347 806 0.02097 798
0.02013 331
I
25 3.19874 25129 1.00333 872
26 (25)1.55112 10043 (-26)6.44695 029 (25)7.87126 49 3.23874 25129 1.00321 011 0.01935 403
(-27)2.47959 626 2712.08588 52 3.27720 40513 1.00309 105 0.01863 281
El
29 3.04888 34461
(-29 9.18368 986
i-30 3.27988 924
28 5.73618 43 3.31424 10884
3.34995 53741
1.00298 050
1.00287 758
0.01796 342
0.01734 046
30 1-31I 1.13099 629 3.38443 81327 1.00278 154 0.01675 925
(3212.65252 85981 -33)3.76998 763 (33)1.47092 26 3.41777 14660 1.00269 170 0.01621 574
86542 -34)1.21612 504 3.45002 95305 1.00260 748 0.01570 637
83693 3.48127 95305 1.00252 837 0.01522 803
76188 3.51158 25608 1.00245 392 0.01477 796
79904 (39)1.74039 42 3.54099 43255 1.00238 372 0.01435 374
47966 (-4:)9.67759 296 40)6.17839 94 3.56956 57541 1.00231 744 0.01395 318
32679 42)2.25511 58 3.59734 35319 1.00225 474 0.01357 438
53091 43)8.45668 42 3.62437 05589 1.00219 534 0.01321 560
61747 45)3.25582 34 3.65068 63484 1.00213 899 0.01287 530
82081 (-47)4.90246 976 47)1.28605 02 3.67632 73740 1.00208 546 0.01255 208
(47)8.15915 28325 (-48)1.22561 744 48 5.20850 35 3.70132 73740 1.00203 455 0.01224 469
42 i 49
52i 6.04152
51 3.34525 26613
1.40500 63063
61178 I(-53)1.65521
-52 12.98931
-50 7.11740 087
083
673 3.99612 90
5112.16152
50
53 9.18649 67
81 3.72571 29557
3.77278
3.74952 71417
76179 1.00193 570
1.00189
1.00198 983 0.01195 668
606 0.01167
0.01140 297
200
43
(54)2.65827 15748 (+5)3.76184 288 55)1.77827 64 3.79551 02284 1.00185 354 0.01115 226
46
I
47
48
;; I -60
-63
-58I13.86662
-57
-62 1.64397 471
1.81731
8.35965
8.05547 607
540
851
084 (63 1.78713 74
58I 4.29046
61
60
56 3.76238
8.66760
8.09115 18
44
82
29 3.83947 81768
3.81773
3.90198
3.88158
3.86074 96734
15102
15811
24506 1.00181 460
1.00166
1.00170
1.00173
1.00177 759
803
210
321 0.01003
0.01090 879
0.01023
0.01045
0.01067 333
283
602
895
51 (64)3.04140 93202 (-65)3.28794 942 (65)2.16668 38 3.92198 96734 1.00163 530 0.00983 596
(11-l)! l/(,1-1)! ()I -a)! 2 Zn(tc-l)! *
p,!= (2,)1,,4,4 fl(II) r(tr)=(2r)~,r"-~,,-)~f, (.)I)
+(u)=ln ~/-f:~(~~) (2#=2.50662 82746 31001
$0)) compiled from I-I. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press,
Bloomington, Ind., 1933, 1935) (with permission).
GAMMA FUNCTION AND RELATED FUNCTIONS 273
GAMMA AND DICAMMA IWiXc'I'IOIvS FOR IKTE(;ER AND IIALF-Ih.rE(;ER VALl!& 'I‘;,],!,. (,.:s
tl rot) l/rot) r(tt+3) JOI) ./I 00 .f:;(l,)
I - 70)2.33924
65)3.28794
68
67 1.23979 942
6.44695 515
993
964 ( 65)2.16668 38
16
12
21 3.96082
3.94159
33.92198
97969 96734
75166
62103
82858 1.00160 530
100154
1.00157
1.00163 383
438
355 0.00983
0 00928 596
0.00946
0.00964 784
363
620
(1 72)4.33193 547 63 3:99821 47288 1:OOlSl 628 0:OOSll 846
73)1.26964 03354 ( 73)9.47993 44 4.01639 65470 1.00148 919 0.00895 514
5.35616 29 4.03425 36899 1.00146 304 0.00879 758
3.07979 37 4.05179 75495 1.00143 780 0.00864 546
1.80167 93 4.06903 89288 1.00141 341 0.00849 852
( 81)1.07199 92 4.08598 80814 1.00138 984 0.00835 648
( 82)6.48559 51 4.10265 47481 1.00136 704 0.00821 912
( 84)3.98864 10 4.11904 81907 1.00134 498 0.00808 619
2.49290 06 4.13517 72229 1.00132 362 0.00795 750
1.58299 19 4.15105 02388 1.00130 292 0.00783 284
( 90)1.02102 98 4.16667 52388 1.00128 286 0.00771 203
( 91)6.68774 50 4.18205 98542 1.00126 341 0.00759 489
4.44735 04 4.19721 13693 1.00124 455 0.00748 125
3.00196 15 4.21213 67425 1.00122 623 0.00737 096
2.05634 36 4.22684 26248 1.00120 845 0.00726 388
1.42915 88 4.24133 53785 1.00119 118 0.00715 986
71 (100 1.19785 71670 (-101 8.34824 074 1.00755 70 4.25562 10927 1.00117 439 0.00705 878
72 i 101I 8.50478 58857 i -102 I 1.17580 856 7.20403 24 4.26970 55998 1.00115 807 0.00696 052
:i 105 4.47011
103 6.12344 54615
58377 -104 2.23707
-106 1.63306 744
868 3.83884 87
5.22292 35 4.28359 31188
44887 1.00114 675 0.00686 495
220
4.29729 1.00112 0.00677 197
75 (107)3.30788 54415 (-108)3.02307 930 (108)2.85994 23 4.31080 66323 1.00111 172 0.00668 148
101 (157)9.33262 15444 (-158)1.07151 029 (158)9.36756 79 4.61016 18527 1.00082 542 0.00495 866
(II-l) ! l/(+1)! (1/-i)! *-& Zn(tr-l)! [C-37)2] ['Ff'l]
,I != (2T)fd’+~~-~~fl (ttj r (t,) = (2r) f,,J~--,>-,y, (1,) +()I) =ln ef:~(~l) (2+2.50662 82746 31001
c(-;)l1
0.000 1.00000 000 0.91893 853 0.00000 000 00
c 1
i-y
+(x) =h x-f3f3(x) .
<x>=nearest integer to x.
n n! n!
100 9.3326 21544 39441 52682 6;10 1408 1.2655 72316 22543 07425
200 7.8865 78673 64790 50355 700 1689 2.4220 40124 75027 21799
300 3.0605 75122 16440 63604 800 1976 7.7105 30113 35386 00414
400 6.4034 52284 66238 95262 900 2269 6.7526 80220 96458 41584
500 1.2201 36825 99111 00687 1000 i 2567 I 4.0238 72600 77093 77354
l?(n+l) Un+l>
Compiled from Ballistic Research Laboratory, A tame of the factorial numbers and their reciprocals
from l! to lOOO! to 20significant digits, Technical Note No. 381, Aberdeen Proving Ground, Md.(1951)
(with permission).
GAMMA FUNCTION AND RELATED FUNCTIONS 277
GAMMA FUNCTION FOR COMPLEX ARGUMENTS Table 6.7
z=l.O
Jln r(z) !I &Tln r(z) Yin r(z)
0.00000 00000 00 5.0 - 6.13032 41445 53 3.81589 85746 15
~~~ - 0.00819
0.00000 00000
77805 00
65 - 0.05732 2940417 - 6.27750 24635 84 3.97816 38691 88
0:2 - 0.03247 62923 18 - 0.11230 22226 44 :*: - 6.42487 30533 35 4.14237 74050 86
- 0.16282 0672168 5:3 - 6.57242 85885 29 4.30850 21885 83
00:: - 0.12528
0.07194 93748
62509 00
21 - 0.20715 58263 16 5.4 - 6.72016 21547 03 4.47650 25956 68
o"*z - 0.19094
0.26729 5499187
00682 14 - 0.24405
0.27274 82989
38104 91
05 - 6.86806 72180 48 4.64634 42978 70
- 7.01613 75979 76 4.81799 41933 05
0:7 - 0.35276 86908 60 - 0.29282 6351187 - 7.16436 7442106 4.99142 03424 89
- 0.44597 87835 49 - 0.30422 56029 76 Pi - 7.31275 12034 30 5.16659 19085 37
- 0.54570 51286 05 - 0.30707 43756 42 5:9 - 7.46128 36194 29 5.34347 91013 53
- 0.65092 31993 02 - 0.30164 03204 68 - 7.60995 96929 51 5.52205 31255 15
- 0.76078 39588 41 - 0.28826 66142 39 - 7.75877 46746 55 5.70228 61315 35
- 0.87459 04638 95 - 0.26733 05805 81 - 7.90772 40468 98 5.88415 11702 39
- - 0.23921 67844 65 - 8.05680 35089 04 6.06762 21500 13
:*:
1:4 - 1.11186 45664
0.99177 27669 59
26 - 0.20430 0724149 - 8.20600 89631 00 6.25267 37967 05
1.5 - 1.23448 30515 47 - 0.16293 97694 80 - 8.35533 65025 11 6.43928 16159 76
- 0.11546 87935 89 - 8.50478 2399125 6.62742 18579 12
::: - 1.3593122484
1.48608 96127 6557 - 0.06219 86983 29 - 8.65434 30931 23 6.81707 14837 44
::g" - 1.61459
1.74464 53960
427617400 - 0.0034166314 77 - 8.8040151829 10 7.00820 81345 02
+ 0.0606128742 95 - 8.95379 54158 79 7.20081 01014 93
0.12964 63163 10 - 9.10368 06798 32 7.39485 62984 36
21" - 2.00876
1.87607 41504
87864 31
71 0.20345 94738 33 - 9.25366 79950 15 7.59032 6235184
0.28184 56584 26 - 9.40375 45067 08 7.78719 99928 77
2'2 - 2.27743
2.14258 42092
81922 96
04 0.3646140489 50 - 9.55393 74783 21 7.98545 82004 68
214 - 2.41323 81411 84 0.45158 81524 41 - 9.7042142849 72 8.18508 20125 03
0.54260 44058 52 - 9.85458 24074 86 8.38605 30880 89
22:; - 2.54990
2.68737 61537
68424 95
50 0.6375109190 46 -10.00503 94267 90 8.58835 35709 62
0.73616 63516 79 -10.15558 30186 86 8.79196 60705 87
22'87 - 2.96448
2.82558 5641191
14617 89 0.83843 89130 96 -10.3062109489 48 8.99687 36442 29
2:9 - 3.1040154399 01 0.94420 54730 39 -10.45692 10687 39 9.20305 97799 25
1.05335 07710 69 -10.6077113103 15 9.41050 83803 12
33:; - 3.24414
3.38482 90223
42995 90
77 1.16576 67132 86 -10.75857 96829 95 9.61920 37472 42
1.28135 17459 32 -10.90952 42693 78 9.82913 05671 62
:*: - 3.52603
3.66772 81104
43067 09
88 1.4000102965 76 -11.06054 32217 92 10.04027 38971 80
314 - 3.80988 12618 23 1.52165 22746 73 -11.21163 47589 48 10.2526191518 09
1.64619 26242 69 -11.36279 71628 04 10.46615 20903 24
::: - 3.95246
4.09546 7126189
13204 51 1.77355 09225 91 -11.51402 87756 02 10.68085 88047 12
1.903651019019 -11.66532 79970 81 10.89672 5708177
zl - 4.38258
4.23884 69752
14660 71
28 2.03642 0709693 -11.81669 32818 48 11.11373 9524157
3:9 - 4.52667 88647 16 2.17179 14436 05 -11.96812 31369 01 11.33188 72758 53
21" - 4.67109
4.81583 95934
29197 96
09 2.30969 80565 73 -12.1196161192 81 11.55115 62762 02
2.45007 85299 47 -12.27117 08338 67 11.77153 41183 09
2.59287 37713 19 -12.42278 59312 81 11.99300 86662 85
i-3' - 4.96086
5.10617 37766
81606 87
63 2.73802 74148 20 -12.57446 01059 08 12.21556 80464 79
414 - 5.25176 30342 30 2.88548 56389 27 -12.72619 20940 29 12.43920 06390 90
3.03519 69999 22 -12.87798 06720 44 12.66389 5070128
44:; - 5.54369
5.39760 64183
62389 84
04 3.1871122793 89 -13.02982 46547 89 12.88964 02037 08
4;i - 5;69002 29483 73 3.34118 43443 27 -1ji18172 28939 51 13.11642 51346 66
3.49736 80186 15 -13.33367 42765 47 13.34423 91814 77
i:; - 5.83657
5.98334 58655
58764 54
32 3.6556199647 12 -13.48567 77234 95 13.57307 18794 55
5.0 - 6.13032 41445 53 3.81589 85746 15 10.0 -13.63773 21882 47 13.80291 29742 30
Linear interpolation will yield about three figures;eight-point interpolation will yield about eight figures.
For z outsidethe range of the table, seeExamples 5-8.
.r=l.l
gin r(z) Jln r(z) Y Vln r(z) 4 In r(z)
- 0.04987 24412 60 0.00000 00000 00 - 5.96893 91493 52 3.96198 63258 60
00~~
0:2
-
-
0.05702 02290
0.07824 35801
38
68
-
-
0.04206
0.08230
65443
97383
76
98
E -
-
6.11415
6.25959
43840
93585
05
61
4.12446 68364
4.28888 73284
90
80
0.3 - 0.1129143470 17 - 0.11905 06275 18 :-; - 6.40526 53566 40 4i4552112743 47
0.4 - 0.16008 21257 99 - 0.15086 79240 09 514 - 6.55114 41480 20 4.62340 34819 04
- 0.21858 96764 09 - 0.1766611398 43 - 6.69722 7953189 4.79343 00232 04
- 0.28718 99839 43 - 0.19566 16788 64 55'2 - 6.84350 94110 69 4.96525 81683 67
- 0.36464 38731 53 - 0.20740 35526 60 517 - 6.98998 15495 70 5.13885 63238 91
- 0144978 83131 87 - 0.21167 10325 55 5.8 - 7.13663 77586 96 5.31419 39750 77
- 0.54157 54093 11 - 0.20843 91333 00 5.9 - 7.28347 17659 19 5.49124 16322 40
- 0.63908 78153 48 - 0.1978178257 67 - 7.43047 76136 25 5.66997 07803 94
:G! - 0.74153 80620 74 - 0.18000 55175 74 - 7.57764 96383 95 5i85035 3832146
1:2 - 0.84825 85646 26 - 0.15525 33222 12 - 7.72498 24519 72 6.03236 40835 50
1.3 - 0.95868 73364 97 - 0.12383 93047 38 - 7.87247 09237 38 6.21597 56726 90
1.4 - 1.07235 26519 67 - 0.08605 08957 00 - 8.02011 01645 61 6.40116 35407 92
- 1.18885 84815 22 - 0.04217 34907 11 6.5 - 8.16789 55118 88 6.58790 33956 67
:-z - 1.30787 15575 95 + 0.00751 65191 79 - 8.31582 25159 69 6.77617 16773 32
1:7 - 1.4291103402 04 0.06275 56777 30 - 8.46388 6927117 6.96594 55256 30
- 1.55233 58336 11 0.12329 53847 15 - 8.61208 46838 95 7.15720 27497 24
::: - 1.67734 40572 49 0.18890 25358 69 - 8.760411902172 7.34992 17993 20
- 1.80395 99248 63 0.25935 93780 23 - 8.90886 48649 60 7.54408 17375 09
2:: - 1.93203 22878 13 0.33446 29085 79 - 9.05744 00129 63 7173966 2215113
- 2.06142 99239 46 0.41402 4032150 - 9.20613 39357 92 7.93664 34464 25
z - 2.19203 82866 29 0.49786 66085 82 - 9.35494 33637 73 8i13500 61862 70
214 - 2.32375 68617 01 0758582 64745 04 - 9.50386 51603 25 8.33473 17082 71
2.5 - 2.45649 70097 26 0.67775 04868 09 7.5 - 9.65289 63148 29 8.53580 17842 76
- 2.59018 01959 43 0.77349 56148 91 - 9.80203 39359 83 8.73819 86648 33
Z -
-
2.72473
2.86010
65306
35591
67
81
0.87292 80949
0.97592 26515
66
07
E - 9.95127 52455
-10.100617572694
81 8.94190
9.14690
50606
41251
84
84
29" - 2.99622 52529 98 1.0823617859 08 E -10.25005 83482 21 9.35317 94376 01
3.0 - 3.13305 11644 50 1.19213 51297 05 8.0 -10.39959 50997 80 9.56071 49872 49
- 3.27053 57144 30 1.30513 8858177 -10.54922 54469 17 9.76949 51583 85
;:: - 3.40863 75892 32 1.42127 51595 43 t:: -10.69894 70966 06 9.97950 47158 43
- 3.5473192273 03 1.54045 17547 76 -10.84875 78390 24 10.19072 87913 49
;:: - 3.68654 63804 17 1.66258 1463194 2: -10.99865 55435 72 10.40315 28704 84
3.5 - 3.82628 77368 25 1.78758 18092 68 8.5 -11.14863 8155138 10.61676 27802 52
3.6 - 3.9665145962 20 1.91537 46664 26 8.6 -11.29870 36905 72 10.83154 46772 22
3.7 - 4.10720 05882 64 2.04588 59340 24 8.7 -11.44885 02353 71 11.04748 50362 14
- 4.24832 14278 81 2.17904 52440 32 8.8 -11.59907 59405 42 11.26457 06394 86
E - 4.38985 47017 40 2.31478 56943 26 8.9 -11.74937 90196 53 11.48278 85664 18
4.0 - 4.53177 96812 84 2.45304 36058 25 -11.89975 77460 43 11.70212 61836 32
2:
-
-
4.67407 71584
4.81672 93009
70
83
2.59375
2.73687
83010
19016
13
54
E!
9:2
-12.050210450183
-12.20073 5517188
11.92257
12.14411
11355
13354
62
15
413 - 4.9597195242 44 2.88232 91437 48 9.3 -12.35133 13844 58 12.36673 49565 33
4.4 - 5.10303 23779 21 3.03007 72080 09 9.4 -12.50199 65394 43 12.59043 04241 06
.x=1.3
Y :1 In r(z) 4ln r(z) Y din r(z) 4 In r(z)
E - 0.10817
0.11383 61080
48095 85
08 0.00000 00000 00 5.0 - 5.645414138133 4.24823 90621 27
- 0.0167199199 34 - 5.78673 23355 37 4.41126 31957 95
0:2 - 0.13070 20636 90 - 0.03225 84033 35 - 5.92835 35606 66 4.57620 66023 67
- 0.04549 95427 81 - 6.07026 64370 51 4.74303 39118 17
0":; - 0.19649
0.15843 1008149
12771 78 - 0.05544 82296 06 - 6.21246 02140 03 4.9117110050 12
0.5 - 0.24420 93680 45 - 0.06126 78750 55 5.5 - 6.35492 47217 66 5.08220 49501 77
- 0.30082 34434 02 - 0.06229 79103 48 - 6ii9765 03105 97 5.25448 39434 72
0:8
E - 0.36553
0.43754 39002
53407 19
27 - 0.05805 28252 04 55': - 6.64062 79133 72 5.4285172533 50
- 0.04820 73993 35 5:s - 617838488113 55 5.60427 51684 12
0.9 - 0.51609 74046 40 - 0.03257 37450 94 5.9 - 6.92730 48028 21 5.78172 89485 09
- 0.01107 52190 48 6.0 - 7.07098 80742 52 5.96085 07788 45
::; - 0.69006
0.60048 45154
62005 05
12 + 010162790894 04 6.1 - 7.21489 11938 62 6.1416137268 52
0.04941 70710 23 - 7.35900 70872 13 6.32399 1701649
:*: - 0.88259
0.78427 03001
13601 02
03 0.08822 25250 96 z*: - 7.50332 90147 58 6.50795 94158 99
1:4 - 0.98458 61322 90 0.13255 01649 50 6:4 - 7.64785 0551098 6.69349 23498 81
1.5 - 1.08986 76158 16 0.18223 70479 17 6.5 - 7.79256 55658 27 6.88056 67176 38
:*; - 1.30898
1.19809 86148
54162 82
04 Oi2371109920 47 - 7.93746 82058 02 7.06915 94350 45
0.29699 65855 44 - 8.08255 28787 24 7.25924 80896 76
1:s - 1.42227 1923714 Oi3617193463 93 - 8.2278142379 13 7.45081 09123 38
1.9 - 1.53773 44011 63 0.43110 85022 51 - 8.37324 7168176 7.64382 67501 64
2.0 - 1.65517 68709 10 0.50499 87656 67 7.0 - 8.51884 67726 68 7.83827 50411 67
2Il - 1.77442 7143191 0.58323 13926 09 7.1 - 8.66460 83606 78 8.03413 57901 50
s-23 - 2.01776
1.89533 34239
14331 34
28 0.66565 47394 67 7.2 - 8.81052 74362 48 8.23138 95458 91
0.75212 4475930 7.3 - 8.95659 96875 66 8.4300173795 19
2:4 - 2.14159 19646 87 0.84250 35670 42 7.4 - 9.10282 09770 73 8.63000 08640 04
2.5 - 2.2667188222 04 0.93666 21049 03 7.5 - 9.24918 73322 19 8.83132 20546 97
- 2.39304 70725 18 1.03447 70464 53 - 9.39569 49368 29 9.03396 34708 43
- 2.52049 15659 37 1.13583 18965 15 :*; - 9.54234 01230 14 9.23790 80780 23
2.8 - 2.64897 56799 18 1.2406163628 56 7:8 - 9.68911 93636 11 9.44313 92714 58
2.9 - 2.77843 02497 03 1.34872 60013 87 7.9 - 9.83602 92650 88 9.64964 08601 22
3-f - 2.90879
3.04000 60402
26554 26
06 1.46006 18633 96 - 9.98306 65608 89 9.85739 70516 25
1.57453 01525 07 -10.13022 8105196 10.06639 24378 12
312 - 3.1720186387 60 1.69204 18960 57 -10.27751 08670 60 10.2766119810 47
1.81251 26335 69 -10.4249119248 88 10.48804 10011 24
;:: - 3.43825
3.30478 31979
64765 94
05 1.93586 21235 97 -10.57242 84612 54 10.70066 51627 91
- 3.57239 88099 07 2.0620140693 37 -10.72005 77580 15 10.91447 04638 39
- 3.70717 37325 19 2.19089 58627 45 -10.86779 71917 09 11.12944 32237 30
;*87 - 3.84254
3.97848 9534695
76469 59 2.32243 83465 44 -11.01564 42292 16 11.34557 00727 24
2.45657 55932 86 -11.16359 64236 64 11.56283 79415 00
3:9 - 4.11497 07016 98 2.59324 47004 59 -11.31165 14105 63 11.78123 40512 20
2: - 4.38944
4.25196 64012
45543 12
38 2.73238 56006 34 -11.45980 6904159 12.00074 59040 23
2.87394 08855 80 -11.60806 06939 74 12.22136 12739 31
3.01785 56433 48 -11.75641 06415 49 12.44306 8198138
t:: - 4.52739
4.66578 32778
37904 30
84 3.16407 73073 22 -11.90485 46773 52 12.66585 49686 64
4.4 - 4.80459 79774 65 3.31255 55163 23 9.4 -12.05339 0797849 12.88971 01243 51
4.5 - 4.9438171850 33 3.46324 19848 78 9.5 -12.20201 7062734 13.11462 2443199
t:: - 5.22340
5.08342 39564
19323 42
94 3.61609 03828 59 -12.35073 15923 02 13.34058 09350 03
3.77105 62237 32 99*! -12.49953 2564949 13.56757 48342 95
3.92809 67607 19 9:8 -12.64841 82148 10 13.79559 35935 62
t:," - 5.36373
5.5044110199
57615 52
31 4.08717 08902 55 9.9 -12779738 68295 12 14.02462 68767 33
5.0 - 5.64541 41381 33 4.24823 90621 27 10.0 -12.94643 6748034 14.25466 45529 28
GAMMA FUNCTION AND RELATED FUNCTIONS 281
GAMMA FUNCTION FOR COMPLEX ilRGUMENTS Table 6.7
.r=1.8
Y .#!‘I r(z) .f In r(z) tiln r(z) ./In r(z)
0":: - 0.07108
0.07476 38729
57386 86
14 0.00000 00000 00 - 4.83045 6845113 4.92989 76263 84
0.02858 6333136 - 4.96226 53555 54 5.09490 86275 80
0.05769 29209 31 - 5.09454 72216 70 5.26176 50781 04
t: - 0.10400
0.08577 55297
76857 09
32 0.08782 58538 91 - 5.22728 53433 89 5.43043 56009 62
0:4 - 0.12929 22486 30 0.11946 40495 57 - 5.36046 35143 73 5.60088 97905 12
265 - 0.20006
0.16140 82029
31015 53
52 0.15304 83729 82
0.18897 35429
0.22758 31014
70
17
??I --- 5.49406
5.62807
5.76248
63619
92920
84380
68
13
56
5.77309 81726
5.94703 21669
6.12266 40498
78
16
86
Kl
0:9
- 0.24498
0.29581 08149
07721 71
- 0.3522150054 25
51 0.2691673612
0.31396 39650
58
50
5’78
5:9
- 5.89728
- 6.03244
06145
32737
63
64
6.29996 69207
6.478914668158
68
.r=1.9
Y tiln I‘(z) 9 In r(z) %ln r(z) X In r(z)
0.00000 00000 00 - 4.66612 81728 77 5.06052 77830 38
::"I - 0.03898
0.04242 42759
16648 23
18 0.03569 47077 36 - 4.79608 44074 24 5.22603 70297 75
0.07184 49288 73 - 4.92654 53878 64 5.39337 36626 27
ia; - 0.05270
0.06974 4359613
5307116 0.10889 51730 33 - 5.05749 30552 47 5.56250 72499 47
0:4 - 0.09340 38158 25 0.14726 87453 39 - 5.1889102823 51 5.73340 82679 93
0.5 - 0.12349 16727 26 0.18735 90383 60 - 5.32078 0812105 5.90604 80662 49
::"7 - 0.15978
0.20201 20244
08372 82
30 0.22952 28050 02 - 5.45308 92008 98 6.08039 88340 38
0.27407 56544 06 - 5.58582 07663 21 6.25643 35684 02
0.32128 97690 64 - 5.71896 15389 41 6.43412 60432 49
::9" - 0.24990
0.30315 35004
95035 09
34 0.37139 36389 55 - 5.85249 82177 50 6.61345 07797 49
1.0 - 0.36147 78527 10 0.42457 34706 81 - 5.9864181289 78 6.79438 30179 35
0.48097 58618 37 - 6.12070 91879 56 6.97689 86894 96
::: - 0.42455
0.49209 86372
6462111
39 0.5407113247 70 - 6.25535 98637 85 7.16097 43917 16
::: - 0.5638171504
0.63943 71834 98
20 0.60385 82827 52 - 6.39035 91465 66 7.34658 73625 14
0.67046 72268 81 - 6.52569 65169 71 7.53371 54565 59
:2 - 0.71869
0.80135 54698
82795 30
42 0.74056 4797147 - 6.66136 19179 75 7.72233 71224 13
0.81415 76239 52 - 6.79734 57285 54 7:9124j 13866 57
1:7 - 0.88717 97447 03 0.89123 58296 55 - 6.93363 87392 01 8.10397 78029 64
1.8 - 0.97595 80247 42 0.97177 6140147 - 7.07023 2129112 8129695 64920 80
1.9 - 1.06749 27687 53 1.05574 45936 43 - 7.20711 74449 04 8.49134 80626 65
2.0 - 1.16160 13318 68 1.14309 88592 34 - 7.34428 65807 56 8.68713 36229 72
2.1 - 1.25811 51641 83 1.23379 01934 57 - 7.48173 17598 49 8.88429 47573 07
1.32776 50714 39 - 7.61944 55170 18 9.08281 35092 45
z-z - 1.45774
1.35687 89195
95259 14
72 1.42496 65323 75 - 7.75742 06825 11 9.28267 23655 74
2: 4 - 1.56059 52554 63 1.52533 52787 28 - 7.89565 03667 87 9.48385 42409 11
2.5 - 1.66529 48176 11 1.6288105662 06 - 8.d3412 79462 62 9.68634 24629 88
1.73533 09179 80 - 8.17284 70499 43 9.89012 07585 45
;:; - 1.77173
1.8798173280
64947 51
00 1.84483 46926 69 - 8.31180 15468 79 10.09517 32398 33
1.95726 05315 67 - 8.45098 55343 75 10.30148 43916 76
;:: - 2.10052
1.98944 39332
23595 80
16 2.07254 77068 08 - 8.59039 33269 14 10.50903 90590 64
3.0 - 2.21298 10520 42 2.19063 63887 13 - 8.7300194457 32 10.71782 24352 78
3.1 - 2.32673 87919 77 2.31146 78475 36 - 8.86985 86090 10 lo:92782 00504 91
2.43498 46022 00 - 9.00990 57226 31 11.1390177608 39
;:; - 2.55788
2.44172 77675
36468 72
15 2.56113 05263 98 - 9.15015 58714 69 11.35140 17379 39
3.4 - 2.67514 6711148 2.68985 09205 60 - 9.29060 43111 75 11.56495 84588 29
3.5 - 2.79346 14569 24 2.82109 25566 19 - 9.43124 64604 23 11.77967 46963 13
3.6 - 2.91277 62346 38 2.95480 37012 40 - 9.57207 78935 85 11.99553 75096 87
3.7 - 3.03304 29224 14 3.09093 41220 91 - 9.71309 43338 13 12.21253 42358 42
;:9" - 3.27625
3.15421 54337
66305 96
10 3.22943 50808 91 - 9.85429 16464 97 12.43065 24807 06
3.37025 93162 16 - 9.99566 58330 75 12.64988 01110 27
4.0 - 3.39912 01294 42 3.51336 10185 24 -10.137213025172 12.87020 52464 75
4.1 - 3.52277 40173 08 3.65869 57993 21 -10.27892 94790 52 13.09161 62520 42
4.2 - 3.64718 27007 49 3.80622 06560 50 -10.4208115703 58 13.31410 17307 41
- 3.77231 39057 84 3.95589 39339 63 -10.56285 5789126 13.53765 05165 78
- 3.89813 73167 71 4.10767 52859 66 -10.70505 87350 54 13.76225 16677 85
4.5 - 4.02462 44269 53 4.26152 56312 41 -10.8474171130 08 13.98789 44603 16
416 - 4115174 84023 59 4.41740 71132 72 -10.98992 77287 64 14.21456 83815 73
4.7 - 4.27948 39577 56 4.57528 30577 67 -11.13258 74849 48 14.44226 31243 75
4:8 - 4.40780 72434 44 4.73511 79308 60 -11.27539 33771 93 14.67096 85811 36
4.9 - 4.53669 57418 38 4.89687 72979 01 -11.41834 24904 66 14.90067 48382 65
5.0 - 4.66612 81728 77 5.06052 77830 38 -11.56143 19955 88 15.13137 21707 60
GAMMA FUNCTION AND RELATED FUNCTIONS 287
GAMMA FUNCTION FOR COMPLEX ARGUMENTS Table 6.7
.,-=2.0
Y %ln r(z) 9 In r(z) 9 In r(z) X In r(z)
0.0 0.00000 00000 00 0.00000 00000 00 - 4.50127 50755 42 5.18929 93415 60
- 0.00322 26151 39 0.04234 57120 74 - 4.62939 88796 82 5.35533 82031 27
- 0.01286 59357 41 0.08509 33372 06 - 4.75805 7022252 5.52318 54439 62
- 0.02885 74027 79 0.12863 61223 10 - 4.88723 13522 76 5.69281 16137 11
- 0.05107 93722 62 0.17335 05507 97 - 5.01690 38831 33 5.86418 81052 00
::!l - 0.36428
0.30434 96090
77010 22
76 0.48375 78429 30 6.0 - 5.80450 07366 29 6.92770 07748 95
0.5447146524 35 6.1 - 5.93722 60439 25 7.11059 33491 13
:*:
1:4
-
-
0.49700
0.42859
0.56926
21701
14442
99322
42
52
58
0.60872
0.67588
0.74624
74700
39160
61166
17
88
63
66’;
6:4
-
-
-
6.07033
6.20381
6.33765
37820
23278
05713
31
98
36
7.29503
7.48100
7.66847
43738
16040
33815
76
81
76
1.5 - 0.64515 55533 76 0.81985 39537 67 6.5 - 6.47183 78858 22 7.85742 86143 76
1.6 -
-
-
0.72443
0.80688
19760
50339
33
42
0.89672
0.97687
1.06028
82178
35612
11909
63
07
26
i-76
6:8
-
-
-
6.60636
6.7412194789
6.87639
41013
46872
16
19
45
8.04784
8.23970
8.43299
67567
77898
22035
00
07
86
:*:
119 - 0.98053
0.89231 03476
37613 69
78 1.14693 12720 53 6.9 - 7.01188 07803 50 8.62768 09788 99
2.0 - 1.07135 98302 14 1.23679 5034104 7.0 - 7.14766 9177118 8.82375 55706 27
1.32983 65907 26 - 7.28375 16419 82 9.02119 78914 05
2: - 1.2602188108
1.16463 96040 76
42 1.4260144920 94 - 7.42012 02668 81 9.21999 02960 14
22:: - 1.45772
1.35795 76568
6696157 48 1.52528 30352 04 - 7.55676 74543 62 9.42011 55664 09
1.62759 33595 36 - 7.69368 59017 46 9.62155 68973 45
44:! - 3.25449
3.37595 2871145 81
29213 3.63551 57202 41 - 9.91625 64956 49 13.01129 53818 23
3.78164 32567 78 -10.05689 46678 12 13.23287 94959 63
2: -- 3.62122
3.49820 74039
88720 59
03 3.92992 69172 45 -10.19770 96994 20 13.45553 44022 19
4.08032 71023 23 -10.33869 78553 49 13.67924 90499 21
4:4 - 3.74497 69383 89 4.23280 53645 81 -10.47985 55166 49 13.90401 26078 95
-
-
3.86942
3.99455
77912
19873
99
65
4.38732
4.54384
43808
79226
43
20 ;:‘6 -10.62117
-10.76266
91758
54322
12
81
14.12981
14.35664
4458193
41900 46
t:;
4.9
-
-
-
4.12032
4.2467163216
4.37370
31366
79930
90
20
a7
4.70234
4.86276
5.02509
08252
89562
91831
48
20
32
x
9:9
-10.904310988175
-11: 04611
-11.18806
26442
72959
29
27
14.58449
14.81334
15.04319
15940
66565
95540
42
09
92
5.0 - 4.50127 58755 42 5.18929 93415 60 10.0 -11.33017 19298 27 15.27404 06485 34
288 GAMMA FUNCTION AND RELATED FUNCTIONS
<y>=nearest integer to y.
GAMMA FUNCTION AND RELATED FUNCTIONS 289
DIGAMMA FUNCTION FOR C( IMPLEX ARGUMENTS Table 6.8
x=1.1 x=1.2
H m(z) .fGZ) !I %qz) .feC(Z) ?/ .W(z) .eJ(z)
-0 42ji5 0.00000 5.0 1.61498 1.45097 0.0 -0.28964 o.ooodo 5.0 1.61756 1.43125
-0.41451 0.14258 5.1 1.63457 1.45332 0.1 -0.28169 0.12620 5.1 1.63705 1.43396
-0.38753 0.28082 1.65617 1.43658
-0.34490 0.41099 :-: 1.65378 1.45557 0.3 -0.26014
0.2 -0.22578 0.24926
0.36640 5.2
5.3 1.67494 1.43910
-0.28961 0.53042 5:s ;.p;
. 1.45774
1.45983 0.4 -0.18064 0.47552 5.4 1.69336 1.44152
0.63764 ::2
-0.22498 1.70933 1.46184
1.72718 0.5 -0.12710 0.57530 5.5 1.71146 1.44386
-0.15426
0.73229 :*I: ;.;z;;;
* 1.46565
1.46378 ;; -0.06753 0.66517 5.6 1.72924 1.44612
0.81484
-0.08023 -0.00412 0.74519 5.7 1.74672 1.44829
-0.00509
0.88630 +0.06130 0.81589 5.8 1.76390 1.45039
0.94792 5:9
+0.06954 1.77893 1.46921
1.46746 4;. 0.12730 0.87806 5.9 1.78079 1.45243
1.0 0.14255 1.00102 1.79561 1.47090 0.19280 0.93260 6.0 1.79740 1.45439
0.21327 1.04687 1.81375 1.45629
2: 0.28131 1.08660 6:2
2-7 1.81201 1.47253
1.82815 1.47411 11.1"
. 0.31960
0.25707 0.98046
1.02252 6.1
6.2 1.82983 1.45813
1.12119 1.84404 1.47565 ::: 0.38012 1.05960 6.3 1.84567 1.45991
::i F%%
. 1.15146 1.85968 1.47713 1.4 0.43846 1.09240 6.4 1.86126 1.46164
0.46829 1.17810 0.49459 1.12153 6.5 1.87661 1.46331
::2 0.52507 1.20169 22
0.57930 1.22269
1.89025 1.47857
1.87508 1.47996 ::i 0.54851
0.60028
1.14752
1.17082
6.6
6.7
1.89173
1.90663
1.46493
1.46651
2 0.63111 1;24148 2
1:9 0.68067 1.25839 6:9
1.91992
1.90519 1.48132 zi 0.64999
0.69774
1.19179
1.21074
6.8
6.9
1.92132
1.93579
1.46803
1.46952
1.93443 1.48263
1.48391 1:9
0.72813 1.27368 7.0 ;-;z;;; :-i 0.74362 1.22794 7.0 1.95006 1.47096
Z:! 0.77363 1.28755 7.1
0.81730 1.30021 7.2
1.48515
1.48635
1197675 1.48752
0.78775
212 0.83022
1.24362
1.25796
7.1
7.2
1.96413 1.47236
%E 1.47372
1.47505
$2 0.85928 1.31179 7.3
214 0.89967 1.32243 7.4
1.99047 1.48866
2.00401 1.48977
5:: 0.87114
0.91060
1.27112
1.28323
7.3
7.4 2:00519 1.47634
2.5 0.93858 1.33224 1.49085 0.94868 1.29442 7.5 2.01852 1.47760
0.97610 1.34131 7.5
::; 2.01736
2.04356 1.49190
2.03054 22 0.98546 1.30478 7.b 2.03167 1.47882
::76 1.01234 1.34972 1.49292 1.02103 1.31441 7.7 2.04465 1.48001
1.04736 1.35753 1.05546 1.32337 7.8 2.05746 1.48117
::98 1.08124 1.36482 ::: 2.05640 1.49489
2.06908 1.49392 219 1.08881
z-i 1.33173 7.9 2.07012 1.48230
3.0 1.11405 1.37162 1.12113 1.33955 8.0 2.08262 1.48341
1.14586 1.37800 i-1" 2.08160 1.49584
2.09397 ::1" 1.15250 1.34688 8.1 2.09496 1.48448
::: 1.17671 1.38398 812 2.10619 1.49767
1.49676 3.2 1.18295 1.35377 8.2 2.10716 1.48553
1.20667 1.38960 1.21254 1.36024 8.3 2.11921 1.48656
;:: 1.23578 1.39489 88:: 2.13019 1.49855
2.11826 1.49940 ::4' 1.24132 1.36635 8.4 2.13111 1.48756
1.26409 1.39989 1.50024 1.26932 1.37211 8.5 2.14288 1.48853
::: 1.29164 1.40461 8.5 2.14198 1.50106 ;:2
2.15363 1.29659 1.37756 8.6 2.15451 1.48949
1.31847 1.40907 1.32315 1.38272 8.7 2.16601 ii44042
:::, 1.34461 1.41331 8:8
it; 2.17654 1.50186
2.16515 1.50265 ::; 1.34905 1.38761 8.8 2.17738 1.49133
3.9 1.37010 1.41732 8.9 2.18780 1.50341 3.9 1.37432 1.39226 8.9 2.18862 1.49222
1.39496 1.42114 1.39898 1.39667 9.0 2.19973 1.49310
i-t 1.41924 1.42478 Z:! 2.20995 1.50416
2.19893 1.50489 2: 1.42306 1.40088 9.1 2.21073 1.49395
4:2 1.44294 1.42824 1.44659 1.40489 9.2 2.22160 1.49478
1.46611 1.43154 z-3 2.22084 1.50631
2.23161 1.50561 2'3 1.46959 1.40871 9.3 2.23236 1.49560
::: 1.48876 1.43469 9:4 2.24228 1.50699 4.4 1.49209 1.41236 9.4 2.24301 1.49640
4.5 1.51092 1.43771
1.53261 1.44059 ;:z 2.25283 1.50832
2.26326 1.50766 22 1.53565 1.41586
1.51410 1.41920 9.5
9.6 2.26397
2.25354 1.49718
1.49794
t-t 1.55384 1.44335 Z-I: 2.28382 yw&
2.27360 2; 1.57743 1.42240
1.55676 1.42547 9.8
9.7 2.28450 1.49943
2.27429 1.49869
418 1.57463 1.44600
4.9 1.59501 1.44854 9:9 2.29395 1:51021 4.9 1.59769 1.42842 9.9 2.29461 1.50015
5.0 1.61498 1.45097 10.0 2.30397 1.51082 5.O 1.61756 1.43125 10.0 2.30462 1.50085
[c-y] [‘-;)5] [‘-;)I] [(-p] [c-y] [‘-;)“I [f-2”“]
290 GAMMA FUNCTION AND RELATED FUNCTIONS
1.5 0.52310 1.06809 6.5 1.87837 1.44810 1.5 0.55336 1.01778 6.5 1.88036 1.43294
1.6 0.57409 1.09605 6.6 1.89344 1.44995 1.6 0.60144 1.04730 6.6 1.89537 1.43502
1.7 0.62333 1.12126 6.7 1.90829 1.45174 1.7 0.64811 1.07409 6.7 1.91017 1.43702
0.67084 1.14409 6.8 1.92293 1.45348 1.8 0.69337 1.09849 6.8 1.92475 1.43898
::: 0.71667 1.16483 6.9 1.93735 1.45517 1.9 0.73722 1.12075 6.9 1.93912 1.44087
2.0 0.76087 1.18373 7.0 1.95158 1.45681 2.0 0.77968 1.14113 7.0 1.95330 1.44271
2.1 0.80353 1.20102 7.1 1.96560 1.45841 2.1 0182078 1.15984 7.1 1.96727 1.44450
225 0.88447
0.84470 1.23148
1.21688 7.3
7.2 1.97944 ;.;fJy;; f-3 0.86058 1.17707 7.2 1.98106 1.44625
1.19296 7.3 1.99467 1.44794
2:4 0.92290 1.24495 7.4 2.00655
1.99309 i46294 2:4 0.89913
0.93647 1.20768 7.4 2.00809 1.44959
2.5 0.96007 1.25743 7.5 2.01984 1.46438 2.5 0.97265 1.22133 7.5 2.02134 1.45119
2.6 0.99604 1.26900 7.6 2.03296 1.46577 2.6 1.00775 1.23402 7.6 2.03442 1.45276
2.7 1.04179 li24585 7.7 2.04733 1.45428
z-i 1.06464
1.03088 1.28980
1.27976 7.7
7.8 2.04591 1.46713
1.46845 2.8 1.07484 1.25689 7.8 2.06008 1.45576
2:9 1.09739 1.29918 7.9 2.07131
2.05869 I.46974 2.9 1.10693 1.26723 7.9 2:07267 1.45721
3.0 1.12917 1.30797 8.0 2.08378 1.47100 3.0 1.13813 1.27693 8.0 2.08510 1.45862
1.16004 1.31621 8.1 2.09610 1.47223 3.1 1.16846 1.28604 8.1 2.09739 1.46000
;:: 1.19005 1.32396 8.2 2.10827 1.47342 3.2 1.19797 1.29461 8.2 2.10952 1.46134
1.21923 1.33126 8.3 2.12029 1.47459 3.3 1.22670 1.30269 8.3 2.12151 1.46266
3:: 1.24763 1.33814 8.4 2.13217 1.47573 3.4 1.25469 1.31032 8.4 2.13337 1.46394
3.5 1.27529 1.34464 8.5 2.14391 1.47685 3.5 1.28196 1.31753 8.5 2.14508 1.46519
3.6 1.30223 1.35080 8.6 2.15552 1.47794 1.30855 1.32436 8.6 2.15666 1.46641
1.32851 1.35663 8.7 2.16700 1.47900 E 1.33450 1.33084 8.7 2.16811 1.46760
33'87 1.35413 1.36216 8.8 2.17834 1.48004 3:s 1.35983 1.33699 8.8 2.17943 1.46877
319 1.37915 1.36742 8.9 2.18956 1.48106 3.9 1.38456 1.34283 8.9 2.19063 1.46991
4.0 1.40357 1.37242 9.0 2.20066 1.48205 4.0 1.40873 1.34840 9.0 2.20170 1.47103
4.1 1.42744 1.37718 9.1 2.21163 1.48302 4.1 1.43235 1.35370 9.1 2.21265 1.47212
4.2 1.45077 1.38172 9.2 2.22249 1.48397 4.2 1.45546 1.35876 9.2 2.22349 1.47319
4.3 1.47358 1.38606 9.3 2.23323 1.48490 4.3 1.47806 1136359 9.3 2.23421 1.47423
4.4 1.49590 1.39020 9.4 2.24386 1.48582 4.4 1.50019 1.36821 9.4 2.24481 1.47525
4.5 1.51775 1.39416 9.5 2.25437 1.48671 4.5 1.52185 1.37263 9.5 2.25531 1.47626
4.6 1.53914 1.39795 9.6 2.26478 1.48758 4.6 1.54307 1.37686 9.6 2.26570 1.47724
1.56010 1.40258 9.7 2.27508 1.48844 4.7 1.56387 1.38092 9.7 2.27598 1.47820
f-i 1.58064 1.40507 9.8 2.28528 1.48927 4.8 1.58425 1.38481 9.8 2.28616 1.47914
4:9 1.60078 1.40841 9.9 2.29537 1.49010 4.9 1.60425 1.38854 9.9 2.29623 1.48006
5.0 1.62052 1.41163 10.0 2.30537 1.49090 5.0 1.62386 1.39213 10.0 2.30621 1.48096
[C-,,2] [(-$W] [(-y-j [‘-p”] [C-y] ['-;'"I ['-;'"I ['-['"I
GAMMA FUNCTION AND RELATED FUNCTIONS 291
DIGAMMA FUNCTION FOR COMPLEX ARGUMENTS Table 6.8
x=1.5 x= 1.6
0.13189 0.44066 5.5 1.71976 1.39047 0.5 0.20790 0.40789 5.5 1.72313 1.37289
0.16935 0.51640 5.6 1.73725 1.39364 0.6 0.24050 0.47942 5.6 1.74051 1.37635
0.21064 0.58668 5.7 1.75445 1.39670 0.7 0.27674 0.54642 5.7 1.75760 1.37969
0.25479 0.65144 5.8 1.77137 1.39965 0.8 0.31581 0.60875 5.8 1.77441 1.38293
0.30091 0.71078 5.9 1.78801 1.40251 0.9 0.35697 0.66642 5.9 1.79095 1.38605
0.34824 0.76494 6.0 1.80439 1.40528 1.0 0.39957 0.71957 6.0 1.80724 1.38908
0.39614 0.81424 6.1 1.82051 1.40796 1.1 0.44305 1.82327 1.39200
0.44411 0.85907 6.2 1.83638 1.41055 :.g yyw& o*7684o
0.81319 22 1.83906 1.39484
0.49175 0.89980 6.3 1.85201 1.41306 0.85423 "6:: 1.85460 1.39759
1.4 0.53878 0.93684 6.4 1.86741 1.41549 114 0157445 0.89183 . 1.86992 1.40025
1.5 0.58497 0.97054 6.5 1.88258 1.41786 1.5 0.61757 0.92629 6.5 1.88501 1.40284
1.6 0.63018 1.00127 t-7" 1.89752 1.42015 1.6 0.66001 0.95790 6.6 1.89989 1.40534
1.7 0.67432 1.02932 . 1.91225 1.42237 1.7 0.70167 0.98693 6.7 1.91455 1.40778
1.8 0.71732 1.05500 1.92677 1.42453 1.8 0.74244 1.01363 6.8 1.92900 1.41014
1.9 0.75916 1.07855 66:: 1.94109 1.42663 1.9 0.78228 1.03824 6.9 1.94326 1.41244
2.0
3::
“o*ZE
0:87772
1.10020
li12015
1.13857
7.0
34
1.95521
1.96914
1.98287
1.42866
1.43065
ii43257
2.0
22.:
0.82115
0.85905
0.89597
1.06096
1.08197
1.10144
7.0
7.1
7.2
1.95731
1.97118
1.98487
1.41467
1.41684
1.41895
0.91499 1.15563 7:3 1.99643 1.43445 2:3 0.93193 1.11953 7.3 1.99837 1.42101
5:: 0.95118 1.17146 7.4 2.00981 1.43628 2.4 0.96694 1.13635 7.4 2.01169 1.42301
1.18618
92
2:7
0.98634
1.02050
1.05370
1.19990
1.21271
;.;
7:7
2.02301
2.03604
1.43805
1.43978
1.44147
2.5
2.6
1.00102
1.03421
1.15204
1.16668
7.5
7.6
2.02485
2.03784
2.05066
1.42496
1.42686
1.42871
2.04891 2.7 1.06653 1.18039 7.7
2.8 1.08598 1.22469 ;a;. 2.06162 1.44312 2.8 1.09801 1.19324 7.8 2.06332 1.43051
2.9 1.11738 1.23592 2.07417 1.44472 2.9 1.12867 1.20530 7.9 2.07583 1.43227
1.14794 1.24647 8.0 2.08657 1.44628 3.0 1.15856 1.21664 8.0 2.08819 1.43398
3:: 1.17769 1.25639 z.; 2.09882 1.44781 3.1 1.18770 1.22733 8.1 2.10040 1.43565
1.20667 1.26574 1.44930 3.2 1.21611 1.23741 8.2 2.11246 1.43728
z*; 1.27457 8:3 ;-;;;;; 1.45075 1.24383 1.24693 2.12439 1.43888
3:4 x:51
. 1.28290 8.4 2:13470 1.45217 33:: 1.27089 1.25594 2.13617 1.44043
1.28931 1.29080 8.5 2.14638 1.45355 3.5 1.29731 1.26448 2.14782 1.44195
;*: 131552 1.29828 8.6 2.15794 1.45491 3.6 1.32311 1.27257 2.15934 1.44344
3:7 1:34112 1.30537 8.7 2.16936 1.45623 3.7 1.34833 1.28026 2.17073 1.44489
3i8 1.36612 1.31212 8.8 2.18065 1.45753 3.8 li37297 1.28757 2.18199 1.44631
3.9 1.39055 1.31853 8.9 2.19182 1.45879 3.9 1.39707 1.29454 2.19313 1.44770
4.0 1.41443 1.32464 9.0 2.20286 1.42065 1.30117 2.20415 1.44905
4.1 1.43779 1.33047 9.1 2.21379 2:: 1.44373 1.30750 2.21504 1.45038
4.2 1.46065 1.33603 9.2 2.22460 1146242 1.4b632 1.31354 2.22583 1.45168
4.3 1.48302 1.34134 9.3 2.23530 1.46358 2; 1.48844 1.31932 2.23650 1.45295
4.4 1.50493 1.34642 9.4 2.24588 1.46471 414 li51012 1.32485 2.24706 1.45420
4.5 1.52639 1.35128 9.5 2.25635 1.46582 4.5 1.53136 1.33014 2.25751 1.45542
4.6 1.54742 1.35594 9.6 2.26672 1.46691 1:552i9 1.33522 2.26785 1.45661
1.56804 1.36041 9.7 2.27698 1.46798 i-76 1.57262 1.34009 2.27809 1.45778
:*s7 1.58826 1.36470 9.8 2.28714 1.46902 418 1.59265 1.34476 2.28822 1.45892
4:9 1.60810 1.36882 9.9 2.29720 1.47004 4.9 1.61232 1.34925 2.29826 1.46005
5.0 1.62756 1.37278 lo.0 2.30716 1.47105
[i-y] ['-:I']
[ C-i)4
1[ (-;I2
1
.f$(1.5+iy) =$ tanh KY-~- 4Y
4&l
292 GAMMA FUNCTION AND RELATED FUNCTIONS
s=1.9
:!t*(z) ./tic4 z*(z) 1 tic4 v X!b(z) -@ti(z) Y
010
0.35618
0.35847
0.00000
0;06870
5yo
5:1 1.66428
1.64585 1.30212
1.29698 0.1
0.4i2i8
0.42480
0.00000
0.06441 ::y 1.65125
1.66948
1.27849
1.28394
0.36528
0.37644
0.39169
0.13681
0.20377 2.:
5:4
1.68240
1.70022
1.71775
1.30707
1.31185
0.43081
0.44068
0.45420
0.12833
0.19130
0.25288
2-G
514
1.68742
1.70506
1.28919
1.29426
0.26908 1.31647 1.72242 1.29916
0.41071 0.33229 ::6' 1.75197
1.73500 1.32522
1.32092 0.5 0.47111
0.49110 0.31269
0.37042 5.5 1.73951 1.30389
0.39306 1.75633 1.30846
0:7
i-2 0.43309
0.45842 0.45110 1.76868 1.32938 i:; 0.51380 0.42583 z.7" 1.77290 1.31288
0.48625 0.50624 ihi
5:9 1.78513
1.80133 1.33730
1.33341 2: 0.56594
0.53887 0.47874
0.52904 5:8 1.78921 1.31715
0.51614 0.55838 5.9 1.80528 1.32129
0.54770 0.60749 1.81728 1.34107 1.0 0.59465 0.57667 6.0 1.82111 1.32530
:?i 0.58053 0.65359 21" 1.83300 1.34473 0.62468 0.62165 1.83671 1.32918
1:2 0.61431 0.69677 1.84848 ::: 0.65572 0.66400 t-t. 1.85208 1.33295
0.64872 0.73714 1.86374 Z% 0.68751 0.70380 1.86723 1.33660
::: 0.68351 0.77483 1.87878 1:35503 2: 0.71980 0.74116 2:: 1.88217 1.34015
6.5
22 0.71846
0.75338
0.78814
0.80999
0.84278 22
1.89361
1.90824
1.92266
1.35826
1.36140
1.36445
::2
0.75239
0.78510
0.81779
0.77618
0.80899
0.83973
6.6
1.89690
1.91143
1.92576
1.34358
1.34692
1’2
1:9
0.82261
0.85669
0.87335
0.90188
0.92851
2:
6:9
1.93688
1.95092
1.36741
1.37029
zl
1:9
0.85033
0.88262
0.86853
0.89551
2;
6:9
1.93990
1.95385
:*E::
1:35639
z-10
2:2
0.89031
0.92342
0.95598
0.95338
0.97664
0.99840
x
712
1.97843
1.96476
1.99192
:-::::1"
1137846
?1"
2:2
0.91459
0.94617
0.97731
0.920E:
0.94454
0.96681
7'*1"
712
1.96761
1.98120
1.99462
1.35937
1.36227
1.36509
0.98795 1.01879 0.98775 2.00786 1.36784
::i 1.01932 1.03792 2: 2.01838
2.00523 1.38355
1.38104 ;:: 1.03814
1.00798 1.00743 ::: 2.02094 1.37052
2.5 1.05008 1.05588 7.5 ;.;;I:; 1.38599 2.5 1.06779 1.02597 7.5 2.03385 1.37313
2.6 1.08022 1.07278 1.04344 7.6 2.04661 1.37567
1.10975 1.08868 ::; 2:05684 1.39070
1.38838 ::; 1.12548
1.09690 1.05992 7.7 2.05921 1.37815
z3 1.13867 1.10367 1.07548 2.07167 1.38056
2:9 1.16698 1.11782 ::; 2.08171
2.06935 1.39518
1.39297 29" 1.18102
1.15352 1.09020 ::: 2.08397 1.38292
3.0 1.19470 1.13119 2.09393 1.39734 3.0 1.20798 1.10413 8.0 2.09613 1.38522
3.1
3.2
1.22184
1.24841
1;14384
1.15583
21"
8;2
2.10600
2.11793
1.39944
1.40149 ::: 1.23442
1.26034
1.11733
1.12985 2: 2.10815
2.12003
1.38746
1.38966
::4'
1.27442
1.29990
1.16719
1.17798 i:: 2.14139
2.12973 1.40546
1.40350
::: 1.31067
1.28575
:%:
. 88:: 2.13178
2.14339
1.39180
1.39389
1.32485
::2 1.34929 1.18823
1.19798 2:
2.15292
2.16432
1.40738
1.40925 z-2
317
1.33510
1.35905
1.38254 1.18379
88’22.15487
2.16623
817 2.17746
1.39593
:%z
1.37324 1.20727 817 2.17560 1.41108
:-‘B
3:9
1.39670
1.41970 :*::::i 88::
2.18675
2.19778
1.41286
1.41461 ::i 1.42818
1.40558
1.19310
1.20200 88::
2.18858
2.19957
1:40179
1.40366
.
4.0 1.44226 1.23265 4.0 1.45036 1.21050 2.21045 1.40548
1.46437 1.24037 Z:i 2.21950
2.20870 1.41800
1.41632 2.22121 1.40727
:*: 1.48606 ii24775 t:: 1.49348
1.47212 1.21864
1.22643 2.23187 1.40902
4:3 1.25482 2'3 2.23019
2.24077 1.41964
1.42124 44:: 1.53505
1.51446 1.24105
1.23389 2.24241 1.41074
4.4 :*:z
. 1.26160 9:4 2.25124 1.42281 2.25284 1.41241
1.54872 1.26810 1.24792 2.26318 1.41406
t:: 1.56885 1.27434 2 2.26160
2.27186 1.42435
1.42586 2: 1.55527
1.57514 1.25452 2.27340 1.41566
1.58861 1;28033 z-i 2.29207
2.28202 1.42878
1.42733 2; 1.59466
1.61385 1.26086 2.28353 1.41724
f-i 1.60803 1.28610 1.26696 2.29356 1.41879
4:9 1.62710 1.29164 9:9 2.30203 1.43020 4:9 1.63270 1.27283 2.30349 1.42030
5.0 1.64585 1.29698 10.0 2.31190 1.43159 5.0 1.65125 1.27849 2.31332 1.42179
[ 1
C-t)6 [ (-;)“I [ (-,121 [(-951 [ (-:)“I [ (-35)4] [ (635)3]
coth 1r,/--1+3!!?
’ Wl+!f2)
7. Error Function and Fresnel Integrals
WALTER GAUTSCHI l
Contents
- (t-XP e-12dt
2*r :+I in erfc x=~~+T ~fl)iJ
( > z n.I
p2
S 0
‘ef2cZt, x=0(.02) 2, 10D
* 2
xeez2 ef dt, xm2=.25(-.005)0, 9D
S0
1 Guest worker, National Bureau of Standards, from The American University. (Presently
Purdye University.)
295
296 ERROR FUNCTION AND FRESNEL INTEGRALS
Page
Table 7.6. (3/I’(1/3))Jzemt3dt (0<&2.3) . . . . . . . . . . . . . 320
0
z=O(.O2)1.7(.04)2.3, 7D
2=0(.02)1,2-‘=1(-.02)0, 15D
Table 7.10. Complex Zeros of the Error Function (l<n< 10) . . . . 329
z,,erf z,=O,n=l(l)lO, 8D
The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K.
Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and
Ruth Zucker in the preparation and checking of the tables.
7. Error Function and Fresnel Integrals
Mathematical Properties
restriction arg t-+a with lal<z as t-+= along the For 1,-t(x), see chapter 10.
z t'
FIQUBE 7.1. y=e”’ m e-“dt. FIQURE 7.2. y=e-” e dt.
s ..I s 0
P=w)6 p=2(1)6
297
298 ERROR FUNCTION AND FRESNEL INTEGRALS
7.1.15
Y $Jy e-'2d~-~~~~$. ..
m z-t
(f.. # 0)
7.1.20
u1("+2)(z)+2zw("t1) (z)+2(n+l)?P(z)=O
(n=O, 1,2, . . .)
7.1.14
2ez2
s(D
*
e-z2dt- 1 ‘I2 ’ -3/2 -.2 . .
7.1.23
&zeJ erfc zml+m$I
Asymptotic
(-1)”
Expansion
’ *3 * (222)"
*’
(27n-1)
&(2)=(-l)” l-3 .(2$:-l) 8, erf (x+iy)=erf x+& [(l-cos 2xy)+i sin 2xy]
PI<1 where
7.2.3
7.1.27
Power Series a
7.2.4
le(z)(~3XlO-' (n=1,2,3,. . .)
Asymptotic Expansion
.6 (-1)“(27+n)!
n!m!(22y”
(~+a, largA<?)
7.3. Fresnel Integrals
Definition
Derivatives
Cl (x)=4; l cos Pdt, G(x)=-&
S o 4
7.3.4
7.2.8 -$ in erfc z=--in-l erfc z (n=O, 1,2, . . .)
S,(+$~ sin Pdt, &(x)=&l ‘@$ dt
7.2.9
Auxiliary Functions
7.3.5
g (era erfc 2) = (- 1) n2nn! erzin erfc 2
(n=O, 1,2, . . .)
.f(z)s[i-S(Z)] COS 6 Z2)-[k-C(Z)] sin 6 22)
Interrelations
Relation to Hermite Polynomials (see chapter 22:
7.2.11 (-1)“i”erfc z+i” erfc (-z)=&J *-* H,(iz: 7.3.7 C(x>=C1 (x&)4?2 6 x2)
ERROR F’UNCTION AND FRESNEL INTEGRALS
301
7.3.14
7.3.8
S(z)=-cos 522 go 1 ynT2"+'
( > * . . . (4n+3) 24r+3
7.3.9 c(z)=;+f(z)sin (a 2)-g(z) cos G 9)
Symmetry Relations
7.3.11 C(z)=$ ~~;~;~;~~)l; CPfl
7.3.17 C(-2)=--c(z), S(-2)=--s(z)
7.3.12 7.3.18 C(iz)=iC(z), S(S)=--is(z)
Value at Infinity
fi +sin/.- 6 z2) SO 1 S-l”& 24n+3
1 1
7.3.20 C(x) -+-7 S(s)+ (-00)
.i 2 2
- Derivatives
7.3.13 '(')=go (2n+1)!(4n+3)
(-1,7+/2)2nf Z4n+3
V
A
l-
Relation to Error Function (see 7.1.1, 7.1.3)
7.3.22
X
.6 1.2 I.8 2.4 3.0 3.6 Relation to Spherical Bessel Functions (see chapter 10)
7.4.3
m
7.3.28
(4m+l)
az$(z)- Sri0(--lP l *3($,+i- S0
e (ga>o, LB>01
mt2?le-atz&
7.4.4
Smt*TS+le-d2&=&
1 * 3 . . . (4n-1) e(,’
Ri”(z)=(-1)” ,
(7rZ*)2n L?Za>O;n=O, 1,2,. . .)
0
1 - e-rt2n-t
p =
rW+3) o 1+ 2 S
( uz2>
,dl(Isrg 4<:) 7.4.6
12
7.3.30
f$‘(z)=(-1)“l * 3 .(;;;:+l) p, S 0
me-d’ cos (2zt)dt=i
d- %e-: (9a>O)
7.4.7 .*
1 m e-lt2n+*
e(g)=r (2n+$) S o 1+ 2t
( rz2>
,dt (lwz zl<z)
S 0
me-d’ sin (2st)dt =i e-I’/a
Sxl vs
o e%!t
Wa>O)
7.3.31 /8”‘[<1, pP’l<l 7.4.8
t,
For z real, R:“(z) and RLg)(x) are less in absolute
value than the first neglected term and of the
same sign.
Rational Approximations ’ (0 <Z 5 m)
mema’&
7.3.32
1+.92&x
f(x~=2+l.792x+3.1042 +4x1 la(z)l52X10-3
S -=L
0 &t+z) &
ear erfc &
[Is
J;;
0
4ar,12dt
-k Ei(ax2)
1
(a>% s>O)
(For more accurate approximations see [i’.l].)
7.4.11
S0-e-a12dLJ!.
t2+2eaI2
erfc
6X (a>%
7.4. Definite and Indefinite Integrals
o- I-
e-t$-jts -“t2dtu e’[l-
7.4.1
S 2
7.4.12
S o1 tZS1=4
e (erf -\lZ)‘] (a>01
integrals, Approximation
[Seealso MTAC 10,
Newsletter, April 1956, Note 10.
173, 1956.1 S yemt2dt
-0 (X-t)*+yZ
=u 9w(x+iy) (x real, y>O)
7.4.15
+; [i-s (;$)1 (a%>O)
?r
* [P-(x2-y2)]e-tZdt (g4”+iY)
s0 t"-z(~-y*)t*+(z*+y*)*=2 y-ix 7.4.25
(5 real, y>O)
‘om$$dt=,rg { [;-C(,/T)]coi (ab)
7.4.16 s
m 2xye-“*dt j-[&S (@)I sin (ab) } (Si’a>O, Wb>O)
S0 t4-2($-y2)t*+(22+y*)*=Z
n-j w(z+;Y>
y--ix
7.4.26
(x red y>O)
m ewaldt
7.4.17 =‘$ {[$S(dF)]cos (ab)
s o &t2+b2) b b
0 1”’
e-a’ erf bt dt=- e4@erfc $
s 0 a -[&CT(@)] sin (ab)} (9%>0, L%?b>O)
s0
0
sin (2at)erfc bt dt=& [l-e-‘“‘“‘*](a>O,~b>O) S 0
e-“la(t)dt=i {[:-A @] cos (f)
(D 7.4.28
S0
e -ar erfd%dt=i Wb+b)>o)
s0
m
e+S(t)dt =i { [$C’(~)] cos (f)
7.4.20
+[& S @] sin @} @?a>O)
Jm eeat erfc .$ dt=a e -2fi (Wa>O, ab>O)
7.4.29
7.4.21
s,- e-“‘C(~~)dt=ze(~~la)r~~
(9u>O)
(Wb>o, ac>o) 7.4.30
7.4.22
m
S 0
e --(II cos (t*)dt =&{ [;-S(&/~)]cos@
Jim e-“‘S(~~)dt=2~~~:a)t~~
(S?a>O)
-[&C(i #)]sing)} (9a>O) 7.4.31 lm{ [;-C(t)]‘+[;4(t)]*}dt=;
7.4.23
m 7.4.32
S 0
e --atsin (t*)dt= &{ [&?(q)] cos @
S e erf (&z+$)+const.
+[+B (Ed:)] sin (:)} (&%>O) (a+3
ERROR FUNCTION AND FRESNEL INTEGRALS
304
7.4.33 7.4.38
e-&Lb’
S .*dx=g [em” erf (ox+:)
S
cos (ax2+2bx+c)dx
+e- 2oberf
( ax-- i I +const. (a+01
=Jg { co9 (p)C[Jz (ax+b)]
7.4.34
-.~~+gjx=
-22 emazz2+S
[w(i+iax)
Se +sin(~)S[JZ(ax+b)]}+const.
7.4.35
Serf xdx=x erf x+’ e-z2+const. sin(ax2+2bx+c)dx
fi S
7.4.36
Numerical Methods
z
Example 3. Compute e+’ e%t to 5s for From Table 7.1 we have $ e-(1.72)2=.058565.
s 0
x=6.5. Thus,
i erfc 1.72 = (.058565)(6.0064X 101*)/1.0087X 1013
With
l/22=.0236686 and linear interpolation
=3.4873X 10-S
in Table 7.5
6.6
e-WJ’
i2 erfc 1.72 = (.O58565)(1.292OX1O11)/1.OO87X1O13
=7.5013x 10-4
S 0
e%t= (.506143)/(6.5) ==.077868.
i3 erfc 1.72 = (.058565) (2.6031 X 1O1o)/1.OO87X 1013
=1.5114x10-4.
Example 4.Compute i2 erfc 1.72 using the
recurrence relation and Table 7.1. Example 6. Compute C(8.65) using Table 7.8.
With x=8.65, l/x= .115607 we have from Table
7.8 by linear interpolation
By 7.2.1, using Table 7.1,
j(8.65) = .036797, g(8.65) = .000159.
i-‘erfc 1.72 = .05856 50.
From Table 4.6
lim -=-
w”x) J;; eZ2ik erfc x (x>O) . u S:(u)
m+w w!%(x) 2
5.20310 58 .4329l 04 .03689 42
5.31898 80 .41673 97 -. 07803 89 .42732 63
6.0893801 .45993&3 .16061 99 691 63 .42718 63
With x= 1.72, m= 15 we obtain 6.43432 70 .39999 44 -. 19432 70 756 60 6 52 .42717 71
4.97691 11 .4699094 .2830889 674 79 9 39 61 .42717 67
&(5.24)=.427177
g2(u)=-[c~+c2(u-zc,) sYe(t)
1
lo $=.41826 00.
co=f2c7d, c1=--g2(u3,
1 * 3 . . . (2&l)
clr+2= -cr+ t--v
J%zo(2uo)k
(k=O, 1, 2, . . .).
Finally,
+32 ’ 215.
52*72
1
5! ’ g2t-13’2 dt=7,33XlO-‘.
Using the 4th formula at the bottom of Table 7.8 using Tables 7.8 and 4.8. Hence
=1.003606-.011259Oi.
where z=zO+ .l (p,+ipJ. Thus, with zo=.4+ .6i,
tI=.45+.65i, S;=.45+.55i, p,k.4, pz=.l, we get Example S, ((a+;)
18. Compute 4).
from Table 7.9
From 7.3.22, 7.3.8, 7.3.18 we have
S%‘w(~~)=$(.522246 +.498591+.487556+.467521)
= .493979 2’ 9
&(z)=-- e@w [(l+i) $1
&!w(s;) =2(.522246 +.498591+.561252+.533157)
= .528812
-!$! e-fz2w[(i-l) $1.
gw(z)=[l-(.4+.1)]{[1-(.4-.1)].522246
+(.4-.1).528812)+(.4+.1)X Jz and making use of 7.1.11,
{[l-(.4-.1)] .493979+(.4-.1).498591}=.509789. 7.1.12, and Table 7.9
(a>O, 2, y real).
Hence from Table 7.9
Using Tables 7.9, 4.4 and 4.6 m
w(.4-1.3i)=4.33342+8.042013. e-(1’4)tP-3r cos (2t)dt= &$?w(2+3i) = .231761.
s 0
308 ERROR FUNCTION AND FRESNEL INTEGRALS
References
Texts L7.221 G. N. Watson, A treatise on the theory of Bessel
[7.1] J. Boersma, Computation of Fresnel integrals, functions, 2d ed. (Cambridge Univ. Press, London,
Math. Comp. 14, 380 (1960). England, 1958).
17.21 A. V. Boyd, Inequalities for Mills’ ratio, Rep. Tables
Statist. Appl. Res. Un. Jap. Sci. Engrs. 6, 44-46
(1959).
[7.3] 0. Emersleben, Numerische Werte des FehIer-
[7.23] M. Abramowitz, Table of the integral
Math. Phys. 30, 162-163 (1951). :=0(.01)2.5,
J ‘e-u3 du, J.
~2 p-12 2 (d-2
.I’ erf 2 .I erf .I:
\R
0. 00 1.12837 91671 0.00000 00000 0.50 0.87\8758 25789 0.52049 98778
0. 01 1.12826 63348 0.01128 34156 0. 51 0.86995 15467 0.52924 36198
0. 02 1.12792 79057 0.02256 45747 0. 52 0.86103 70343 0.53789 86305
0.03 1.12736 40827 0.03384 12223 0. 53 0.85204 34444 0.54646 40969
0.04 1.12657 52040 0.04511 11061 0.54 0.84297 51813 0.55493 92505
0.05 1.12556 17424 0.05637 19778 0.55 0.83383 66473 0.56332 33663
0. 06 1.12432 43052 0.06762 15944 0. 56 0.82463 22395 0.57161 57638
0.07 1.12286 36333 0.07885 77198 0.57 0.81536 63461 0.57981 58062
0. 08 1.12118 06004 0.09007 81258 0. 58 0.80604 33431 0.58792 29004
0. 09 1.11927 62126 0.10128 05939 0. 59 0.79666 75911 0.59593 64972
0.10 1.11715 16068 0.11246 29160 0. 60 0.78724 34317 0.60385 60908
0. 11 1.11480 80500 0.12362 28962 0. 61 0.77777 51846 0.61168 12189
0. 12 1.11224 69379 0.13475 83518 0. 62 0.76826 71442 0.61941 14619
0. 13 1.10946 97934 0.14586 71148 0. 63 0.75872 35764 0.62704 64433
0.14 1.10647 82654 0.15694 70331 0.64 0.74914 87161 0.63458 58291
0. 15 1.10327 41267 0.16799 59714 0. 65 0.73954 67634 0.64202 93274
0.16 1.09985 92726 0.17901 18132 0.66 0.72992 18814 0.64937 66880
0.17 1.09623 57192 0.18999 24612 0.67 0.72027 81930 0.65662 77023
0.18 1.09240 56008 0.20093 58390 0.68 0.71061 97784 0.66378 22027
0.19 1.08837 11683 0.21183 98922 0. 69 0.70095 06721 0.67084 00622
0.20 1.08413 47871 0.22270 25892 0.70 0.69127 48604 0.67780 11938
0.21 1.07969 89342 0.23352 19230 0.71 0.68159 62792 0.68466 55502
0.22 1.07506 61963 0.24429 59116 0.72 0.67191 88112 0.69143 31231
0.23 1.07023 92672 0.25502 25996 0. 73 0.66224 62838 0.69810 39429
0.24 1.06522 09449 0.26570 00590 0.74 0.65258 24665 0.70467 80779
0.25 1.06001 41294 0.27632 63902 0.75 0.64293 10692 0.71115 56337
0.26 1.05462 18194 0.28689 97232 0.76 0.63329 57399 0.71753 67528
0. 27 1.04904 71098 0.29741 82185 0.77 0.62368 00626 0.72382 16140
0.28 1.04329 31885 0.30788 00680 0.78 0.61408 75556 0.73001 04313
0.29 1.03736 33334 0.31828 34959 0. 79 0.60452 16696 0.73610 34538
0.30 1.03126 09096 0.32862 67595 0. 80 0.59498 57863 0.74210 09647
0.31 1.02498 93657 0.33890 81503 0. 81 0.58548 32161 0.74800 32806
0.32 1.01855 22310 0.34912 59948 0.82 0.57601 71973 0.75381 07509
0. 33 1.01195 31119 0.35927 86550 0.83 0.56659 08944 0.75952 37569
0. 34 1.00519 56887 0.36936 45293 0.84 3.55720 73967 0.76514 27115
0.35 0.99828 37121 0.37938 20536 0. 85 0.54786 97173 0.77066 80576
0. 36 0.99122 10001 0.38932 97011 0.86 0.53858 07918 0.77610 02683
0. 37 0.98401 14337 0.39920 59840 0.87 0.52934 34773 0.78143 98455
0. 38 0.97665 89542 0.40900 94534 0. 88 0.52016 05514 0.78668 73192
0. 39 0.96916 75592 0.41873 87001 0. 89 0.51103 47116 0.79184 32468
0.40 0.96154 12988 0.42839 23550 0. 90 0.50196 85742 0.79690 82124
0.41 0.95378 42727 0.43796 90902 0. 91 0.49296 46742 0.80188 28258
0.42 0.94590 06256 0.44746 76184 0.92 0.48402 54639 0.80676 77215
0. 43 0.93789 45443 0.45688 66945 0.93 0.47515 33132 0.81156 35586
0.44 0.92977 02537 0.46622 51153 0.94 0.46635 05090 0.81627 10190
0.45 0.92153 20130 0.47548 17198 0.95 0.45761 92546 0.82089 08073
0.46 0.91318 41122 0.48465 53900 0.96 0.44896 16700 0.82542 36496
0. 47 0.90473 08685 0.49374 50509 0.97 0.44037 97913 0.82987 02930
0.48 0.89617 66223 0.50274 96707 0. 98 0.43187 55710 0.83423 15043
0.49 0.88752 57337 0.51166 82612 0.99 0.42345 08779 0.83850 80696
0.50 0.87878 25789 0.52049 98778
[(-y1
1.00 0.41510 74974 0.84270 07929
[ 1
C-5513
c(-y1
See Exa~nple 1.
$= 0.88622 69255
ERROR FUNCTION AND FRESNEL INTEGRALS 311
ERROR FUNCTION AND ITS DERIVATIVE Table 7.1
[(-;)l1
1.50 0.11893 02892 2.00 0.02066 69854
c-y
[ I I-
c 1
C-5614
s = 0.88622 69255
312 ERROR FUNCTION AND FRESNEL INTEGRALS
2;13
2.14
2.15
2.60
2.61
2.62
2.63
2.64
2.65
II - 3 1.2416
1.1783
1.3080 455
1.0607
1.1181 075
500
090
764
3.10
3.11
3.12
3.13
3.14
3.15
3.60
3.61
3.62
3.63
3.64
3.65
2.16 2.66 3.16 3.66
2.67 3.17 3.67
2.68 3.18 3.68
2769 3.19 3.69
2.20 2.70 3.20
2.21 2.71 3.21
2.22 2.72 3.22 3:72 (- 6)1.1028 445
2.23 2.73 3.23 3.73
2.24 2.74 3.24 3.74
2.25 2.75 3.25
2.26 2.76 3.25 3.76
2.27 (- 3)6.5249 776 2.77 3.27 3.77
2.28 2.78 3.28 3.78
2.29 2.79 3.29 3.79
2.30 2.80 3.30 3.80
2.31 2.81 3.31 3.81
2.32 2.82 3.32 - 5)X8428 397 3.82
2.33 2.83 3.33 I - 5)1.7242 768 3.83
2.34 2.84 3.34 (- 5)1.6130 192 3.84
2.35 - 3)4.5088 292 2.85 3.85
2.36
2.37
2.38
2.39
$4; (- -
- 3)3.7301
3 3.5556
3.3886
092
487
700
2.86
2.87
2.88
2.89
2.90
2.91
3.37
3.38
3.39
3.86
3.87
3.88
3.89
3.90
3.91
II
- 77)4.1221
(-
3.0245
3.2689 624
3.5324
3.8162
7)2.7979
971
796
013
867
245
2:42 - 3 3.2288 871 2.92 3.92
2.43 - 3 3.0760 230 2.93 3;93
2.44 I- 3 I 2.9298 098 2.94 3.94
3.45 3.95 (- 7 1.8896 240
I
3.46 3.96
3.47 3.97
3.48 3.98
3.49 3.99 - 7 I 1.6128
1.3754
1.4895
1.7459 557
458
098
135
2.50 (- 3)2.1782 842 3.00 (- 4)1.3925 305 3.50 (- 6)5.3994 268 4.00 (- 7)1.2698 235
F~O.88622 692%
ERROR FUNCTION AND FRESNEL INTEGRALS 313
DERIVATIVE OF THE ERROR FUNCTION Table 7.2
0
II
6.04 -16)1.6169 533 6.54 -19)2.9990 603 7.04
6.05 -16)1.4328 188 6.55 -19)2.6310 921 7.05 t-22)2.9304 450 7.55 t-25)1.9796 292
6.06 -16 1.2693 992 6.56 -19 2.3078 100 7.06 -22 1.9179
1.6645
2.2094
2.5448 736
491
450
057 7.56
6.07 -16 1 1.1243 934 6.57 -19 2.0238 447 7.07 7.57
6.08 -17)9.9575 277 6.58 -19 1.7744 651 7.08 7i58 -251112573 541
6.09 -17)8.8165 340 6.59 -19 I 1.5555 031 7.09 7.59 -25)1.0803 765
iI
6.25 -17 1.2241 281 6.75 7.25 7.75
6.26 -17 1.0801 812 6.76 7.26 7.76
6.27 -18 9.5297 064 6.77 7.27 7.77
6.28 -18 8.4057 325 6.78 7.28 7.78
6.29 -18 7.4128 421 6.79 7.29 7.79
z
8.00 8.50 9.00 I -36)7.4920 734 9.50
8.01 8.51 9.01 -36 I 6.2572 800 9.51
8.02 9.02 -36 5.2249 519 9.52
8.03 9.03 -36 4.3620 651 9.53
8.04 9.04 -36 3.6409 535 9.54
8.05 t-29)8.1112 334 8.55 (-32)2.0157 780 9.05 -36)3.0384 441 9.55
8.06 9.06 -36j2.5351 317 9.56
8.07 9.07 -36j2.1147 690 9.57
8.08 8i58 -32 1.2057 541 9108 -36j1.7637 559 9.58
8.09 (-29)4.2531 077 8.59 -32 1.0155 245 9.09 -36)1.4707 105 9.59 -40)1.2918 638
8.10 (-29j3.6173 797 9.10 -36)1.2261 088 9.60 -40)1.0662 907
8.11 9.11 -36 1.0219 837 9.61
8.12 9.12 -37 1 8.5167 148 9.62
8.13 8:63 -33)510997 438 9.13 -37)7.0959 960 9.63
8.14 (-29j1.8891 933 8.64 -33)4.2908 734 9.14 -37)5.9110 925 9.64
8.15 8.65 -33)3.6095 760 9.15 -37 4.9230 619 9.65 (-41)4.0725 570
8.16 8.66 -33 3.0358 465 9.16 -37 14.0993 592 9.66
8.17 8.67 -33 2.5527 988 9.17 -37)3.4127 918 9.67
8.18 8.68 -33 2.1461 817 9.18 -37)2.8406 437 9.68
8.19 8.69 -33 I 1.8039 709 9.19 -37)2.3639 423 9.69 i-41)1.8788 710
8.20 8.70 -33)1.5160 228 9.20 -37)1.9668 449 9.70 -41)1.5477 017
8.21 8.71 -3311.2737 818 9.21 -37)1.6361 251 9.71
8.22 8.72 -33'1.0700 339 9.22 -37 1.3607 427 9.72
8.23 8.73 -34 8.9869 668 9.23 -37 1.1314 847 9.73
8.24 8.74 ( -34 i 7.5464 360 9.24 -38 I 9.4066 395 9.74
8.25 8.75 -34)6.3355 422 9.25 -38)7.8186 802 9.75 -42)5.8524 252
8.26 8.76 -34j5.3178 836 9.26 -38 6.4974 888 9.76 -42 4.8150 968
8.27 8.77 -34j4.4627 957 9.27 -38 1 5.3984 710 9.77 -42 3.9608 401
8.28 8.78 -34)3.7444 525 9.28 9.78 I -42 I 3.2574 873
8.29 8.79 -34)3.1411 074 9.29 9.79 (-42)2.6784 979
8.30 8.80 9.30 9.80
8.31 8.81 9.31 9.81
8.32 8.82 9.32 9.82
8.33 8.83 -38j1.7684 718 9.83
8.34 8.84 -38)1.4672 880 9.84
8.35 9.35 -38 1.2171 545 9.85 (-43)8.2436 338
8.36 9.36 -38 1.0094 602 9.86
8.37 9I37 -39 8.3703 932 9.87
8.38 9.38 -39 I 6.9392 997 9.88
8.39 9.39 -39)5.7517 311 9.89 (-43)3.7428 271
8.40 -31)2.5623 380 8.90 -35)4.4873 418 9.40 -39 4.7664 456 9.90 -43)3.0708 096
8.41
8.42
8.43
8.44
I(
-31 2.1658
-31 1.8303
-31 I 1.5465
-31)1.3064
657
736
399
586
8.91
8.92
8.93
8.94
-35 3.7552
-35 3.1420
-35 2.6283
-35 I 2.1982
711
030
611
476
9.41
9.42
9.43
9.44
-39 3.9491
-39 3.2713
-39 I 2.7093
-39)2.2434
520
439
286
186
9.91
9.92
9.93
9.94
-43 2.5189
-43 2.0658
-43 1.6939
-43 I 1.3886
477
489
130
628
8.45 8.95 -3511.8381 516 9.45 (-39)1.8572 574 9.95 -43)1.1381 922
8.46 9.46 9.96
8.47
8.48
8.49
8.50
I
(-32)4.7280 139
8;98
8.99
9.00
-3511.0734
-36)8.9687
(-36)7.4920
315
435
734
9.47
9:48
9.49
9.50 (-40)7.2007 555
9.97
9.98
9.99
10.00 (-44)4.1976 562
$f=O.88622 69255
316 ERROR FUNCTION AND FRESNEL INTEGRALS
See Example 2.
[ 1
C-i)1
II
C-i)3
1
(- 2)5:68138
1. 0
:-:
1:3
1.4
1. 5 (- 3j8.02626
:*;
1: 8
1. 9 (- 3j1.26566
;:1"
9;
2: 4 (- 4j2.22250
- 6)9.52500
92
2: 7
2. 8
2.9
22
33.87
3:9 I- 9
8I 4.58945
8)5.10148
1.98190
1.04329
2.32831
4. 0
4.1
i-f
4: 4
t-6'
4: 7
4.8
II
-13
-14 4.02809
-12 2.35705
8.76348
1.14567
3.19826
4.9
5. 0 (-13)2.62561 (-14)5.61169 (-14)1.38998 (-15)3.83592
pr (;+I)]-’
(-1)5.64189 58355 (-1)2.50000 00000 (-2)9.40315 97258 (-2)3.12500
See Examples 4 and 5.
318 ERROR FUNCTION AND FRESNEL INTEGRALS
1.00000
n=5 n=6
1.00000
11 =I0
1.00000
0”::
I I
- 1 6.28971
- 1 3.91490
i*%
0:4 I I-
- lj2:41089
1)1.46861
I- 111.65569 - 2)7.95749
if:
0: 7
i:!
I-
-
(-
- 2)4.64127
2)2.67626
2)1.52533
3)8.59126
1. 0
1'::.
2:
1.5 (- 3)1.19278 (- 4)2.89186
::;
1':: I-
- 5j4;04407
5)2.04244
22'10
2: 2
$1'4
f:'6
;*i
2: 9
;:1"
E
3:4
- 9)2.47236
3::
2; -~0)1.84200 (-11)7.30331
3:9 -11)7.48503 (-11)2.91245
4. 0
4.1
-13)3.82601
-13)1.37691
-14j4: 87328
I-13)1.78294
-1416.31544
-14j2.20038
2.3'
4: 4
22
4:7
4. a
4. 9
5. 0 (-15)1.15173 (-16)3.70336
I-15 16.71719
-17
-16 2.11065
6.46126
2.10125
1.95316
(-18)6.51829 (-18)2.61062
pr (;+1)]-l
2: e-2.2
s0=rt;lt
’ x-2 xe-=
s zet2c/t <x>
0.00 0.00000 00000 1.00 0.53807 95069 0.250 0.6026080777
0.02 0.01999 46675 1. 02 0.53637 44359 0.245 0.60046 6027 ;
0.04 0.03995 73606 1.04 0.53431 71471 0.240 0.59819 8606
0.06 0.05985 62071 1.06 0.53192 50787 0.235 0.59588 1008 ;
0.08 0.07965 95389 1.08 0.52921 57454 0.230 0.59351 6018 2
0.10 0.09933 59924 1.10 0.52620 66800 0.225 0.59110 6724
0.12 fl.11885 46083 1.12 0.52291 53777 0.220 0.58865 6517
0.14 0.13818 49287 1.14 0.51935 92435 0.215 0.58616 9107
0.16 0.15729 70920 1.16 0.51555 55409 0.210 0.58364 8516
0.18 0.17616 19254 1.18 0.51152 13448 0.205 0.58109 9080
0.20 0.19475 10334 1.20 0.50727 34964 0.200 0.57852 5444
0.22 0.21303 68833 1.22 0.50282 85611 0.195 0.57593 2550
0.24 0.23099 28865 1.24 0.49820 27897 0.190 0.57332 5618
0.26 0.24859 34747 1.26 0.49341 20827 0.185 0.57071 0126
0.28 0.26581 41727 1.28 0.48847 19572 0.180 0.56809 1778
0.30 0.28263 16650 1.30 0.48339 75174 0.175 0.56547 6462
0.32 0.29902 38575 1.32 0.47820 34278 0.170 0.56287 0205
0.34 0.31496 99336 1.34 0.47290 38898 0.165 0.56027 9114
0.36 0.33045 04051 1.36 0.46751 26208 0.160 0.55770 9305
0.38 0.34544 71562 1.38 0.46204 28368 0.155 0.55516 6829
0.40 0.35994 34819 1.40 0.45650 72375 0.150 0.55265 7582
0.42 Oii~739241210 1.42 0.45091 79943 0.145 0.55018 7208
0.44 0.38737 52812 1.44 0.44528 67410 0.140 0.54776 0994
0.46 0.40028 46599 1.46 0.43962 45670 0.135 0.54538 3766
0.48 0.41264 14572 1.48 0.43394 20135 0.130 0.54305 9774
0.50 0.42443 63835 1.50 0.42824 90711 0.125 0.54079 2591
0.52 0.43566 16609 1.52 0.42255 51804 0.120 0.53858 5013
0.54 0.44631 10184 1.54 0.41686 92347 0.115 0.53643 8983
0. 56 0.45637 96813 1.56 0.41119 95842 0.110 0.53435 5529
0.58 0.46586 43551 1.58 0.40555 40424 0.105 0.53233 4747
0. 60 0.47476 32037 1.60 0.39993 98943 0.100 0.53037 5810
0.62 0.48307 58219 1.62 0.39436 39058 0.095 0.52847 7031
0.64 0.49080 32040 1.64 0.38883 23346 0.090 0.52663 5967
0.66 0.49794 77064 1.66 0.38335 09429 0.085 0.52484 9575
0.68 0.50451 30066 1.68 0.37792 50103 0.080 0.52311 4393
0.70 0.51050 40576 1.70 0.37255 93490 0.075 0.52142 6749
0.72 0.51592 70382 1.72 Oi36725 83182 0.070 0.51978 2972
0.74 0.52078 93010 1.74 0.36202 58410 0.065 0.51817 9571
0.76 0.52509 93152 1.76 0.35686 54206 0.060 0.51661 3369
0.78 0.52886 66089 1.78 0.35178 01580 0.055 0.51508 1573
0.80 0.53210 17071 1.80 0.34677 27691 0.050 0.51358 1788
0.82 0.53481 60684 1.82 0.34184 56029 0.045 0.51211 1971
0.84 0.53702 20202 1.84 0.33700 06597 0.040 0.51067 0372
0.86 0.53873 26921 1.86 0.33223 96091 0.035 0.50925 5466
0.88 0.53996 19480 1.88 0.32756 38080 0.030 0.50786 5903
0.90 0.54072 43187 1.90 0.32297 43193 0.025 0.50650 0473 6
0.92 0.54103 49328 1.92 0.31847 19293 0.020 0.50515 8078
0.94 0.54090 94485 1.94 0.31405 71655 0.015 0.50383 7717 T:
0.96 0.54036 39857 1.96 0.30973 03141 0.010 0.50253 8471
0.98 0.53941 50580 1.98 0.30549 14372 0.005 0.50125 9494 ::
0.53807 95069 2.00 0.30134 03889 0.000 0.50000 0000 m
See Example 3.
[ 1
C-l)4
<x> =nearest integer to 2.
[c-p1
Compiled from J. B. Rosw, Theory and application of Jo” rp2dx and so2 e-@@dy so” c~‘t/.t~.
Mapleton House, Brooklyn, N.Y., 1948; and B. Lohmander and S. Rittsten, Table of the
function y=ePzz so2rt*dt,Kungl. Fysiogr. Stillsk. i Lund Forh. 28,45-52, 1958 (with permission).
320 ERROR FUNCTION AND FRESNEL INTEGRALS
3
Table 7.6 -J” e-t3at
r! 0
03
- 3 JZe-Qt
3
2
_ J” e-Pdt __3 J” eddt
X
r1 0 r! 0 X r! 0
0 0 0
0. 00 0.8000000 0.70 0.‘72276 69 1.40 0.398973 54
0. 02 0.02239 69 0.72 0.73842 49 1.42 0.99109 36
0. 04 0.04479 31 0.74 0.75360 34 1.44 0.99229 70
0. 06 0.06718 72 0.76 0.76829 12 1.46 0.99335 97
0.08 0.08957 63 0.78 0.78247 88 1.48 0.99429 49
0.10 0.11195 67 0.80 0.79615 78 1.50 0.99511 49
0.12 0.13432 36 0.82 0.80932 16 1.52 0.99583 14
0.14 0.15667 11 0.84 0.82196 48 1.54 0.99645 52
0.16 0.17899 22 0.86 0.83408 41 1.56 0.99699 62
0.18 0.20127 90 0. 88 0.84567 73 1.58 0.99746 38
0.20 0.22352 24 0,90 0.85674 42 1.60 0.99786 63
0. 22 0.24571 24 0.92 0.86728 62 1.62 0.99821 16
0.24 0.26783 80 0.94 0.87730 62 1.64 0.99850 65
0.26 0.28988 71 0.96 0.88680 89 1.66 0.99875 75
0. 28 0.31184 70 0.98 0.89580 05 1.68 0.99897 03
[ 1
55)‘3
[ 1
c-:,5)7
[Ic-y1
95116
h Example 8.
[ 1 [ 1
C-54)2 C-i)8
[. 1
C-f3
322 ERROR FUNCTION AND FRESNEL INTEGRALS
9.
[ I
1
Cc,)=-+f(x) sin ix2 -8(r) cos 2TTZ C&(U)=! +fi(u) sin uyz(u) cos u
2 ( ) t 1 2
1
S(X)=--fG) cos (5”)-g(x) sin (ixz) s~(~)=A-~~(u)
2 cos u-g2(u) sin II
2
324 ERROR FUNCTION AND FRESNEL INTEGRALS
J t-i)6 1
1
[k--1
1 1
C(r)=-2 +f(x) sin ;22 -g(,r) cos $9 Cz(tr)=-+fz(u) sin u-gz(u) cosu
( > ( ) 2
1
s(X)=--f(x) COS 5.9 -g(r) sin Fz19 sz(u,=&/z(u) co.5u-g2(u) sin u
2 (2 > (2 )
<o=nearest integer to 2.
.
ERROR FUNCTION AND FRESNEL INTEGRALS 325
ERROR FUNCTION FOR COMPLEX ARGUMENTS Table 7.9
w(t)=e-2’ erfc (-iz) z=x+iy
“w(.z~z~(z) %w(z) .fW(Z) .oAw(z) Iw(z)
x=0.3 x-o.4
1.000000 0.000000 0.990050 0.112089 0.960789 0.219753 0.913931 0.318916 0.852144 0.406153
0.896457 0.000000 0.888479 0.094332 0.864983 0.185252 0.827246 0.269600 0.777267 0.344688
0.809020 0.000000 0.802567 0.080029 0.783538 0.157403 0.752895 0.229653 0.712146 0.294653
0.734599 0.000000 0.129331 0.068410 0.713801 0.134739 0.688720 0.197037 0.655244 0.253613
0.670788 0.000000 0.666463 0.058897 0.653680 0.116147 0.632996 0.170203 0.605295 0.219706
0.615690 0.000000 0.612109 0.051048 0.601513 0.100782 0.584333 0.147965 0.561252 0.191500
0.567805 0.000000 0.564818 0.044524 0.555974 0.087993 0.541605 0.129408 0.522246 0.167880
0.525930 0.000000 0.523423 0.039064 0.515991 0.077275 0.503896 0.113821 0.487556 0.147975
0.489101 0.000000 0.486982 0.034465 0.480697 0.068235 0.470452 0.100647 0.456579 0.131101
0.456532 0.000000 0.454731 0.030566 0.449383 0.060563 0.440655 0.089444 0.428808 0.116714
0.427584 0.000000 0.426044 0.027242 0.421468 0.054014 0.413989 0.079864 0.403818 0.104380
0.401730 0.000000 0.400406 0.024392 0.396470 0.048393 0.390028 0.071628 0.381250 0.093752
0.378537 0.000000 0.377393 0.021934 0.373989 0.043542 0.368412 0.064510 0.360799 0.084547
0.351643 0.000000 0.356649 0.019805 0.353691 0.039336 0.348839 0.058329 0.342206 0.076538
0.338744 0.000000 0.337876 0.017951 0.335294 0.035671 0.331054 0.052936 0.325248 0.069538
0.321585 0.000000 0.320825 0.016329 0.318561 0.032463 0.314839 0.048210 0.309736 0.063393
0.305953 0.000000 0.305284 0.014905 0.303290 0.029643 0.300009 0.044051 0.295506 0.057978
0.291663 0.000000 0.291072 0.013648 0.289309 0.027154 0.286406 0.040377 0.282417 0.053186
0.278560 0.000000 0.278035 0.012536 0.276470 0.024948 0.273892 0.037118 0.270346 0.048931
0.266509 0.000000 0.266042 0.011547 0.264648 0.022987 0.262350 0.034217 0.259186 0.045139
0.255396 0.000000 0.254978 0.010664 0.253732 0.021236 0.251677 0.031626 0.248844 0.041748
0.245119 0.000000 0.244745 0.009874 0.243628 0.019669 0.241783 0.029304 0.239239 0.038706
0.235593 0.000000 0.235256 0.009165 0.234251 0.018260 0.232592 0.027217 0.230300 0.035968
0.226742 0.000000 0.226438 0.008526 0.225531 0.016991 0.224033 0.025335 0.221963 0.033498
0.218499 0.000000 0.218224 0.007949 0.217404 0.015845 0.216047 0.023633 0.214172 0.031263
0.210806 0.000000 0.210557 0.007427 0.209813 0.014806 0.208582 0.022090 0.206879 0.029234
0.203613 0.000000 0.203387 0.006952 0.202710 0.013862 0.201589 0.020687 0.200039 0.027389
0.196874 0.000000 0.196668.0.006520 0.196050 0.013002 0.195028 0.019409 0.193613 0.025706
0.190549 0.000000 0.190360 0.006125 0.189796 0.012216 0.188861 0.018241 0.187566 0.024168
0.184602 0.000000 0.184429 0.005764 0.183912 0.011498 0.183056 0.017172 0.181868 0.022759
0.179001 0.000000 0.178842 0.005433 0.178368 0.010839 0.177581 0.016192 0.176491 0.021466
0.5 0.354900 0.342872 0.318884 0.349266 0.284638 0.351299 0.252654 0.349611 0.223262 0.344868
0.6 0.345649 0.308530 0.313978 0.316128 0.283540 0.319910 0.254784 0.320368 0.228026 0.318022
0.335721 0.278445 0.307816 0.286815 0.280740 0.291851 0.254895 0.293927 0.230578 0.293453
it: 0.325446 0.252024 0.300807 0.260847 0.276693 0.266757 0.253461 0.270040 0.231385 0.271015
0:9 0.315064 0.228759 0.293259 0.237800 0.271752 0.244295 0.250858 0.248462 0.230826 0.250549
0.304744 0.208219 0.285402 0.217306 0.266189 0.224168 0.247381 0.228967 0.229205 0.231897
::: 0.294606 0.190036 0.277407 0.199046 0.260213 0.206108 0.243266 0.211343 0.226767 0.214902
0.284731 0.173896 0.269401 0.182742 0.253985 0.189878 0.238695 0.195398 0.223710 0.199416
:.'3 0.275174 0.159531 0.261476 0.168151 0.247628 0.175271 0.233813 0.180957 0.220192 0.185299
1:4 0.265967 0.146712 0.253697 0.155066 0.241233 0.162100 0.228733 0.167863 0.216340 0.172423
1.5 0.257128 0.135242 0.246112 0.143305 0.234870 0.150205 0.223542 0.155975 0.212253 0.160668
0.248665 0.124954 0.238752 0.132711 0.228592 0.139441 0.218309 0.145167 0.208014 0.149927
::; 0.240578 0.115702 0.231635 0.123147 0.222436 0.129684 0.213086 0.135326 0.203684 0.140103
0.232861 0.107361 0.224775 0.114495 0.216428 0.120822 0.207912 0.126353 0.199315 0.131106
::: 0.225503 0.099824 0.218176 0.106650 0.210587 0.112760 0.202818 0.118158 0.194947 0.122858
0.218493 0.092998 0.211839 0.099523 0.204926 0.105411 0.197827 0.110662 0.190608 0.115286
::i 0.211816 0.086801 0.205760 0.093035 0.199452 0.098700 0.192953 0.103795 0.186324 0.108325
0.205457 0.081162 0.199935 0.087116 0.194166 0.092562 0.188208 0.097495 0.182112 0.101919
:*: 0.199402 0.076021 0.194356 0.081706 0.189072 0.086936 0.183599 0.091706 0.177985 0.096015
2:4 0.193634 0.071324 0.189014 0.076753 0.184165 0.081773 0.179131 0.086378 0.173954 0.090567
2.5 0.188139 0.067024 0.183901 0.072208 0.179444 0.077024 0.174805 0.081467 0.170024 0.0'35532
2.6 0.182903 0.063080 0.179008 0.068031 0.174903 0.072651 0.170623 0.076933 0.166201 0.080873
0.177910 0.059456 0.174324 0.064186 0.170538 0.068617 0.166582 0.072742 0.162487 0.076557
:*; 0.173147 0.056118 0.169840 0.060639 0.166342 0.064890 0.162681 0.068863 0.158883 0.072553
2:9 0.168602 0.053041 0.165546 0.057363 0.162310 0.061440 0.158916 0.065266 0.155389 0.068834
3.0 0.164261 0.050197 0.161434 0.054331 0.158435 0.058243 0.155285 0.061926 0.152005 0.065375
0.5 0.196636 0.337720 0.172820 0.328777 0.151751 0.318584 0.133288 0.307609 0.117233 0.296240
0.203461 0.313397 0.181177 0.306990 0.161171 0.299261 0.143369 0.290613 0.127644 0.281392
E 0.207990 0.290847 0.187245 0.286517 0.168379 0.280846 0.151366 0.274180 0.136134 0.266823
0:s 0.210664 0.270016 0.191423 0.267378 0.173725 0.263418 0.157578 0.258431 0.142949 0.252681
0.9 0.211846 0.250823 0.194049 0.249556 0.177513 0.247012 0.162268 0.243439 0.148310 0.239067
0.211837 0.233171 0.195407 0.233009 0.180002 0.231630 0.165667 0.229244 0.152418 0.226046
::i 0.210881 0.216954 0.195734 0.217678 0.181414 0.217253 0.167977 0.215857 0.155452 0.213656
0.209182 0.202067 0.195228 0.203494 0.181938 0.203847 0.169373 0.203272 0.157569 0.201914
:*: 0.206902 0.1'38403 0.194053 0.190384 0.181733 0.191366 0.170003 0.191471 0.158906 0.190821
1:4 0.204177 0.175862 0.192347 0.178275 0.180933 0.179762 0.169997 0.180425 0.159585 0.180367
1.5 0.201115 0.164349 0.190222 0.167092 0.179651 0.168980 0.169465 0.170099 0.159709 0.170534
0.197806 0.153773 0.187772 0.156765 0.177983 0.158969 0.168500 0.160457 0.159369 0.161300
:*; 0.194320 0.144054 0.185073 0.147226 0.176008 0.149674 0.167183 0.151458 0.158641 0.152637
1:8 0.190717 0.135113 0.182189 0.138412 0.173792 0.141045 0.165579 0.143063 0.157593 0.144516
1.9 0.187043 0.126883 0.179172 0.130262 0.171390 0.133033 0.163746 0.135234 0.156282 0.136908
0.183335 0.119298 0.176064 0.122723 0.168849 0.125590 0.161733 0.127931 0.154757 0.129781
2; 0.179623 0.112302 0.172901 0.115744 0.166206 0.118674 0.159580 0.121118 0.153059 0.123108
0.175930 0.105842 0.169710 0.109277 0.163493 0.112243 0.157320 0.114761 0.151224 0.116858
:-: 0.172276 0.099870 0.166513 0.103280 0.160737 0.106260 0.154982 0.108827 0.149281 0.111003
2:4 0.168674 0.094343 0.163330 0.097713 0.157958 0.100689 0.152591 0.103285 0.147256 0.105519
0.165136 0.089222 0.160175 0.092541 0.155175 0.095499 0.150165 0.098107 0.145172 0.100378
$2 0.161669 0.084472 0.157060 0.087732 0.152402 0.090660 0.147722 0.093265 0.143045 0.095558
0.158281 0.080061 0.153993 0.083254 0.149649 0.086143 0.145274 0.088735 0.140892 0.091037
:*;: 0.154975 0.075960 0.150981 0.079082 0.146927 0.081925 0.142834 0.084493 0.138725 0.086794
2:9 0.151753 0.072142 0.148030 0.075191 0.144243 0.077982 0.140411 0.080519 0.136555 0.082809
3.0 0.148618 0.068585 0.145144 0.071558 0.141602-. 0.074293 0.138012 0.076794 0.134391 0.079065
‘
3.0 0.112878 0.088283 0.109439 0.089170 0.106067 0.089898 0.102767 0.090479 0.099544 0.090921
-.
see Examples 12-19. w(x)=&+ zt e-22 Jo=et2 dt
4”
w(-x+iy)=-$i$J w(x-iy)=2e+z2 (cos2xy+i sin Sxy)-w(xtiy)
; 2.24465 616
1.45061 928 1.88094 514
2.61657 300 : 4.15899 840
4.51631 940 4.43557
4.78044 144
764
: 2.83974 105
3.33546 074 3.17562
3.64617 810
438 98 5.15876
4.84797 791
031 5.10158 264
5.40333 804
5 3.76900 557 4.06069 723 10 5.45219 220 5.68883 744
erf z,=erf (-z,)=erf zn=erf (--in)=0
From H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 260,209-211,
1955 (with permission).
Contents
Page
Mathematical Properties. ................... 332
Notation .......................... 332
8.1. Differential Equation ................. 332
8.2.Relations Between Legendre Functions .......... 333
8.3.Values on the Cut ................... 333
8.4.Explicit Expressions .................. 333
8.5.Recurrence Relations ................. 333
8.6.Special Values .................... 334
8.7.Trigonometric Expansions ............... 335
8.8.Integral Representations ................ 335
8.9.Summation Formulas ................. 335
8.10. Asymptotic Expansions ................ 335
8.11. Toroidal Functions .................. 336
8.12. Conical Functions ................... 337
8.13. Relation to Elliptic Integrals .............. 337
8.14. Integrals ....................... 337
Numerical Methods ...................... 339
8.15. Use and Extension of the Tables ............ 339
References ........................... 340
Table 8.1. Legendre Function-First Kind P,(z) (z(1) ....... 342
z=o(.ol)l, n=0(1)3, 9, 10, 5-8D
Table 8.2. Derivative of the Legendre Function-First Kind P;(z)
(251). ........................... 344
z=o(.ol)l, n=1(1)4, 9, lo, 5-7D
Table 8.3. Legendre Function-Second Kind &(s) (XI 1) ...... 346
~=o(.ol)l, n=0(1)3, 9, 10, 8D
Table 8.4. Derivative of the Legendre Function-Second Kind &i(z)
(x51). ........................... 348
2=0(.01)1, n=0(1)3, 9, 10, 6-SD
Table 8.5. Legendre Function-First Kind Pn(z) (z 2 1) ....... 350
z-1(.2)10, n=0(1)5, 9, 10, exact or 6s
Ttible 8.6. Derivative of the Legendre Function-First Kind P:(z)
(z>l). ........................... 351
x=1(.2)10, n=1(1)5, 9, 10, 6s
Table 8.7. Legendre Function-Second Kind &(z) (z> 1) ...... 352
2=1(.2)10, n=0(1)3, 9, 10, 6s
Table 8.8. Derivative of the Legendre Function-Second Kind Q:(Z)
(x21). ........................... 353
2=1(.2)10, n=0(1)3, 9, 10, 6s
The author acknowledges the assistance of Ruth E. C:apuano, Elizabeth F. Godefroy,
David S. Liepman, and Bertha II. Walter in the preparation and checking of the tables
and examples.
1 National Bureau of Standards.
331
8. Legendre Functions
Mathematical Properties
E(x) for (-l)R (2n+l)b--m)! e(x) (For P:(z), p=O, Legendre polynomials, see
J z(n+?n)! chapter 22.)
p:(z) for P:(z), XX(s) for Q:(z) Mz>l) 8.1.2
Q;(z) for e@4%) PC(z)= & [srF(-v, v+1; 1-p; 2)
(Additional forms may be obtained by means of the transformation formulas of the hyper-
geometric function, see [S.11.)
(12”1<1)
8.1.4 P:(z)=%f(z2-1)-b
8.1.7 e-‘rrQ:(z)=*12r(z*-1)--tp
(12*1<1)
8.3.2
PrI (x)=e*lifi=P:(xfiO) *
8.3.3
=i*-‘e,-i~ff”[e-1”‘6):(s+iO) *
8.1.9 W{P,(z), Q&z)} =-(2*-l)--’ -e~f~rQ:(s-iO) ] *
8.3.4
8.2. Relations Between Legendre Functions
&:(x)=~e~f~‘[e~~fc~&:(x+iO)+e~*~”&:(x-iO)]
Negative Degree
(Formulas for P:(z) and &y’(z) are obtained with
8.2.1 P!y-l(z) =Pf(z) the replacement of z-1 by (l--z)e*‘“, (~‘-1) by
(l-~*)e*~~, z-/-l by zf 1 for z=zfiO.)
8.2.2
Q’-.-l(z)= {-7refp’cos wrP~(z) 8.4. Explicit Expressions
*
8.4.5
Degree h + ) and Order v+ )
p&) =)(a+ 1) P*(Z) =i(h+- 1)
az>o
=i(3 cos 2e+ 1)
8.4.6
Q*(z) =9*(z) In (2) Q&J) =
8.2.8
-- 32
2
8.3. Values on the Cut (Both P: and &: satisfy the same recurrence
relations.)
(--1<2<1) Varying Order
8.3.1 8.5.1
P~(x)=~[e~f~“P~(x+iO)+e-~f~“P~(x--iO)] P:+‘(z) = (z’- 1)-‘{ (v-c()zPW -(v+LoPL(4 I
*see pa23 xl.
334 LEGENDRE FUNCTIONS
8.5.2 8.6.9
(z’-l) c!!y= (v+/.L) (v-/.tc(sl) (z’-l)P:-‘(~) p;J(+ 2 I’* (z;-;“4 {[z+(z”-l)““]~++
0?r V
--Pex4
Varying Degree -[z+(z”-l)y--v--f}
8.5.3
8.6.10
(Y-/.L+l)P~+I(Z)=(2~+l)@!(~)-(~+P)R!-*(~)
~~(~)=~(~~)*~*(~*-1)--1~4[~+(~*-1)*~*]-~--t
8.5.4 (~“-1) ~~=,,P:(,)-(,+,)p:_I(,)
8.6.11
Varying Order and Degree
&,‘(4=-~(2~)~13 kk&1’4[2+(2-~)1/2]-V-+ *
8.5.5 P~+,(~)=P:-~(2)+(2~+1)(~~-1)~~~-~(~)
8.6.17
&~(2)=(-1)yl-q~m ‘9
=P&(x)+ ...
II-ztt +5(n-2)
8.6.8
P!(z)=(z*-1)-“4(2a)-~~*{[Z+(~Ll)u*]v+t
+[z+(22-l)1’2]--v-~} W-*(z)=0
*See page II.
LEGENDRE FUNCTIONS 335
v=o, 1
bP;‘(cos e)
8.6.21
dV 1 v=o
=-tan 46-2 cot 40 In (cos $0)
dP;‘(cos e>
8.6.22
C bV “3,
=-3
1 tan 50 sin* +f?+sin
8.7. Trigonometric
e In (cos se)
Expansions (O<e<a)
r(v+Pi-l) 2 (P+3Mv+P+l)k
8.7.2 Q:(COSe) =7r1’*2r(siti ep ____- cos [(v+p+2k+W4
r(v+g) I;=0 Wv+2h
For fixed z and v and L%I.L-+CO,8.10.1-8.10.3 are asymptotic expansions if z is not on the real
axis between --03 and -1 and +a and + 1. (Upper or lower signs according as YzZO.)
8.10.1 P;(z)=
-- e
8.10.2 Q:(z)=fe”“”
z-1 -tr
--
( z+1 >
F(--Y, v+1; 1+/J; ++32>
1
With ccreplaced by -F, 8.1.2 is an asymptotic expansion for P;+‘(z) for fixed z and Y and 9 ~+OD
if z is not on the real axis between --a~ and - 1.
For fixed z and p and L%?v+w, 8.10.4 and 8.10.6 are asymptotic expansions if z is not on the real
axis between - 0~ and - 1 and +a and + 1; 8.10.5 if z is not on the real axis between - 0~ and + 1.
r(v+P+l) 2+(22--l)+
8.10.4 ~:(2)=(27r)-~(~~--l)-~‘~ [z+ (z’-l)qy+*F(3+c(, 3--/J; g+v;
r(v+#) 2(22-l), )
+~e-r~‘[z-(22-l)~]V+~F(~+Lc, &P;*+v;
r(v+P+l) --2+(22-l))
8.10.5 &:(z)=ef~“(&r)~(z2-1)-1’4 [z--(ZZ-l)q”+*F(~+~, 3--c(;$+v; 2(22--l)+ >
rb+g>
The related asymptotic expansion for P!+(z) may be derived from 8.10.4 together with 8.2.1.
r(v+p+l) T
8.10.8 &$(cos e>= r(v+S) ’ cos [(v++)e+%+~l+O(v-l)
(2 >
(Only special properties are given; other properties and representations follow from the earlier
sections.)
8.1~2 C-+(cosh
~)=r(n-m++pm~r(m+t)
S
r(n+m+t>(sinh drn * (sin CPR&J
0 (cash q+cos cpsinh v)~+~+*
Q-~WA(COS 0) = f i sinh XT
S /--
0 1 ~(COSII t+c05i e) 8.13.8
8.13.9 P-,(c0s e) =i K
8.12.5
~-t+,A(-~~~ e)
8.13.10 Q-,(x)=K (@) *
=c [Q-t+a(~~~ e>+Q+&os e)]
* 8.13. Relation to Elliptic Integrals 8.13.11 ~~~~)=~[2~(J~)-K(~~)]
(see chapter 17) (S&>O)
8.14. Integrals
8.14.1 ,= P~(~>Q,(~)dz=[(p-v)(p+v+l)l-’ (aP>9v>o)
J
8.14.4
S -I’ f’.W’,(x)dx=; [(P-v)(p+v+l)]-’ {2 sinrvsinrp[#(v+l)-$(p+l)]+*sin (rp-TV)}
r2-2(sin #‘(v+l) +
8.14.5
S ;l [PAz)12dz=
*v)~
r2b-t 3)
8.14.6
S -,’ Q~(2>Q~(~>d~=[(p-v)(p+v+l)]-‘{[~Mv+ I)-+(p+ I>][1 +cos pr cos ml-
(p+v+l#o; v, pf-1,
$r sin (vr-
-2, -3,
pi)} *
. . .>
8.14.1 _,(Q”(z)l’d~=(2v+l)-‘{
’ Ed--c’(v+I)[I+(cosv+]} (vz-1. -2, -3,. . .)
s
338 LEGENDRE FUNCTIONS
8.14.10
1
-‘Q(s)py(z)(&(-l)” l-(-l)‘+“(n+m)!
8.14.13
S ’ [Pg(z)]Yz=(n+~)-‘(n+m)!/(n-m)!
-1
(z--n)(z+n+1)(?%-7n)! 8.14.14
Is
s
1 (l-~~)-‘[~(2)]%2=(n+m)!/m(n--m)!
8.14.11
S 1 ~(2)Py(2)d2=0
-1
(I #n>
8.14.15
-1
8.14.12
r
.
1e(z)P:(z)
-1
(1-22)-‘&x=0 Wm)
S 0
l P,(x)x~dx=
**2-P-T(l+p)
ro+~P-3v)~(lP+3v+~)
8.14.16
2-CI?rr(3CY+&I)r(3a-3~)
S o* (sin t)a-lP;M(cos t)dt=
r<a+a~+~v>r!3~-3v)r(~~+4v+l)r(3~-3v+~) (ab~P)>o)
8.14.17
P;qz)=(22-1)-fm I* * * zP,(z)(dz)”
s1 s1
P 0
8.14.18
Q;“(z)=(-l)m(22-l)-tm . . . smQ.(z)C-W”
s/ s 2.0
P”( COSB)
I .o
.5
0 X
.2 .4 .6 .6 1.0
/
:5
-I .o
-1.5
Fmum 8.1. P.(cob 0). n=0(1)3. FIQTJRE 8.2. P,!(z). n=1(1)3, $51.
LEGENDRE FUNCTIONS 339
P,(X) /
16'
- \ ‘\\,\
‘\ ‘\\
10-3
- ‘\ ‘Y,
‘\ ‘L,
ii/I
I 2
I
3
I
4
I
5
I
6
I
7
I
8
I
9
I
IO
DX ‘\ ” ‘.
‘\
*\ .
PO \ I\ . ‘. Q2
-.
FIQURE 8.3. P,(z). n=0(1)3, z 11. IO-4
= ‘\ \
\\
P”(X)
\ \
A 1. 0,
I I I I I I I‘\” ) I Y
lO-5
I 2 3 4 5 6 7 8 9 IO ‘.
Numerical Methods
8.15. Use and Extension of the Tables
Computation of P,(z)
Plo(x) =&4(-252+1386Ox2-12Ol2Oz*+36O36OxE-4375SOzs+l84756z1")
(n+l)P,+1(~)=(2n+l)xPn(x) -nP,-l(X)
For coefficientsof other polynomials,see chapter 22.
LEGENDRE FUNCTIONS 343
LEGENDRE FUNCTION-FIRST KIND P,(r) Table 8.1
Po(z)=l PI(X) =2
o.to
0.51
arccos z
60.00000 00
59.33617 03
pzw
-0.12500
-0.10985
P3(4
-0.43750
-0.43337
00
25
P9@)
-0.26789
-0.25900
856
667
PI,(Z)
-0.18822 86
-0.21010 83
0.52 58.66774 85 -0.09440 -0.42848 00 -0.24682 215 -0.22914 92
0.53 57.99454 51 -0.07865 -0.42280 75 -0.23139 939 -0.24498 73
0.54 57.31636 11 -0.06260 -0.41634 00 -0.21283 321 -0.25728 92
56.63298 70 -0.04625 -0.40906 25 -0.19126 025 -0.26575 85
55.94420 22 -0.02960 -0.40096 00 -0.16686 000 -0.27014 28
55.24977 42 -0.01265 -0.39201 75 -0.13985 552 -0.27023 97
54.54945 74 +0.00460 -0.38222 00 -0.11051 366 -0.26590 30
53.84299 18 0.02215 -0.37155 25 -0.07914 497 -0.25704 92
0.60 53.13010 24 0.04000 -0.36000 00 -0.04610 304 -0.24366 27
0.61 52.41049 70 0.05815 -0.34754 75 -0.01178 332 -0.22580 16
0.62 51.68386 55 0.07660 -0.33418 00 +0.02337 862 -0.20360 19
0.63 50.94987 75 0.09535 -0.31988 25 0.05890 951 -0.17728 16
0.64 50.20818 05 0.11440 -0.30464 00 0.09430 141 -0.14714 41
0.65 49.45839 81 0.13375 -0.28843 75 0.12901 554 -0.11358 05
0.66 48.70012 72 0.15340 -0.27126 00 0.16248 693 -0.07707 01
0.67 47.93293 52 0.17335 -0.25309 25 0.19412 981 -0.03818 08
47.15635 69 0.19360 -0.23392 00 0.22334 410 +0.00243 30
%. 46.36989 11 0.21415 -0.21372 75 0.24952 270 0.04403 37
0.70 45.57299 60 -0.19250 00 0.27205 993 0.08580 58
0. 71 44.76508 47 t ::z -0.17022 25 0.29036 111 0.12686 31
0.72 43.94551 96 0:27760 -0.14688 00 0.30385 323 0.16625 89
0.73 43.11360 59 0. 29935 -0.12245 75 0.31199 698 0.20299 76
0. 74 42.26858 44 0.32140 -0.09694 00 0.31430 004 0.23605 08
0.75 41.40962 21 0.34375 -0.07031 25 0.31033 185 0.26437 45
0.76 40.53580 21 -0.04256 00 0.29973 981 0.28693 19
0.77 39.64611 11 it ;8696;50 -0.01366 75 0.28226 712 0..3027179
0.78 38.73942 46 0:41260 +0.01638 00 0.25777 224 0.31078 93
0.79 37.81448 85 0.43615 0.04759 75 0.22625 012 0.31029 79
0.80 36.86989 76 0.46000 0.08000 00 0.18785 528 0.30052 98
0. 81 35.90406 86 0.48415 0.11360 25 0.14292 678 0.28094 87
0.82 34.91520 62 0.50860 0.14842 00 0.09201 529 0.25124 52
0.83 33.90126 20 0.53335 0.18446 75 +0.03591 226 0.21139 19
0.84 32.85988 04 0.55840 0.22176 00 -0.02431 874 0.16170 50
0. 85 31.78833 06 0.26031 25 -0.08730 820 0.10291 23
0.86 30.68341 71 0.30014 00 -0.15134 456 +0.0362291
0.87 29.54136 05 0.34125 75 -0.21433 544 -0.03655 86
0.88 28.35763 66 0.38368 00 -0.27376 627 -0.11300 29
0. 89 27.12675 31 0.42742 25 -0.32665 610 -0.18989 29
0.90 25.84193 28 0.47250 00 -0.36951 049 -0.26314 56
0. 91 24.49464 85 0.51892 75 -0.39827 146 -0.32768 58
0.92 23.07391 81 0.56672 00 -0.40826 421 -0.37731 58
0.93 21.56518 50 0.61589 25 -0.39414 060 -0.40457 43
0.94 19.94844 36 0.66646 00 -0.34981 919 -0.40058 29
18.19487 23 0.85375 0.71843 75 -0.26842 182 -0.35488 03
16.26020 47 0.77184 00 -0.14220 642 -0.25524 34
14.06986 77 : %i 0.82668 25 +0.03750 397 -0.08749 40
11.47834 09 Of94060 0.88298 00 0.28039 609 +0.16470 81
8.10961 44 0.97015 0.94074 75 0.59724 553 0.52008 90
0.00000 00 1.00000 1.00000 00 1.00000 000 1.00000 00
(-Z)4
II 1 [ C-32
1 II 1
c-y
1'29)=4(-1+322) Pz(z)=F(-3+5~2)
2
Pg(x)=& (1260-18480x2+7207224-10296Ch++486202*)
PI&Z) =~4(-252+1386~2-12O12~~+36036O.xs-43758W+l8475fW')
(n+l)Pn+1@) = (2n+l)zP&) -nPn-l(2)
For coefficients
of otherpolynomials,
seechapter22.
344 LEGENDRE FUNCTIONS
P',(x)=1 P;(x)=32
2 I%(x) Pi(x) Pi(x) Piob)
0.00 -1.50000 0.00000 00 2.46093 75 0.00000 00
0.01 -1.49925 -0.07498 25 2.45011 64 0.27023 41
0.02 -1.49700 -0.14986 00 2.41773 75 0.53765 93
0.03 -1.49325 -0.22452 75 2.36405 34 0.79949 17
0. 04 -1.48800 -0.29888 00 2.28948 35 1.05299 82
0. 05 -1.48125 -0.37281 25 2.19461 13 1.29552 05
0.06 -1.47300 -0.44622 00 2.08018 11 1.52449 98
0.07 -1.46325 -0.51899 75 1.94709 32 1.73750 05
0. 08 -1.45200 -0.59104 00 1.79639 87 1.93223 25
0.09 -1.43925 -0.66224 25 1.62929 31 2.10657 29
0.10 -1.42500 -0.73250 00 1.44710 87 2.25858 73
0.11 -0.80170 75 1.25130 64 2.38654 80
0.12 1: ;g; -0.86976 00 1.04346 68 2.48895 24
0.13 -1:37325 -0.93655 25 0.82528 00 2.56453 90
0.14 -1.35300 -1.00198 00 0.59853 47 2.61230 18
0.15 -1.33125 -1.06593 75 0.36510 73 2.63150 28
0.16 -1.30800 -1.12832 00 +0.12694 88 2.62168 25
0.17 -1.28325 -1.18902 25 -0.11392 76 2.58266 81
-1.25700 -1.24794 00 -0.35546 01 2.51458 04
E. -1.22925 -1.30496 75 -0.59555 27 2.41783 68
0.20 -1.20000 -1.36000 00 -0.83208 96 2.29315 33
-1.41293 25 -1.06295 03 2.14154 35
E 1;. :;;g -1.46366 00 -1.28602 54 1.96431 51
0:23 -1:10325 -1.51207 75 -1.49923 18 1.76306 37
0.24 -1.06800 -1.55808 00 -1.70052 94 1.53966 43
0.25 -1.03125 -1.60156 25 -1.88793 72 1.29625 99
-0.99300 -1.64242 00 -2.05954 92 1.03524 77
it:; -0.95325 -1.68054 75 -2.21355 15 0.75926 26
0:28 -0.91200 -1.71584 00 -2.34823 78 0.47115 77
0.29 -0.86925 -1.74819 25 -2.46202 63 +0.17398 30
0.30 -0.82500 -1.77750 00 -2.55347 51 -0.12903 87
-0.77925 -1.80365 75 -2.62129 80 -0.43453 90
-0.73200 -1.82656 00 -2.66437 95 -0.73903 23
0.33 -0.68325 -1.84610 25 -2.68178 96 -1.03894 72
0.34 -0.63300 -1.86218 00 ~2.67279 74 -1.33065 96
0.35 -0.58125 -1.87468 75 -2.63688 47 -1.61052 81
0.36 -1.88352 00 -2.57375 82 -1.87493 10
p;. I:* ::;;; -1.88857 25 -2.48336 07 -2.12030 43
-0:41700 -1.88974 00 -2.36588 14 -2.34318 21
0:39 -0.35925 -1.88691 75 -2.22176 52 -2.54023 74
0.40 -0.30000 -1.88000 00 -2.05172 01 -2.70832 36
0.41 -0.23925 -1.86888 25 -1.85672 35 -2.84451 75
0.42 -0.17700 -1.85346 00 -1.63802 69 -2.94616 13
0.43 -0.11325 -1.83362 75 -1.39715 86 -3.01090 51
0.44 -0.04800 -1.80928 00 -1.13592 50 -3.03674 96
0.45 +0.01875 -1.78031 25 -0.85640 91 -3.02208 63
0.46 0.08700 -1.74662 00 -0.56096 76 -2.96573 83
0.47 0.15675 -1.70809 75 -0.25222 53 -2.86699 80
0.48 0.22800 -1.66464 00 +0.06693 30 -2.72566 30
0.49 0.30075 -1.61614 25 0.39337 29 -2.54206 98
0.50 0.37500 -1.56250 00 0.72372 44 -2.31712 34
qw
[ 1 [ 159‘3
[ (-f)3
1 [ 1
(-!)5
P&=+(-3+1w)
PI,(z)=~2(126O-5544o22+36O36Ch?-72O'72Ox~+43758Oa+)
P~o(~)=&4(277~O-48O48Ch~+216216&t?-35OO64W'+184756W)
n+l [zP&) -Pn+1(41
Pi%(z)= 1-22
LEGENDRE FUNCTIONS 345
DERIVATIVE OF THE LEGENDRE FUNCTION-FIRST KIND P:(x) Table 8.2
Pi(x)=1 P;(x)=32
x Pi(x) P;(x) P;(x) Pie(x)
0.50 0.37500 - 1.56250 00 0.72372 44 - 2.31712 34
0.51 0.45075 - 1.50360 75 1.05439 75 - 2.05232 40
0.52 0.52800 - 1.43936 00 1.38160 24 - 1.74978 82
0.53 0.60675 - 1.36965 25 1.70137 21 - 1.41226 67
0.54 0.68700 - 1.29438 00 2.00958 86 - 1.04315 43
0.55 0.76875 - 1.21343 75 2.30201 29 - 0.64649 54
0.56 0.85200 - 1.12672 00 2.57431 87 - 0.22698 16
0.57 0.93675 - 1.03412 25 2.82213 05 + 0.21005 92
0. 58 1.02300 - 0.93554 00 3.04106 49 0.65868 10
0.59 1.11075 - 0.83086 75 3.22677 77 1.11234 92
0.60 1.20000 - 0.72000 00 3.37501 44 1.56397 82
0. 61 1.29075 - 0.60283 25 3.48166 60 2.00598 31
0.62 1.38300 - 0.47926 00 3.54283 00 2.43034 08
0.63 1.47675 - 0.34917 75 3.55487 57 2.82866 68
0.64 1.57200 - 0.21248 00 3.51451 63 3.19230 45
0.65 1.66875 - 0.06906 25 3.41888 50 3.51243 07
0.66 1.76700 + 0.08118 00 3.26561 84 3.78017 74
0.61 1.86675 0.23835 25 3.05294 51 3.98677 13
0.68 1.96800 0.40256 00 2.77978 03 4.12369 16
0.69 2.07075 0.57390 75 2.44582 82 4.18284 84
0.70 2.17500 0.75250 00 2.05168 93 4.15678 18
0.71 2.28075 0.93844 25 1.59891 66 4.03888 45
0.72 2.38800 1.13184 00 1.09043 73 3.82364 72
0.73 2.49675 1.33279 75 + 0.53008 28 3.50693 03
0.14 2.60700 1.54142 00 - 0.07667 36 3.08626 20
0.15 2.71875 1.75781 25 - 0.72287 14 2.56116 49
0.76 2.83200 1.98208 00 - 1.39984 93 1.93351 26
0.77 2.94675 2.21432 75 - 2.09708 32 1.20791 71
0.78 3.06300 2.45466 00 - 2.80201 52 + 0.39215 05
0.79 3.18075 2.70318 25 - 3149987 45 - 0.50239 96
0.80 2.96000 00 =. 4.17348 81 - 1.46023 77
0.81 :* Ei 3.22521 75 - 4.80308 26 - 2.46122 91
0.82 3:54300 3.49894 00 - 5.36607 64 - 3.48002 97
0.83 3.66615 3.78127 25 - 5.83686 10 - 4.48547 21
0.84 3.79200 4.07232 00 - 6.18657 35 - 5.43990 91
0.85 3.91875 4.37218 75 - 6.38285 68 - 6.29851 03
0.86 4.04700 4.68098 00 - 6.38961 06 - 7.00851 07
0.81 4.17675 4.99880 25 - 6.16612 97 - 1.50840 93
0.88 4.30800 5.32576 00 - 5.66983 23 - 7.72711 51
0.89 4.44075 5.66195 75 - 4.84997 54 - 7.58303 90
0.90 4.57500 6.00750 00 - 3.65335 89 - 6.98312 79
0.91 4.71075 6.36249 25 - 2.02101 73 - 5.82184 03
0.92 4.84800 6.72704 00 + 0.11150 20 - 3.98006 04
0.93 4.98675 7.10124 75 2.81447 18 - 1.32394 73
0.94 5.12700 7.48522 00 6.16433 35 + 2.29628 14
0.95 5.26875 1.87906 25 10.24405 70 1.04163 58
0.96 5.41200 8.28288 00 15.14351 59 13.11571 11
0.97 5.55675 8.69677 75 20.95987 66 20.70612 01
9.12086 00 27.79800 16 30.04600 25
"0*9'9"
. 2. E5" 9.55523 25 35.77086 77 41.38561 43
1.00 6.00000 45.00000 00 55.00000 00
(-;)3
c-y
[ 1 (-y
[I [I
P;(z)=; (- 60+140s2)
P;(z)=& (1260-554&0s2+360360z4-720'720s~+437580z8)
P’lo(~)=&~ (27720-48048022+2162160~4-350064@.c~+184756'h*)
P:(x)=g2 [xPn(x)
-P,+,(x)]
346 LEGENDRE FUNCTIONS
Qo(x)=t In (2)
(~+1)&~+~(~)=(2~+1)xQn(x)-nQn-1(x)
Qo(4 =:arctanhx (Tab 11e 4.17) is included here for completeness.
LEGENDRE FUNCTIONS 347
LEGENDRE FUNCTION-SECOND KIND Table 8.3
Q&4 QloW
0.250 QoC4
0 54930 614 -0.72534Ql(x)693 -0.81866Qdx)
327 -0.19865Q3W477 -0.11616 303 +0.29165 814
0.51 0:56272 977 -0.71300 782 -0.82681 587 -0.22745 494 -0.16231 372 0.25442 027
0.52 0.57633 975 -0.70030 333 -0.83440 647 -0.25628 339 -0.20711 759 0.21286 243
0.53 0.59014 516 -0.68722 307 -0.84141 492 -0.28510 113 -0.24999 263 0.16748 08i7
0.54 0.60415 560 -0.67375 597 -0.84782 014 -0.31386 748 -0.29035 406 0.11884 913
0.55 0.61838 131 -0.65989 028 -0.85360 014 -0.34253 994 -0.32762 069 0.06761 47'0
0.56 0.63283 319 -0.64561 342 -0.85873 186 -0.37107 413 -0.36122 172 +0.01449 441
0.57 0.64752 284 -0.63091 198 -0.86319 116 -0.39942 362 -0.39060 386 -0.03973 1414
Oi58 0.66246 271 -0.61577 163 -0.86695 267 -0.42753 983 -0;41523 901 -0.09422 6310
0.59 0.67766 607 -0.60017 702 -0.86998 970 -0.45537 186 -0.43463 218 -0.14810 594
0.60 0.69314 718 -0.58411 169 -0.87227 411 -0.48286 632 -0.44832 986 -0.20044 847
0.61 0.70892 136 -0.56755 797 -0.87377 622 -0.50996 718 -0.45592 864 -0.25030 577
0.62 0.72500 509 -0.55049 685 -0.87446 461 -0.53661 553 -0.45708 410 -0.29671 648
0: 63 0.74141 614 -0.53290 783 -0.87430 597 -0.56274 938 -0.45151 989 -0;33872 031
0. 64 0.75817 374 -0.51476 880 -0.87326 492 -0.58830 338 -0.43903 693 -0.37537 391
0.65 0.77529 871 -0.49605 584 -0.87130 380 -0.61320 855 -0.41952 271 -0.40576 815
0.66 0.79281 363 -0.47674 300 -0.86838 239 -0.63739 196 -0.39296 048 -0.42904 673
0.67 0.81074 313 -0.45680 211 -0.86445 768 -0.66077 634 -0.35943 834 -0.44442 606
0.68 0.82911 404 -0.43620 245 -0.85948 352 -0.68327 969 -0.31915 810 -0.45121 636
0.69 0.84795 576 -0.41491 053 -0.85341 027 -0.70481 480 -0.27244 363 -0.44884 377
0.70 0.86730 053 -0.39288 963 -0.84618 438 -0.72528 868 -0.21974 878 -0.43687 329
0; 71 0.88718 386 -0.37009 946 -0.83774 785 -0.74460 199 -0.16166 443 -0.41503 236
0. 72 0.90764 498 -0.34649 561 -0.82803 775 -0.76264 823 -0.09892 467 -0.38323 471
0.73 0.92872 736 -0.32202 902 -0.81698 546 -0.77931 296 -0.03241 178 -0.34160 431
0.74 0.95047 938 -0.29664 526 -0.80451 593 -0.79447 280 +0.03684 038 -0.29049 884
0.75 0.97295 507 -0.27028 369 -0.79054 669 -0.80799 424 0.10764 474 -0.23053 218
0.76 0.99621 508 -0.24287 654 -0.77498 679 -0.81973 225 0.17866 149 -0.26259 54$
0.77 1.02032 776 -0.21434 763 -0.75773 539 -0.82952 866 0.24840 151 -0.08787 565
1.04537 055 -0.18461 097 -0.73868 011 -0.83721 016 0.31523 275 -0.00787 146
E98
. 1.07143 168 -0.15356 897 -0.71769 507 -0.84258 586 0.37739 063 +0.07559 560
0.80 1.09861 229 -0.12111 017 -0.69463 835 -0.84544 435 0.43299 312 0.16037 522
0.81 1.12702 903 -0.08710 649 -0.66934 890 -0.84555 002 0.48006 146 0.24398 96
0.82 1.15681 746 -0.05140 968 -0.64164 264 -0.84263 849 0.51654 781 0.32364 35 17
0.83 1.18813 640 -0.01384 678 -0.61130 745 -0.83641 078 0.54037 123 0.39624 661
0.84 1.22117 352 +0.02578 575 -0.57809 671 -0.82652 589 0.54946 418 0.45844 913
0.85 1.25615 281 0.06772 989 -0.54172 080 -0.81259 105 0.54183 191 0.50669 726
0. 86 1.29334 467 0.11227 642 -0.50183 576 -0.79414 886 0.51562 828 0.53731 190
0; 87 1.33307 963 0.15977 928 -0.45802 786 -0.77065 991 0.46925 273 0.54659 757
0.88 1.37576 766 0.21067 554 -0.40979 212 -0.74147 880 0.40147 508 0.53099 253
0.89 1.42192 587 0.26551 403 -0.35650 171 -0.70582 022 0.31159 776 0.48727 156
0.90 1.47221 949 0.32499 754 -0.29736 306 -0.66270 962 0.19967 037 0.41282 291
0.91 1.52752 443 0.39004 723 -0.23134 775 -0.61090 890 +0.06677 934 0.30602 901
0.92 1.58902 692 0.46190 476 -Oil5708 489 -0i54880 000 -0.08454 828 +0.16680 029
0.93 1.65839 002 0.54230 272 -0.07268 272 -0.47419 336 -0.24975 925 -0.00265 4215
0.94 1.73804 934 0.63376 638 +0.02458 593 -0.38399 297 -0.42137 701 -0.19666 273
0.95 1.83178 082 0.74019 178 0.13888 288 -0.27356 330 -0.58752 240 -0.40421 502
0.96 1.94591 015 0.86807 374 0.27707 112 -0.13540 204 -0.72921 201 -0.60564 435
0.97 2.09229 572 1.02952 685 0.45181 370 +0.04408 092 -0.81464 729 -0.76587 179
0.98 2.29755 993 1.25160 873 0.69108 487 0.29436 613 -0.78406 452 -0.81720 734
0.99 2.64665 241 1.62018 589 1.08264 984 0.70624 831 -0.48875 677 -0.59305 105
1.00 m co
f:n+l)Qn+l(x) = (2%+1)x&n(x)-nn&n-I(X)
348 LEGENDRE FUNCTIONS
PO(X) = 1 P1(x)=x
pm p3w P4W p5w p9w PlO(X)
1.00 1.00 1.00000 1.00000 1.00000 1.00000
1.66 2.52 4.04700 6.72552
2.44 4.76 9.83200
3.34 7. a4
4.36 il.88 I 234 I 1.06544
1.13789
2.alllo
6.65436
5.50 17.00 1 5.53750
22:20 23.32 1 8.47120
% 30.96 2 1.23927
%*Z 9:64 40.04 2 1.74952
2:a 11.26 50.68 II 2 2.39887
3.0 13.00 63.00
14.86 77.12
3-g 16.84 93.16
3:6 la.94 111.24 II 67 1.59814
3.01437
5.46578
9.57313
8.09745
3. a 21.16 131.48
::2"
z!
4: a
23.50
25.96
28.54
31.24
34.06
37.00
154.00
178.92
206.36
236.44
269.28
305.00
3
3
4
4
II 4
4
7.51150
9.65154
1.22500
1.53765
1.91071
2.35250
II II78 3.41632
7.90944
5.25060
2.17406
1.16994 8a9 4.33189
1.05524
2.68690
6.82993
1.62597
II
5. 0
40.06 343.72 4 2.87205
55:: 43.24 385.56 4 3.47916
46.54 430.64 4 4.18440
2:: 49.96 479.08 II 4 4.99917 9 2.38657
1.60047
3.50362
7.23884
5.06985
53.50 531.00
t/
6. 0
57.16 586.52
2:: 60.94 645.76
64.84 708.84
2:: 68.86 775.88 a9 9.01781
2.81890
2.14858
1.62372
1.21596
73.00 847.00 5 1.29367
;:2" 77.26 922.32 5 1.49122
81.64 1001.96 5 1.71215
764 86.14 1086.04 5 1.95846
7: a 90.76 1174.68 II 5 2.23227
a. 0
a. 2
i-2
a: a
9.0
95.50
100.36
105.34
110.44
115.66
121.00
1268.00
1366.12
1469.16
1577.24
1690.48
lao9.00
5
5
5
5
II5
5
2.53583
2.87149
3.24171
3.64912
4.09643
4.58649
II II
10 1.23283
2.92387
1.91848
2.37430
1.54212
11 1.16898
12 6.10897
7.62030
1.43817
9.45994
126.46 1932.92 5 5.12230
z:: 132.04 2062.36 5 5.70699
137.74 2197.44 5 6.34383
9':; 143.56 2338.28 II5 7.03621
10. 0 149.50 2485.00 (4)4.33754 (5)7.78769 (10)9.29640 (12)1.76laa
From National Bureau of Standards, Tables of associated Legendre functions. Columbia Univ. Press, New
York, N.Y., 1945 (with permission).
LEGENDRE FUNCTIONS 351
DERIVATIVE OF THE LEGENDRE FUNCTION-FIRST KIND Pk(z) Table 8.6
P;(z)=1 P&)=32
I II
2 1.01688
2 1.92723 1174282 4 5.24824
II2 3.30168 5.33445 5)1.85808
5 1.39531
5 3.25362
I
5 6.94480
6 1.38132 6 5.50068
7
5 7.29317
1.42939
1.48267
3.36028
6 2.59296
3 2.95500 6 4.63721
II 3
3
3
3
3.86184
I
4.96025
6.27516
7.83305
6
7
7
7
7.95819
1.31805
2.11632
3.30652
II 78 7.52431
1.10110
5.04229
2.23988
1.58313
8
8
9
9
9
4.19097
6.57653
1.00955
1.51918
2.24508
4 2.39550
II 4
4
4
4
2.80816
3.27172
3.79020
4.36775
II
4
4
4
4
4
I
5.00869
5.71746
6.49870
7.35714
8.29772
10
10
10
10
10
1.72421
2.32397
3.10217
4.10354
5.38214
5 1.59602
tl 5
5
5
5
5
1.76260
1.94187
2.13445
2.34099
2.56215
(2)7.485
II II
4 1.35580
1.63974
1.54109
1.44647
1.26900
(4)1.74250
5
5
5
5
(5)3.91127
2.79860
3.05102
3.32013
3.60663
(10)8.40642 (12)1.77028
From National Bureauof Standards,Tablesof associated
Legendrefunctions. ColumbiaUniv. Press,
New York,N.Y., 1945(with permission).
352 LEGENDRE FUNCTIONS
33'20
3:4
it:",
5.0
z-24
II
(-1 2.02733
-1 1.94732
1.87347
II -5 9.56532
-5 8.14823
6.98500
5: 6 -1 1.80507 -5 6.02278
5.8 -1 1.74153 -5 5.22117
7. 0 II -1
-1
-1
1.43841
1.39792
1.35967
II
-3
-3
-3
6.88725
6.50550
6.15475
Ii
-5
-5
-5
2.43500
2.17277
1.94497
;:fI (-1
8.0
i::
-1 11:28915
32346
II
-3 5.83171
5.53353
-3 5.25771
-3 4.76469
5.00208
-5 1.57242
1.74631
II -5 1.41968
-5 1.28507
1.16606
II
-13 5.37876
-13 4.19350
3.28941
i::
9.0
E
9:6
-3 4.54386
4.33807 -6 1.06054
-5 9.66707
I
-13 2.59524
2.05891
-13 I 1.64205
-13 1.06011
1.31620
-14 8.57794
9. 8 -14 6.97159
10.0 (-1)1.00335 (-3)3.35348 (-4)1.34486 (-6)5.77839 (-14)5.69010 (-15)2.71639
From National Bureau of Standards, Tables of associatedLegendre functions. Columbia Univ. Press, New
York,N.Y., 1945 (with permission).
LEGENDRE FUNCTIONS
I
- 8 4.51200
- 8 1 2.11821
220
4: 4
II
4.6
II-5 7.82792
5.27543
3.66172
4.38019
6.40058 I -10
-11 I 1.28985
5.43056
2.43819
3.61188
8.29696
-12 1.36497
-11 3.40566
5.31340
2.21848
8.43598
t-20
6: 4
::t
II
-13 3.19817
-12 4.59703
9.83782
1.46703
6.68395
II
-4
-4
-4
1.71573
1.53040
1.36949
I-13
-13
-13
2.24909
1.59779
1.14602
II
7:
Pi! 8 -4 1.10651
1.22923 -14 I 6.05494
8.29452
Ei
8:4
II
-3
-3
-3
1.32691
1.23104
1.14421
1.06538
-5
-5
-5
9.98765
9.03846
8.19960
7.45601
-4 9.93646 -5 6.79498
Contents
Page
Mathematical Properties .................... 358
Notation. .......................... 358
Bessel Functions J and Y. .................. 358
9.1. Definitions and Elementary Properties ......... 358
9.2. AsymptoCc Expansions for Large Arguments ...... 364
9.3. Asymptotic Expansions for Large Orders ........ 365
9.4. Polynomial Approximations. ............. 369
9.5. Zeros. ....................... 370
Modified Bessel Functions I and K. .............. 374
9.6. Definitions and Properties .............. 374
9.7. Asymptotic Expansions. ............... 377
9.8. Polynomial Approximations. ............. 378
Kelvin Functions. ...................... 379
9.9. Definitions and Properties .............. 379
9.10. Asymptotic Expansions ............... 381
9.11. Polynomial Approximations ............. 384
Numerical Methods ...................... 385
9.12. Use and Extension of the Tables. .......... 385
References. .......................... 388
Table 9.1. Bessel Functions-Orders 0, 1, and 2 (0 5x5 17.5) .... 390
Jo@), 15D, JIW, JzP), Y&3, YIW, 1011
Y&J>, 8D
x=0(.1)17.5
Bessel Functions-Modulus and Phase of Orders 0, 1, 2
(lO<zI a). ................... 396
z*M&), e,(z) -2, 8D
n=0(1)2, s-‘=.l(-.Ol)O
Bessel Functions-Auxiliary Table for Small Arguments
(05212). .................... 397
Yo(cc)-i Jo(z) In z, 2[Yl(z)--f JI(z) In 21
x=0(.1)2, 8D
Page
Table 9.3. Bessel Functions-Orders 10, 11, 20, and 21 (0 1x_<20) . . 402
x-‘“J~o(x), x-llJIT,,(z), Z’“Y~O(Z>
x=0(.1)10, 8s or 9s
JlOb), Jll@), YlO(X>
x= 10(.1)20, 8D
x-‘“J*&), lc-21J21(.x)) 2mY20(4
x=0(.1)20, 6s or 7s
Bessel Functions-Modulus and Phase of Orders 10,11,20,
and21 (2O<x<a). . . . . . . . . . . . . . . . . 406
zfM&(z), &k4 --z
n=lO, 11, 8D
n=20,21, 6D
x-‘=.05(-.002)0
Table 9.4. Bessel Functions-Various Orders (0 <n<lOO). , . . . . 407
J*(x), YJx), n=0(1)20(10)50, 100
x=1, 2, 5, 10, 50, 100, 10s
Table 9.5. Zeros and Associated Values of Bessel Functions and Their
Derivatives (OIn18, 1 <s<20) . . . . . . . . . . . 409
j?w J?x.L.A ; L, J&b,,), 5D (10D for n=O)
YY,,, YXYw); Yyb,7 Y,(yk,,), 5D (8D for n=O)
s=1(1)20, n=0(1)8
2=0(.2)20, 5s to 7s
Modified Bessel Functions-Auxiliary Tabie for Large
Arguments (205x5 ~0). . . . . . . . . . . . . . . 427
In { x+e-~IIo(x) } , In { x+e-‘II1 (x) } , In {a-‘x~e”KIo(x) }
ln{xie-“Izo(x)}, In{ x~e-z121(x)}, In{7r-1xfe”K20(x)}
s-‘=.05(-.001)0, 8D, 6D
The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K.
Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and
Ruth Zucker of the National Bureau of Standards, and N. F. Bird, C. W. Clenshaw, and
Joan M. Felton of the National Physical Laboratory in the preparation and checking of the
tables and graphs.
9. Bessel Functions of Integer Order
Mathematical Properties
,I’
,‘l,,lXl
I’
I
:
/
-DX
FIGURE 9.4. Contour lines of the modulus and phase of the Hankel Function HA” (x+iy)=M&Q. From
E. Jahnke, F. Emde, and F. L&h, Tables of higher functions, McGraw-Hill Book Co., Inc., New
York, N.Y., 1960 (with permission).
360 BESSEL FUNCTIONS OF INTEGER ORDER,
9.1.8 Ye(z) --iHAl) -~iH$~‘(z) -(2/r) In z Y&> =f J’ cos (2 cos 0) {r+ln (22 sin2 0) } a%
0
9.1.9 9.1.20
9.1.21
9.1.10 JY(z)=(w” 2 &$;;1)
9.1.11
*-?a T
Y,(&+ ,(3W n-1(n-k--l)! (fz”>” =- a dr ~080 cos (ne)de
-a k3 k! r s0
2 9.1.22
+- a In (3z)Jn (2)
J.,z,=~J COS(Z sin e-ve)de
-- sin(m) -
e-zslDh r-vcdt (la-g .zl<&r)
?r s 0
where #(n) is given by 6.3.2.
Y,(z) =; J S~II(Z sin e--ve)de
9.1.12 J,,(z) = 1
9.1.13 -~~m(e~‘+e-.’ cos (v7r)}e-*s**tdt (jarg2/<$7r)
2 2 $2”
Yo(z>=- ?r Iln (3z>+rVo(z)+- I-(1!)2 9.1.23
*
(-)“uv+P+2k+l) (iz”)”
k=Or(V+k+l)r(C(+k+l)r(V+CC+k+l) k!
9.1.15
9.1.25
W(J”(Z), J-,(z) 1 =J”+l(z)J-,(z)+J”(~)J-~,+l,(~) -+*i
HI”(z)=$ _ ezslnht--r~dt (jargzl<$r)
= -2 sin (wr)/(?rz) s (D
9.1.16 m-r:
H:)(z)=-: _ ezsl*+-vtdt (largzl<+r)
mu& Y”c4 1=J”+lM y&9 -J&9 Yv+1(d s OD
=2/(m) 9.1.26
9.1.17
W{Hj”(z), H~2’(z)}=H~~~(z)H~2~(~)-H~1~(z)H~,2!~(~) In the last integral the path of integration must
= -4i/(rz) lie to the left of the points t=O, 1, 2, . . . .
BESSEL FUNCTIONS OF INTEGER ORDER 361
and
9.1.27
4
9.1.34 pySY-pJY=- u2ab
Analytic Continuation
9.1.46 1=Jo(z)+2J&)+2J&)+2J&)+ . . .
9.1.47
cos z=J,,(z)-2J&)+2J&)-2J&)+ . . .
1
a,=- 2 p.+,+; p.-1-; P, 9.1.48 sin z=2J,(z)-2J&)+2J&)- . . .
362 BESSEL FUNCTIONS OF INTEGER ORDER
In the following 8= z& and V,(z), g,(z) are any Expressions in Terms of Hypergeometric Functions
9.1.74
Multiplication Theorem LdYY
Gegenbaue?‘s addition theorem
~v(AZ)=Afv 2 (v(A2-1)k(w $f?“*,n(z)
ka0 k! If u, v are real and positive and 0 +Y 5 r, then w, x
(IA”-ll<l> are real and non-negative, and the geometrical
relationship of the variables is shown in the dia-
If %‘= J and the upper signs are taken, the restric- gram.
tion on X is unnecessary. The restrictions Ive*‘al< 1~1 are unnecessary in
This theorem will furnish expansions of %?,(rete) 9.1.79 when %= J and v is an integer or zero, and
in terms of 5ZVflll(r). in 9.1.80 when %Z= J.
Addition Theorems Degenerate Form (u= m):
Neumann’s
9.1.81
eir “““~=I’(~)($v)-~ ‘& (u+k)inJ,+r(v)C~“(c~s a)
(YZO, -1, . . .)
The restriction Ivj<lu] is unnecessary when Neumann’s Expansion of an Arbitrary Function in a
%?= J and v is an integer or zero. Special cases are Series of Bessel Functions
GC?”
(4 -&(y+k)
-=2q7(4
WY
%$dv a
C’;’ (cos a)
The more general form of expansion
9.1.86 f(z>=hJ.(z>+2 g1 %Jv+&)
(Y#O,-1, . . ., lveif~l<luI>
BESSEL FUNCTIONS OF INTEGER ORDER
364
also called a Neumann expansion, is investigated 9.2.6
in [9.7] and [9.15] together with further generaliza- Y,(z) =42/(7rz) {P(v, 2) sin X+ Q(v, 2) cos X}
tions. Examples of Neumann expansions are
9.1.41 to 9.1.48 and the Addition Theorems. Other Wg zl<r)
examples are 9.2.7
9.1.87 ~~lyz)=~~j{~(v, z)+~&(v, 2)}efx
-3 g (-I”
(n+2k)J,+&)
k(n+k)
Pb, (II-1)(P-99)
ego(-1”~g$i=l- 2!(&)2
4-got-1”(v,
2k+1)
9.1.89
Q(v, (2z)2x+1
-r--l (P-l)(P-9h-25)+
82 3!(8~)~ ’ ’ ’
9.2. Asymptotic Expansions for Large
Arguments If v is real and non-negative and z is positive, the
remainder after k terms in the expansion of P(v, z)
Principal Asymptotic Forms
does not exceed the (k+l)th term in absolute
When v is fixed and [z]+o, value and is of the same sign, provided that
k>tv-a. The same is true of Q(v,z) provided
9.2.1 that k>!p---f.
Jy(z)=~q(Fg {cos (z-)~-~~)+e’~“O(lzl-‘)} Asymptotic Expansions of Derivatives
(larg zl<r) With the conditions and notation of the pre-
.
9.2.2 ceding subsection
YY(z)=Jm{sin (z-)v,-t~)+e’~“O(IZI-‘)} 9.2.11
(Ias 4<4 J:(z)=J2/(*2){ -R(v, 2) sinx--S(v, 2) co9 x.}
9.2.3
H~l)(z)‘VJ~e”‘-t~~-t~) (--*<a% 2<27r)
(kg 4<r)
9.2.12
9.2.4 Y:(z) =JG) { R(v, 2) cos x- S(V, 2) sin x}
H~z)(Z),J~)e-“‘-1’“-1”’ (-2n<arg z<7r) (be 4-C~)
9.2.13
Hankel’s Asymptotic Expansions
H!‘)‘(z)=J2/(?rz){iR(v, z)-S(V, z)}efx
When v is fixed and Iz]-+ao (--?r<arg z<27r)
9.2.5 9.2.14
For real v and positive x The general term in the last expansion is given by
9.2.1’7 - 1 . 1 .3 . . . (2k-3)
2.4 - 6 . . . (2k)
MY=IH~l)(x))=~{J~(x)+Y~(x)}
&=arg Hi’)(z) =arctan { YV(z)/Jy(z)} x,(P-1)b9). . . w-3;~l b-@k+1)@k--1)21 *
9.2.18
N”=py(x)I=~{J:2(x)+y:2(x)} 9.2.31
Ip,=arg Hz’)‘(z)=arctan { Yi(z)/Ji(z)}
p+3 p2+46p--63
9.2.19 Jy(z)=My cos &, Yp(x)=A4v sin B,, 4Px-(+t) “+2(4x)+ 6(4~)~
9.2.20 J:(x)=Nv cos (p,, Y:(x) = NV sin cpV. p3+185p2-2053p+1899
+ + . . .
5(4x)5
In the following relations, primes denote differ-
entiations with respect to x. If v 2.0, the remainder after k terms in 9.2.28 does
9.2.21 M39:=2/(?rx) A$p~=2(~-v”)/(7rx3) not exceed the (k+ 1) th term in absolute value
and is of the same sign, provided that k>v-3.
9.2.22 Nf=M;2+A4;fI:2=M:2+4/(~xM,)2
9.2.23 ($-v2)M~M:+x2N,N;+xN;=o
9.3. Asymptotic Expansions for Large Orders
9.2.24
Principal Asymptotic Forms
tan ((py-ey)=M,BI/M:=2/(*xM,M:)
M,N,sin ((py-ev)=2/(1rx) In the following equations it is supposed that
V-+m through real positive values, the other vari-
9.2.25 ZM;‘+xM:+ (x2-S)M,--4/(fM;)=O ables being fixed.
9.2.26
9.3.1
i7?w”‘+x(45c2+ 1-422)w’+ (4v2- l)w=O, w=xw
9.2.27
9.3.3 9.3.9
J, (v set 13)= q(t)=1
v,(t)= (3t-5ta;/24
J2/(7rv tan p) {cos (vtanp-v/3-&r)+O (v-l)} uz(t)=(81t2-462t4+385te)/1152
KKP<3*) u3(t)= (30375t3-3 69603ts+7 65765t’
Yy(v set @)= -4 25425t9)/4 14720
1/2/ (7rv t,an p) {sin (v tan /3-v/3--in) +O(v-‘) 1 u4(t)=(44 65125t4-941 21676te+3499 22430P
-4461 85740t1°+1859 10725t12)/398 13120
(O<P<h>
9.3.4 For I and u,(t) see [9.4] or [9.21].
9.3.11
JL (v sech a) -
9.3.6
9.3.12
YL(v sech CX)
where
9.3.13
vo(t)= 1
q(t‘l=(-9t+7tS)/24
v,(t)=(-l35t2+594t4--455t~)/1152
In the last two equations { is given by 9.3.38 and v3(t) = (-42525t3+4 51737P--8 835752’
9.3.39 below. $4 75475t9)/4 14720
9.3.14
vh(t)=u,(t)+t(t2-1){ ~uM(t)+tu;-,(t)}
Debye’s Asymptotic Expansions (k=l, 2, . . .)
(ii) If /3 is fixed, O<p<$r and v is large and
(i) If LY:is fixed and positive and v is large and positive
positive
9.3.15
9.3.7 Jy(v set fij= 42/(7rv tan p){L(v, P) cos \k
+Mb, P) sin *I
9.3.16
Y”(v set p)=J2/(7rv tan /3){L(v, 8) sin +
--M(v, a> cos *}
9.3.8 where \k=v(t#an p-/3)--&r
9.3.19
&(v set fl)=J(sin 2/3)/(m){ -N(v, P) sin \k
-O(v, a> cos \E}
9.3.20
The corresponding expansions for HJ’) (V + ZV)
Y:(v set P)=J(sin 2@)/(m){N(v, /3) cos * and IP(v+zv~‘~) are obtained by use of 9.1.3
-O(V, p) sin \E} and 9.1.4; they are valid for -+3n<arg v<#?T and
where -#?r<arg v<&r, respectively.
9.3.21 9.3.27
N(v, /I) 'v 2 v2ky? l-9
Ji(v+d3) --$ Ai’ (-21132) {1+x - h&+
-
k=l vZki3
135 cot2 /3+594 cot4 b-j-455 cot6 /3
=1+ 11529 -...
+g Ai (-21/32) 5 In(z)
kc0 vZkJ3
9.3.22
O(,,, ,,j)+ vzk+d.h;t 8)=g cot b-t-1 Cot3 8-. .. 9.3.28
-$ Bi (-21132) g0 $$
When z is fixed, Iv/ is large and jarg VI<+
9.3.23 where
Jv(v+d~3)-~ Ai (-21132) {l+eja} 9.3.29
k-1 Vs’3
h&)=-&o 2+; 2
1159
h,(z)=& SO-+0 z7+z z4-- z
115500
where
9.3.25 9.3.30
3 1
x++ lo(z) =- z3--
5 5
f3(.2) =- 957 28
.x$3-- 1
7000 3150 225
368 BESSEL FUNCTIONS OF INTEGER ORDER
9.3.37
Ai (e2rt/3y2/3~)
v1/3
e2*1/3& r (e2~1/3y2/3t)
+ v5/3
p,=- I213 -.00118 48596..., the branches being chosen so that { is real when
10 23750= z is positive. The coefficients are given by
/92=.00043 78 . . ., f13=-.00038 . . .
9.3.40
Yo"1, r,=~o=.00730 15873 . . .,
ak(l)=g C(8f-3a’2U2k-8{ (1-z2)-tj
b,(c)= -~+~{24(15~2)3,2-s(1181)lj
+*i’(v”“s) g+ a,(s))
v5/3
k=O v
=-- 5 +’
5
4852 (-~)~~24(za-l)312+8(~2-l)~
9.3.36
Y&z)ti- ( E2 >1’4{Bi$y3r)
goty Tables of the early coefficients are given below.
For more extensive tables of the coefficients and
+Bi’(v2/31) 2 a,(r)) for bounds on the remainder terms in 9.3.35 and
v5/3
k=O 3k 9.3.36 see [9.38].
BESSEL FUNCTIONS OF INTEGER ORDER 369
Uniform Expansions of the Derivatives For f>lO use
With the conditions of the preceding subsection
a,(+; p-.104p-2, a,(3-)=.003,
9.3.43
p+.146{- 1
, d,(l)=.OOs.
+Ai’ (3’“~) 5 dx(p))
$13 VXk
For {<-lo use
k=O
bo(S‘)-~r2, a1(~)=.000,
lt]<5X10-8
and & is given by 9.3.13 and 9.3.14. For bounds
on the remainder terms in 9.3.43 and 9.3.44 see 9.4.2 o<x53
[9.38].
= = = Yo(x)=(2/r) h(jx)Jo(x)+.36746 691
r boW COG-1 di W +.60559 366(~/3)~-.74350 384(x/3)4
-~ -- -- .- --- +.25300 117(x/3)‘-.04261 214(x/3)’
0 0.0180 -0.004 0. 1587 0.007
-. 004 . 1785 . 009 +.00427 916(x/3)1o-.OOO24 846(x/3)12+e
B : 0278
0351 -. 001 . 1862 .007
+. 002 . 1927 . 005 lel<1.4X10-B
: : 0366
0352 .003 . 2031 . 004
: : 0311
0331 . 004 . 2155 . 003
. 004 . 2284 . 003 9.4.3 3<x<a
ii : 0278
0294 . 004 . 2413 . 003
. 004 . 2539 . 003 Jo(x) =x-y0 cos e, Yo(x)=x-*f, sin 0,
.004 . 2662 . 003
1: : 0265
0253 . 004 . 2781 .003
f,=.79788 456-.OOOOO 077(3/x)-.00552 740(3/x)”
= = =
-.00009 512(3/~)~+.00137 237(3/x):)’
-I ho(r) al(r) cow d,(r)
-- -- _---
--l--0 0.0180 -0. 004 0. 1587 0. 007
-.00072 805(3/~)~+.00014 476(3/x)6+e
J~(x)=x-+$ cos 01, Yl(x) =x-*jl sin e1 If uVis a zero of %‘i (z) then
f,= .79788 456+ .OOOOO156(3/x)+ .01659 667(3/x)’
9.5.5 U,(u,,=~ %c,(u.)=~ Vv+l(G)
+ .00017 105(3/x)3- .00249 511(3/x)*
+.00113 653(3/x):)“-.00020 033(3/~)~+a The parameter t may be regarded as a continuous
Itl<4X 10-B variable and pr, u, as functions dt), u,(t) of t. If
these functions are fixed by
8,=x-2.35619 449+.12499 612(3/x)
9.5.6 P"(O) =o, 40) =jL, 1
+ .00005 650(3/~)~- .00637 879(3/x)3
then
+.00074 348(3/x)*+.00079 824(3/x)5
-.00029 166(3/~)~+e 9.5.7
jv,8=ds), Yv,1=PAS-% (s=l,2, . . .)
lel<9X 10-B
9.5.8
For expansions of Jo(x), Ye(x), Jl(x), and Y1(x)
in series of Chebyshev polynomials for the ranges ji,s=uv(s-l>, y:,s=QY(s-~) (s=l, 2, * . .)
O<x<8 and O<S/x<l, see 19.371.
9.5. Zeros
9.5.9 u:(,J=($ $)-+, w7,,=($$ $)-*
Real Zeros Infinite Products
When v is real, the functions JP(z), J:(z), Y,(z) z2
9.5.10 &I” -
and Y;(z) each have an infinite number of real J’(“)‘r(vfl) ,!I ( l-x >
zeros, all of which are simple with the possible
exception of z=O. For non-negative v the sth
9.5.11 J;(z)=% ii (1-g) (v>O)
positive zeros of these functions are denoted by r-1
BESSEL FUNCTIONS OF INTEGER ORDER 371
McMahon’s Expansions for Large Zeros
where P=(s++$)~ forj,,,, /3=(s+&-$)a for Y”,~. With P=(t++v-i)?r, the right of 9.5.12 is the
asymptotic expansion of p"(t) for large t.
9.5.13
‘, s, yy,
J,,, I ~-~‘-~8~,3-4(7~2+82a--9)_32(83r3+2075~2-3o39~+353~~
3 (8P’)3 15(8~?)~
64(6949r4+2 96492p3-12 48002p2f74 1438Op-58 53627)
105(8@‘)’ -. ..
where P’=(s+%v--f)s forjl,,, P’=(s++-2)~ for Y:,~, P’=(t+$v+t)s for u,,(t). For higher terms in
9.5.12 and 9.5.13 see [9.4] or [9.40].
Asymptotic Expansions of Zeros
and Associated Values for Large Orders Uniform Asymptotic Expansions of Zeros and
Associated Values for Large Orders
9.5.14
jv,l~v+1.85575 71~“~+1.03315 Ov-“3 m fkW
9.5.22 j.,S-vz([)+z ,zrc-l with {=v-2i3a8
- .00397v-‘- .0908v-5/3+ .043v-7’3f . . .
9.5.15 9.5.23
yp,l-v+ .93157 68vlt3+ .26035 l~-“~
+.01198v-‘-.0060v-5’3-.001v-“3+ . . . Jxj”.J---$~ {1+2 Y}
k=l
9.5.16
with c=vq2f3aa
j:, l -v+ .80861 65~“~$ .07249 OV-"~
- .05097v-‘+.oo94v-5’s+ . . .
9.5.24 jl,,-vz(l)+zI - ,zr-l
gk(i-) with r=v+J3a:
9.5.17
y:,1~+1.82109 80~“~+.94000 7~-“~ 9.5.25
-.05808v-‘-.0540v-5’3+ . . .
J Y(j’ “, J)-& (a’) , ho { l+e T}
Gk(r) with l=vS2J3a:
9.5.18 +3
k=l
J:(j,,)--1.11310 28~-~‘~/(1+1.48460 6vT2j3
+ .43294v-4f3- .1943v-2+ .019v-s’s+ . . . ) where a,, a.: are the sth negative zeros of Ai(
Ai’ (see 10.4), z=z({) is the inverse function
9.5.19 defined implicitly by 9.3.39, and
y:(y,, I>w.95554 86~-~‘~/(1+ .74526 l~-~‘~
+.10910v-4’3-.0185v-2- .003v-8’3+ . . .) 9.5.26
Mf)=14u(1--2)I~
9.5.20
Jy(j:, I) m-67488 51v-1’3(1-.16172 3~-~‘~ fi(r)=32(r>Ih(~)j2b,(r)
+ .02918v-4’3-.0068v-2+ . . .) mw =%--‘4l){W) 12COW
9.5.21 where b,(l), co([) appear in 9.3.42 and 9.3.46.
Y,(y:, 1)w-.57319 40~-“~(1- .36422 OV-~‘~ Tables of the leading coefficients follow. More ex-
+ .09077v-4’s+ .0237v-2-c . . . ) tensive tables are given in [9.40].
Corresponding expansions for s=2, 3 are given The expansions of yy, S, YV(yy,J, y:, Sand Y,(y:, 3
in [9.40]. These expansions become progressively corresponding to 9.5.22 to 9.5.25 are obtained by
weaker as s increases; those which follow do not changing the symbols j, J, Ai, Ai’, a, and a: to
suffer from this defect. y, Y, -Bi, -Bi’, 6, and b: respectively.
372 BESSEL FUNCTIONS OF INTEGER ORDER
=
f,W FI(I) (-ShllW (-SMS)
--
0. 0143 -0.007 -0. 1260 -0.010
1.000000
1. 166284
1.347557
1. 25992
1.22070
1. 18337
.0142
.0139
-. 005 -.
-.
1335
1399
-. 010
-. 009
“:88:.004
1. 14780 .0135 2-E: -. 1453 -. 009 .005
:. E% 1. 11409 -. 003 -. 1498 -. 008
1: 978963 1.08220 0: E -0.002 -0. 1533 -0.008 0: 8:x
=
z(S) h(S) flW PI W s1w Q-J
w G,(I)
--
1. 978963 0.0126 -0. 1533 0.006
: EG -.. g22 -l-o:8;:
2.217607
2. 469770
2.735103
1: 02367
0.99687
:E
. 0110
-.
-.
001
001
-.
-.
-.
1301
1130
0998 -:E
:ii$
3. 013256 .97159 .0105 -. 001 -. 0893 -. 001 .002
BXi%!?
5: 661780
0. 84681
.82972
. 81348
0.0078
.0075 -I
-.
;g;
0464
6.041525 . 79806 : FE% -. 0436
6.431269 .78338 .0065 -. 0410
0.0062
2X
4.4
0. 76939
. Ei:
- 0.0386
-. 0365
4. 6 .73115 r: ;;;g
4.8 .71951 -. 0311
8.968548 0.70836
-I-“: ;;g
9. 422900
9.885820
10.357162
.2E
. 67758
-.
-.
0270
0258
5. 8 10. 836791 .66811 -. 0246
11.324575 0. 65901
11. 820388 . 65024
12.324111
iti 12. 835627
6: 8 13.354826
7. 0 13.881601 0.61821
- - - -
Complex Zeros of J,(s)
c-r)-+ z(r)-8(-r): (--r)+W fl(S) Sl(S)
-I I 1 I- When u> - 1 the zeros of J”(z) are all real. If
O. 40 1.528915 1. 62026 0.0040 -2 w; Y< - 1 and Y is not an integer the number of com-
:% 1. 541532
1.551741 1.65351 ..0029 plex zeros of J”(z) is twice the integer part of
3 k . KfEr: 1:
; y3;
71607 : y;
0006 -. 0033
1: ym;
t-v) ; if the integer part of (-v) is odd two of
these zeros lie on the imaginary axis.
0. 15 1.568285 1.72523 0.0003 -0.0014 If ~20, all zeros of J:(z) are real.
.E 1. 570703
1.570048 1. 73002 . 0001
0000 I: g;‘:
. 00 1.570796 :. . %ii . 0000 -* 0000
lf*(!3I=.OOl, I~2(!31=.0~4 (Oh-<4 When Yis real the pattern of the complex zeros of
lga(r)l=.ool, IG2(s)I=.ooo7 Cl<---r<a) Y”(z) and Y:(z) depends on the non-integer part
I(-~)5gd~)I=.002, I(-~)4G&-)l=.ooo7 of Y. Attention is confined here to the case v=n,
(OS-ls‘<l) a positive integer or zero.
T..--m-- -----
BESSEL FUNCTIONS C”
,I UYI’J!iti&kc unlJr;n
There are n zeros of each function near the Modified Bessel Functions I and K
finite curve extending from z=-n to z=n; the 9.6. Definitions and Properties
asymptotic expansions of these zeros for large n
are given by the right side of 9.522 or 9.5.24 Differential Equation
with p=n and f=e-2rg/k-2/aa8 or pe-2+*&-2&:. dzw dw
9.6.1 22p+Z --(z2+v2)w=o
d2
Zeros of Cross-Products
Solutions are I&z) and K(z). Each is a regular
If Y is real and X is positive, the zeros of the
function of z throughout the z-plane cut along the
function
negative real axis, and for fixed z( #O) each is an
9.5.27 J”(Z) Y”(XZ)---J”(XZ) Y”(Z) entire function of v. When v= f n, I,(z) is an
entire function of 2.
are real and simple. If X>l, the asymptotic Iv(z) ($3’~ 2 0) is bounded as 2+0 in any bounded
expansion of the sth zero is range of arg 2. Iv(z) and I-42) are linearly inde-
pendent except when v is an integer. K(z) tends
9.5.28 P n-PyQPd-2P3
fl+s+ /33 06 +*-* to zero as jzj-+ao in the sector jarg 21<337,
and for all values of v, I"(2) and KY(z) arelinearly
where with 4v2 denoted by cc, independent. I"(z), K(2) are real and positive
9.5.29 when Y>-1 and z>O.
jT3=sr/(X- 1)
--?L-1 ,=(lr-l)(~--25~(x3-l)
‘- 8X 6(4X)3(X-1)
T,(p-1)(p2-l14/l+1073)(x6-l)
211
5(4X)yh- 1)
,&+3)X-w)
8X(X-- 1)
,=(~~+46~-63)~~-(p-1)(~-25)
(
6(4X)3(X-l)
5(4X)s(X-l)r=(p3+185~2-2053p+1899)X6
-(/b-l) (/.&-114c(+1073)
BESSEL FUNCTIONS OF INTEGER ORDER 375
9.6.5
Yv(zet*f)=e*(Y+l)rfl,(z)- (2/7r)e-fv”tK,(z)
(--a<arg zIh>
9.6.6 I-n(z>=In(z>, K-,(z)=K,(z)
Most of the properties of modified Bessel
functions can be deduced immediately from those
of ordinary Bessel functions by application of
these relations.
Ascending Series
9.6.10 odd=" go
s k(r;;;;l)
.
9.6.11
K,(z)+($z)-" ns (n-;yl)! (-iz">"
k-0 *
+ (-->"+I ln WI&)
9.6.3
I,(~)=e-fvr~J,(zet*~) (-r<arg 253~) Wronskians
~,r(~)=e~~*~~~J,(~-~~*‘~) 9.6.14
&<mz z<d
W{ I"(Z), I+(z)) =I"(z)I~(,+l)(z)--I"+l(z)l-,(z)
9.6.4
=-2 sin (vT)/(~z)
K,(~)=3?rieC”~H~‘(ief”%) (---<al-g zi3d 9.6.15
K,(z)=-$A? -I,*tH!a)(ze-t”f)(--<arg 25~) W{K"(Z), I"(Z)}=I"(Z)K"+I(Z)+l"+~(Z)K"(Z)=l/Z
376 BESSEL FUNCTIONS OF INTEGER ORDER
9.6.18
Set2
Formulas for Derivatives
(32)” = COI 0 sin2V 0 &
l’(Z)=?rv(Y+f) 0 9.6.28
ca2)”
l (1-P) v-fe*rr& <av>-+
=Ayv+g -1
(>
s
I”(z)=:
s, e’ cone cos (vO)d49 %“!E’
(2)
=fi~,-kc4
9.6.29
+(;)z”-k+2cz,
-- sin (VT) ODe-2 oontll-v’&
‘1F 0 S (b-g 4<+4
9--v-,+4(4
+**’+s”+kk)
1 (k=O,1,2,. . .)
9.6.21
(Y real)
(I~‘yI<L XX)
9.6.23
?d(~z)” o=e-rco*r sinh2? dt Generating Function and Associated Series
K&)=r(v+t)
s
d(&2)”
‘r<YSa)l S -
emrr(t2-1)‘+ dt 9.6.33 e~‘(‘+“‘)= 5 tkIk(z)
kas-oa
O#O)
k-l kb+k)
C--P [g/w] =
p-1 9.6.54 Ko(z)=- (In (~z)+~)Io(2)+2 8 ‘q
m
Zeros
9.7.10
9.7.3
When v++ 03, these expansions hold uniformly
rf3 GL- 1) 01+15) with respect to z in the sector (arg 21 <&r-e,
m)“&11- 82 + 2! (82) ’ where e is an arbitrary positive number. Here
JP-l)oc--9)Gc+w
+ ..4 Wg4<b)
3!(8~)~ 9.7.11 t=l/&p, ~=~+ln L-
9.7.4 1+4+9
1+x+cc+3 01-l) (rfl5) and z&), vk(t) are given by 9.3.9, 9.3.10, 9.3.13
K:(z) -- &e-y
J 2! (82) 2 and 9.3.14. See [9.38] for tables of II, uk(t),
vk(t), and also for bounds on the remainder
+(p-1)Gc-g)b+35)+ * - .) (larg zl<#~) terms in 9.7.7 to 9.7.10.
3!(8~)~
The general terms in the last two expansions
can be written down by inspection of 9.2.15 and 9.8. Polynomial Approximations *
9.2.16. In equations 9.8.1 to 9.8.4, t=x/3.75.
If Y is real and non-negative and z is positive
the remainder after k terms in the expansion 9.8.1 -3.75 5xs3.75
9.7.2 does not exceed the (k+ 1)th term in absolute &,(z)=1+3.51562 29t2+3.0899424t4+1.20674 92te
value and is of the same sign, provided that + .26597 32t8+ SO360768t’O+ .00458 13P2+ t
k_>v-3.
lt\<1.6XlO-’
9.7.5
9.8.2 3.75 5x<-
xhPIo(x) = .39894 228 + .01328 592 t-l
+.00225 315t-2-.00157 565t-a
; l-3 b-w-9)~ . . *)
2.4 t .00916 281t-4-.02057 706t-s
&I4
+.02635 537t-6-.01647 633t-1
(la%?4<+7d
+.00392 377t-8fc
9.7.6
\e~<L9XlO-’
9.8.3 -3.75 sx 53.75
-- 1.- 1 b-1) (r-45) + . . *
) x-‘I,(x) =$+ .87890 594t2+.51498 869t4
2.4 (22)4 + .15084 934te + .02658 733t8
+.00301 532Pf.00032 411t’*+a
The general terms can be written down by (e)<8XlO-’
inspection of 9.2.28 and 9.2.30. 9.8.4 3.75 <x<cQ
xk=I,(x) = .39894 228- .03988 024t-’
Uniform Asymptotic Expansions for Large Orders - .00362 018t-2+.00163 801 t-a
- .01031 555t-4+ .02282 967t-b
9.7.7 I.(vzg- ey’ jl+gl Y} -.02895 312t-0+.01787 654t-’
j!G (1+22)1’4
- .00420 059 t-++ e
9.7.8 le(<2.2XlO-7
herx=1 (tx”)”
In this and the following section v is real, x is (ix”)” --*
real and non-negative, and n is again a positive (2!)2 +m--
integer or zero.
(+xy (+xy”- * * *
bei x=ax* -- (3!)2 +m
Definitions
9.9.1 9.9.11
n-1
berY xfi bei, x=Jy(xe3*f’4) =ey**JV(xe-*f’4) ker, x=$($x)-” 2 cos { (~wl-$k)~j
=etv”i~v(xe”‘“) ,e3v*i/21v(xe-3W4)
9.9.16
kei, x=-$(3x)-” ngia sin { ($n+t&}
B ab er’ x=ber, x+bei, x
x(n-k-l)! 112 bei’ x=-berl x+be& x
($8)k-ln (3x) bei, x-5 her, x
k! 9.9.17
l/z ker’ x=ker, x+keil x
+MY go sin { (Sn+34*1
@ kei’ x=-kerl x+kei, x
x I+(k+lk)r;~ktk+l) 1 oti>”
! Recurrence Relations for Cross-Producta
If
where #(n> is given by 6.3.2.
9.9.18
9.9.12 p,=bee x+beif x
ker x=-ln (3x) ber x+$t bei x q,=ber, x bei: x-her: x bei, x
rV=berr x her: x+bei, x bei: x
+go t-1” :rk;j2 (t’)”
.s,=be$ x+beiia x
kei x=--In (3x) bei x+r ber x then
.a)
+g l-1” {$y-$ w)“+’ 9.9.19
P.+l=P”-1-T rr
Functions of Negative Argument
qv+1= -; P”+r,=--q,4+2r,
In general Kelvin functions have a branch
point at x=0 and individual functions with argu- (v+l)
ments xe*‘: are complex. The branch point is T”+l= ----z&I- P”+l+qv
absent however in the case of berY and bei, when Y
is an integer, and sv=; p.+,+; a.&$ p,
9.9.13 and
j”+l+j”-l=-@ x (.frgv~ In the following jy, gV are any one of the pairs
given by equations 9.9.15 and jf, g: are either the
same pair or any other pair.
fi=& cf”+1+g”+1T~“-1-!7J”-1)
9.9.21
j+=+ U”+l+g”+1)
S xl+“j~~=2c”
Jz (j
“+l--g”+J=--~ I+” (5 S.-d)
where
9.9.15
Sx*-"@x,x~
@(j"-l-g"-l)=xl-'
(;9.+g:>
9.9.23
f,=ber, x j,=bei, x
g,=bei, x1 g.= -berV x 1 S x(j”g:-g”fl)dx=~ 2Jgq vxf”+l+s”+l)
-s:(j”+l- g”+1)-j”(~+l+gF+1)+g”(j~+~-g~+l) 1
=; x(flft-j”~‘+g:gf-g”s:‘)
BESSEL FUNC’I’IONS OF INTEGER ORDER 381
9.9.24 Zeros of Functions of Order Zero 6
= = =
z(j”g:+gvjz)dz=; ~‘(2j”s~-j47~+1 ber x bei x ker x kei x
s -- .- _-
-j”+lg2-1+2g”fr-g~-lff+l-g,+l~-l~ 1st zero 2.84892 5. 02622 1. 71854 3, 91467
9.9.25 2nd zero 7. 23883 9.45541 6. 12728 8. 34422
3rd zero 11. 67396 13. 89349 10.56294 12, 78256
Sx(f".+gay)dx=x(j"g:-f:gl)
4th zero 16. 11356 18.33398 15. 00269 17. 22314
5th zero 20. 55463 22. 77544 19.44381 21. 66464
=-(x/:/1I2)(frf~+l+g"g"+l--f,g~+l+f,+lg~) ber’ x
=
bei’ z
=
ker’ x
=
kei’ x
9.9.26 -- .- --
Sxj"gdx=;
~2(2j~g"-j~-lg~+,-j"+lg,_l) 1st zero 6.03871 3. 77320 4.93181
2nd zero 10. 51364 8.28099 f %i 9.40405
3rd zero 14.96844 12. 74215 11: 63218 13.85827
19.41758 17. 19343 16. 08312 18. 30717
9.9.27 sl::r: 23. 86430 21. 64114 20. 53068 22. 75379
- - -
Sx(-E-g:)dx=;
~(~-j"-lj~+l-g3+g"-lg"+l) 9.10. Asymptotic
Asymptotic Expansions for
Expansions
Large Arguments
Ascending Series for Cross-Producta
9.9.28 When v is fixed and x is large
berf, x+beit x= 9.10.1
0
1 WS>“”
(ix)2’ 3 r (v+k+l) r (v+2k+ 1) k! ber, x=zx{ j,(x) cos a+gv(x) sin a}
9.10.13
ber’ x=M, cos (o~--~T), bei’ 2 = Ml sin (e,- tr)
9.10.14
2M:=(v/x)M,+M,+~cos (e,,-e,-$T)
= - (v/~)M,- M,-, cos (e,-,--e,-tT)
9.10.15
FIGURE 9.11. In MO(x), b(x), In NO(~) and 40(x).
e;= (M,+,IM,) sin (fI,+I-&-$?r)
= - (MJM,) sin (e,-,-e,-)lr)
Equations 9.10.11 to 9.10.17 hold with the symbols
b, M, e replaced throughout by k, N, 4, respectively.
In place of 9.10.10
6 The coefficients of these terms given in [9.17] are in-
correct. The present results are due to Mr. G. F. Miller. 9.10.20 N-p= Ny, 4-Y=&+ wr
BESSEL FUNCTIONS OF INTEGER ORDER
383
Asymptotic Expansions of Modulus and Phase 9.10.29
When Y is fixed, x is large and r=49 ber x ber’ x+bei x bei’ x-e L-3 !.
212 42 8x
9.10.21
15 1 45 1 315 1
-- 6442 x2
--- 512 ?+819242 p+ ’ - *
_ G- 1)w+ 14/J-399)
614442
9.10.22
75 1 2475 1
In My=?-+ In (27rx)-- r-l --1 (p-1)(/?-25) 1 +25642 --+. x4
i?+S192 . .
42 842 5 38442 ?i?
9.10.31
JrW-13)
128 k&x+kei2 x-&e--r42 l--!-!. ’ ’
442 x+64 2
9.10.23
?f+jj-$-;+
p-l 1 --p-l 1
16 Z
+ 25642
- 33 2---
1 8192 x4
1797 -+
1 . . .>
9.10.32
-G-w--25) 1
38442 >+o ($5) ker x kei’ x-ker’ x kei x--g e-zd2 L-!. i
42 8x
9.10.24
+- 9 ---
1 39 s+slaaJa
1 --L+.
75 . .
N,= ,-WT{I+% ;+&I&? 2 6442 x2 512 x4
9.10.33
+(P-w2+14P-399)
614442 ker x ker’ x+ kei x kei’ x m -Fx e -zd2 $+ik
(
9.10.25
h N,=-g+f ln 0& +-cc-1 ;+
1 (cc-1)(/e-25)
384,i2
1
i?
JP-l)(P--13) 9.10.34
128 ker’2 x+kei’2 x-g em242 1 +&t +& f
9.10.26
+k-l)kW
38442 Asymptotic Expansions of Large Zeros
For v=O these expressions give the 6th zero of 9.11.3 O<x58
each function; for other values of v the zeros ker x=-In (h) ber x-&r bei x-.57721 566
represented may not be the sth.
-59.05819 744(x/8)4+171.36272 133(x/8)8
Uniform Asymptotic Expansions for Large Orders
-60.60977 451(x/8)12+5.65539 121(x/8)”
When v is large and positive - .19636 347 (x/S)‘O+ .00309 699 [x/8)24
9.10.37 -.00002 458(x/8)2*+a
9.11.4 0<218
9.10.38
kei x---ln($x)bei x-&r ber s-j-6.76454 936(x/8)2
ker, (~x)++i kei, (vx)
-142.91827 687(x;/8)‘+124.23569 650(x/8)l”
-21.30060 904(x/8)“+1.17509 064(r/8)‘8
-.02695 875(x/8)22+.OOO29 532(~/8)‘~+c
9.10.39
(tj<3x10-9
her: (vx)+si bei: (vx)
9.11.5 -8<x<8
a-
9.10.40 her’ ~=~[-4(x/8)~+14.22222 222(x/8)’
ker: (vx)+i kt:iI (vx) -6.06814 810(~/8)‘~+.66047 849(x/8)”
-.02609 253(~/8)‘~+.00045 957(x/8)22
-.OOOOO 394(x/8)20]+c
where
~e~<2.1x10-*
9.10.41 [=&FT?
and u,(t), c*(l) are given by 9.3.9 and 9.3.13. All
fractional powers take their principal values. 9.11.6 -812_<8
bei’ z=z[$- 10.66666 SS~(X/S)~
9.11. Polynomial Approximations
+11.37777 772(~/8)~-2.31167 514(x/8)12
9.11.1 -85x18 +.14677 204(x/8)“--00379 386(x/8)”
ber x=1-64(2/8)‘+113.77777 774(x/8)* +.00004 609(x/8)24]+c
-32.36345 652(x/8)12+2.64191 397(x/8)“’
IcI<7xlO-*
-.08349 609(x/8)“+.00122 552(x/8)“’
- .OOOOO 901 (x/S)“+t
(cl<lXlO”Q 9.11.7 O<x<8
9.11.2 -8Sx_<8 ker’ x=--In (4%) ber’ z--2+ ber s+t~ bei’ x
bei x= 16(~/8)~- 113.77777 774(x/8)e +x[-3.69113 734(~/8)~.+21.42034 017(x/8)’
+72.81777 742(s/8)*O-10.56765 779(x/8)” -11.36433 272(~/8)‘~+1.41384 780(x/8)‘”
+.52185 615(x/8)‘*.-.01103 667(~/8)~~ -.06136 358(~/8)~~+.00116 137(~/8)~’
+.OOOll 346(x/8)2e+c -.OOOOl 075(x/S)““]+b
~c~<SX~O-~ Icl<SXlO-*
BESSEL FUNCTIONS OF INTEGER ORDER 385
9.11.8 O<x<8 where
kei’ x=--In (ix) bei’ x-x-l bei x-tr ber’ x 9.11.11
(e4~<3x10-'
where
9.11.14
t#~(x)=(.70710 68+.70710 68;)
9.11.10 81x< = +(-.06250 Ol-.OOOOO Oli)(8/x)
+(-.00138 13+.00138 1li)(8/x)2
her x+i bei x-z (ker xfi kei x)=g(x)(l+cJ
+(.OOOOO 05+.00024 52i)(8/~)~
Numerical Methods
9.12. Use and Extension of the Tables n Trial valuea 541.66)
9 0 .ooooo
Example 1. To evaluate J&.55), n=O, 1, 2, 8 1 .oooOO
. ., each to 5 decimals. 7 10 .00003
The recurrence relation 6 89 .00028
6 679 .00211
4 4292 .01331
Jn-l(4 +Jn+1(4 = (W4J,(4 3 21473 .06661
can be used to compute Jo(x), 51(z):), J&c), . . ., 2 78829 .24453
1 181957 .56442
successively provided that n<x, otherwise severe 0 166954 .48376
accumulation of rounding errors will occur.
Since, however, J,,(x) is a decreasing function of n
when n>x, recurrence can always be carried out We normalize the results by use of the equation
in the direction of decreasing n. 9.1.46, namely
Inspection of Table 9.2 shows that J,,(l.55)
JO(X)+~J~(X)+~J~(X)+ . . . =I
vanishes to 5 decimals when n>7. Taking arbi-
trary values zero for Jo and unity for Ja, we compute
by recurrence the entries in the second column of This yields the normalization factor
the following table, rounding off to the nearest
integer at each step. l/322376=.00000 31019 7
386 BESSEL FUNCTIONS OF INTEGER ORDER
and multiplying the trial values by this factor we Remarks. (i) An alternative way of computing
obtain the required results, given in the third YO(x), should J,,(x), Jz(r), J&c), . . ., be avail-
column. As a check we may verify the value of tble (see Example l), is to use formula 9.1.89.
J,(1.55) by interpolation in Table 9.1. The other starting value for the recurrence,
Remarks. (i) In this example it was possible Y1(z), can then be found from the Wronskian
to estimate immediately the value of n=N, say, :elation Jl(z) Y,,(x) - J,,(x) Y1(x) =2/(7rx). This is a
at which to begin the recurrence. This may not :onvenient procedure for use with an automatic
always be the case and an arbitrary value of Nmay :omputer.
have to be taken. The number of correct signifi- (ii) Similar methods can be used to compute the
cant figures in the final values is the same as the modified Bessel function K,(x) by means of the
number of digits in the respective trial values. recurrence relation 9.6.26 and the relation 9.6.54,
If the chosen N is too small the trial values will except that if z is large severe cancellation will
have too few digits and insufficient accuracy is occur in the use of 9.6.54 and other methods for
obtained in the results. The calculation must evaluating K,,(Z) may be preferable, for example,
then be repeated taking a higher value. On the use of the asymptotic expansion 9.7.2 or the poly-
other hand if N were too large unnecessary effort nomial approximation 9.8.6.
would be expended. This could be offset to some Example 3. To evaluate J,(.36) and Y,(.36)
extent by discarding significant figures in the trial each to 5 decimals, using the multiplication
values which are in excess of the number of theorem.
decimals required in J,,. From 9.1.74 we have
(ii) If we had required, say, Jo(1.55), J1(1.55), m
. . ., Jlo(l.55), each to 5 significant figures, we go (X z) =x ak%Yk(z) , where aR = WW~l)‘(W.
would have found the values of J,,(l.55) and k-0
0. 4 3.110
As a check we may verify that +21
0. 6 3.131 -12
JY’- J’Y=2/(75s). +9
0. 8 3.140 -7
Remarks. This example may also be computed +2
using the Debye expansions 9.3.15, 9.3.16, 9.3.19, 1. 0 3.142(x)
and 9.3.20. Four terms of each of these series are
required, compared with two in the computations Interpolating for l/X=.667, we obtain
above. The closer the argument-order ratio is to (x-l)a:“=3.134 and thence the required root
unity, the less effective the Debye expansions @b=6.268.
become. In the neighborhood of unity the expan- Example 7. To evaluate ber, 1.55, bei, 1.55,
sions 9.3.23, 9.3.24, 9.3.27, and 9.3.28 will furnish n=o, 1, 2, . * ., each to 5 decimals.
results of moderate accuracy; for high-accuracy We use the recurrence relation
work the uniform expansions should again be used.
= 1/(294989-22011i)=(.337119+.025155i)x10-6,
n Real Imag. ber,,z be&,x
ial valuer is1 valuer obtained from the relation
--
t-y x . 00000
. xX138:
jo(marf/4) +2Ja(dy +2J4(~3rf’4) + . . . = 1.
-7
$50:
+8i
- 475
-:
-.
+.
:?A!;
00003
00181
-. 00003
2: gyg
Adequate checks are furnished by interpolating
- 4447 - 203 -. 01494 -. 00180
+ 14989 + 17446 +. 04614 +. 06258 in Table 9.12 for ber 1.55 and bei 1.55, and the
+11172 - 88578 -. 29580
- 197012 $_: 8;;;‘: +. 36781 use of a simple sum check on the normalization.
+2s1539 +. 91004 +. 59461 Should ker’s and kei,x be required they can be
--
C $106734 + 207449 f. 30763 +. 72619 computed by forward recurrence using formulas
- 9.9.14, taking the required starting values for
The values of ber,,x and bei,,x are computed by n=O and 1 from Table 9.12 (see Example 2). If
multiplication of the trial values by the normal- an independent check on the recurrence is required
ieing factor the asymptotic expansion 9.10.38 can be used.
References
Texts [9.13] F. W. J. Olver, The asymptotic expansion of Bessel
functions of large order. Philos. Trans. Roy.
[9.1] E. E. Allen, Analytical approximations, Math. Sot. London A247, 328-368 (1954).
Tables Aids Comp. 8, 246-241 (1954). [9.14] G. Petiau, La theorie des fonctions de Bessel
[9.2] E. E. Allen, Polynomial approximations to some (Centre National de la Recherche Scientifique,
modified Bessel functions, Math. Tables Aids Paris, France, 1955).
Comp. 10, 162-164 (1956). [9.15] G. N. Watson, A treatise on the theory of Bessel
[9.3] H. Bateman and R. C. Archibald, A guide to tables functions, 2d ed. (Cambridge Univ. Press,
of Bessel functions, Math. Tables Aids Comp. 1, Cambridge, England, 1958).
205-308 (1944). [9.16] R. Weyrich, Die Zylinderfunktionen und ihre
[9.4] W. G. Bickley, Bessel functions and formulae Anwendungen (B. G. Teubner, Leipzig, Germany,
(Cambridge Univ. Press, Cambridge, England, 1937).
1953). This is a straight reprint of part of the [9.17] C. S. Whitehead, On a generalisation of the func-
preliminaries to [9.21]. tions ber x, bei z, ker x, kei x. Quart. J. Pure
[9.5] H. S. Carslaw and J. C. Jaeger, Conduction of heah Appl. Math. 42, 316-342 (1911).
in solids (Oxford Univ. Press, London, England, [9.18] E. T. Whittaker and G. N. Watson, A course of
1947). modern analysis, 4th ed. (Cambridge Univ.
[9.6] E. T. Copson, An introduction to the theory of Press, Cambridge, England, 1952).
functions of a complex variable (Oxford Univ.
Press, London, England, 1935). Tables
[9.7] A. Erdelyi et al., Higher transcendental functions, [9.19] J. F. Bridge and S. W. Angrist, An extended table
~012, ch. 7 (McGraw-Hill Book Co., Inc., New of roots of 5;(z) Yi(&r) -J:(&r) Y;(z) =O, Math.
York, N.Y., 1953). Comp. 16, 198-204 (1962).
[9.8] E. T. Goodwin, Recurrence relations for cross- [9.20] British Association for the Advancement of Science,
products of Bessel functions, Quart. J. Mech. Bessel functions, Part I. Functions of orders
Appl. Math. 2, 72-74 (1949). zero and unity, Mathematical Tables, vol. VI
[9.9] A. Gray, G. B. Mathews and T. M. MacRobert, (Cambridge Univ. Press, Cambridge, England,
A treatise on the theory of Bessel functions, 2d 1950).
ed. (Macmillan and Co., Ltd., London, England; [9.21] British Association for the Advancement of Science,
1931). Bessel functions, Part II. Functions of positive
[9.10] W. Magnus and F. Oberhettinger, Formeln und integer order, Mathematical Tables, vol. X
S&e fiir die speziellen Funktionen der mathe- (Cambridge Univ. Press, Cambridge, England,
matischen Physik, 2d ed. (Springer-Verlag; 1952).
Berlin, Germany, 1948). [9.22] British Association for the Advancement of Science,
[9.11] N. W. McLachlan, Bessel functions for engineers, Annual Report (J. R. Airey), 254 (1927).
2d ed. (Clarendon Press, Oxford, England, 1955). [9.23] E. Cambi, Eleven- and fifteen-place tables of Bessel
[9.12] F. W. J. Olver, Some new asymptotic expansions functions of the first kind, to all significant orders
for Bessel functions of large orders. Proc. (Dover Publications, Inc., New York, N.Y.,
Cambridge Philos. Sot. 48, 414-427 (1952). 1948).
BESSEL FUNCTIONS OF INTEGER ORDER 389
[9.24] E. A. Chistova, Tablitsy funktsii Besselya ot functions with imaginary argument and their
deistvitel’nogo argumenta i integralov ot nikh integrals).
(Izdat. Akad. Nauk SSSR, Moscow, U.S.S.R., [9.34] Mathematical Tables Project, Table of
1958). (Table of Bessel functions with real f.(z)=nl(%z)-nJ.(z). J. Math. Phys. 23,
argument and their integrals). 45-60 (1944).
[9.25] H. B. Dwight, Tables of integrals and other mathe- [9.35] National Bureau of Standards, Table of the Bessel
matical data (The Macmillan Co., New York, functions Jo(z) and J1(z) for complex arguments,
N.Y., 1957). 2d ed. (Columbia Univ. Press, New York, N.Y.,
This includes formulas for, and tables of Kelvin 1947).
functions. [9.36] National Bureau of Standards, Tables of the Bessel
[9.26] H. B. Dwight, Table of roots for natural frequencies functions YO(z) and Yi(z) for complex arguments
in coaxial type cavities, J. Math. Phys. 27, (Columbia Univ. Press, New York, N.Y., 1950).
8449 (1948). [9.37] National Physical Laboratory Mathematical Tables,
This gives zeros of the functions 9.5.27 and 96.39 vol. 5, Chebyshev series for mathematical func-
for n=0,1,2,3. tions, by C. W. Clenshaw (Her Majesty’s Sta-
[9.27] V. N. Faddeeva and M. K. Gavurin, Tablitsy tionery Office, London, England, 1962).
funktsii Besselia J,(z) tselykh nomerov ot 0 [9.38] National Physical Laboratory Mathematical Tables,
do 120 (Izdat. Akad. Nauk SSSR, Moscow, vol. 6, Tables for Bessel functions of moderate or
U.S.S.R., 1950). (Table of J.(z) for orders 0 to large orders, by F. W. J. Olver (Her Majesty’s
120). Stationery Office, London, England, 1962).
[9.28] L. Fox, A short table for Bessel functions of integer [9.39] L. N. Nosova, Tables of Thomson (Kelvin) functions
orders and large arguments. Royal Society and their first derivatives, translated from the
Shorter Mathematical Tables No. 3 (Cambridge Russian by P. Basu (Pergamon Press, New
Univ. Press, Cambridge, England, 1954). York, N.Y., 1961).
[9.29] E. T. Goodwin and J. Staton, Table of J&o,J), [9.40] Royal Society Mathematical Tables, vol. 7, Bessel
Quart. J. Mech. Appl. Math. 1, 220-224 (1948).
functions, Part III. Zeros and associated values,
[9.30] Harvard Computation Laboratory, Tables of the edited by F. W. J. Olver (Cambridge Univ. Press,
Bessel functions of the first kind of orders 0
Cambridge, England, 1960).
through 135, ~01s. 3-14 (Harvard Univ. Press,
The introduction includes many formulas con-
Cambridge, Mass., 1947-1951).
nected with zeros.
[9.31] K. Hayashi, Tafeln der Besselschen, Theta, Kugel-
und anderer Funktionen (Springer, Berlin, Ger- [9.41] Royal Society Mathematical Tables, vol. 10,
ma.ny, 1930). Bessel functions, Part IV. Kelvin functions, by
[9.32] E. Jahnke, F. Emde, and F. Loach, Tables of A. Young and A. Kirk (Cambridge Univ. Press,
higher functions, ch. IX, 6th ed. (McGraw-Hill Cambridge, England, 1963).
Book Co., Inc., New York, N.Y., 1960). The introduction includes many formulas for
[9.33] L. N. Karmazina and E. A. Chistova, Tablitsy Kelvin functions.
funktsii Besselya ot mnimogo argumenta i 19.42) W. Sibagaki, 0.01 % tables of modified Bessel
integralov ot nikh (Izdat. Akad. Nauk SSSR, functions, with the account of the methods used
Moscow, U.S.S.R., 1958). (Tables of Bessel in the calculation (Baifukan, Tokyo, Japan, 1955).
390 BESSEL FUNCTIONS OF INTEGER ORDER
yocd Ydx)
520 -0.30851
-0.32160
76252
24491
0.14786
0.11373
Ylk)
31434
64420
0.36766 288
0.36620 498
55:: -0.33125
-0.33743
09348
73011
0.07919
0.04454
03430
76191
0.36170 876
0.35424 772
Z:i -0.34016 78783 +0:01012 72667 0.34391 872
-0.33948 05929 -0.02375 82390
::2 -0.33544
-0.32815
41812
71408
-0.05680
-0.08872
56144
33405
0.33084
0.31515
0.29702
123
646
614
:2
5: 9
-0.31774
-0.30436
64300
59300
-0.11923
-0.14807
41135
71525
0.27663
0.25417
122
029
-0.28819 46840 -0.17501 03443 0.22985 790
2:: -0.26943 49304 -0.19981 22045 0.20392 273
-0.24830 99505 -0.22228 36406 0.17660 555
2: -0.22506 17496 -0.24224 95005 0.14815 715
6: 4 -0.19994 85953 -0.25955 98934 0.11883 613
-0.17324 24349 -0.27409 12740 0.08890 666
i-2 -0.14522 62172 -0.28574 72791 0.05863 613
6: 7 -0.11619 11427 -0.29445 93130 to.02829 284
-0.08643 38683 -0.30018 68758 -0.00185 639
29" -0.05625 36922 -0.30291 76343 -0.03154 852
7. 0 -0.02594 97440 -0.30266 72370 -0.06052 661
+0.00418 17932 -0.29947 88746 -0.08854 204
::: 0.03385 04048 -0.29342 25939 -0.11535 668
0.06277 38864 -0.28459 43719 -0.14074 495
77:: 0.09068 08802 -0.27311 49598 -0.16449 573
7.5 0.11731 32861 -0.25912 85105 -0.18641 422
0.14242 85247 -0.24280 10021 -0.20632 353
3 0.16580 16324 -0.22431 84743 -0.22406 617
0.18722 71733 -0.20388 50954 -0.23950 540
5 0.20652 09481 -0.18172 10773 -0.25252 628
8.0 0.22352 14894 -0.15806 04617 -0.26303 660
0.23809 13287 -0.13314 87960 -0.27096 757
i-21 0.25011 80276 -0.10724 07223 -0.27627 430
8:3 0.25951 49638 -0.08059 75035 -0.27893 605
8. 4 0.26622 18674 -0.05348 45084 -0.27895 627
0.27020 51054 -0.02616 86794 -0.27636 244
256 0.27145 77123 +0.00108 39918 -0.27120 562
0.26999 91703 0.02801 09592 -0.26355 987
Ei 0.26587 49418 0.05435 55633 -0.25352 140
8: 9 0.25915 57617 0.07986 93974 -0.24120 758
0.24"3 66983 0.10431 45752 -0.22675 568
;*10 0.23833 59921 0.12746 58820 -0.21032 151
9:2 0.22449 36870 0.14911 27879 -0.19207 786
0.20857 00676 0.16906 13071 -0.17221 280
E 0.19074 39189 0.18713 56847 -0.15092 782
0.17121 06262 0.20317 98994 -0.12843 591
x-2 0.15018 01353 0.21705 89660 -0.10495 952
9:7 0.12787 47920 0.22866 00298 -0.08072 839
0.10452 70840 0.23789 32421 -0.05597 744
82 0.08037 73052 0.24469 24113 -0.03094 449
10. 0 0.05567 11673 0.24901 54242 -0.00586 808
[C-i)4
1 [ 2n 1
(-;I4
[ 1
c-t)4
Y,+1(2)=zY,(z)-Y,-1(2)
394 BESSEL FUNCTIONS OF INTEGER ORDER
[ 1
C-i!3
[ 1
c-:13
2n
[ 1
c-;13
Y~+l(x)=~Y,(x)-Y,~l(x)
396 BESSEL FUNCTIONS OF INTEGER ORDER
2n
[ 1
C-92
J,+.~(z)=~ Jn(z)-Jn-I(Z)
<z>=nearest integer to 2.
BESSEL FUNCTIONS OF INTEGER ORDER 397
BESSEL FUNCTIONS-ORDERS 0,l AND 2 Table 9.1
%+1(x)=$ Yn(2)-FL-1(~)
Table 9.1
BESSEL FUNCTIONS-AUXILIARY TABLE FOR SMALL ARGUMENTS
(1; g.;;;;
-3 9:0629
-2 1.4995
II-2 23197 - 5)8.5712
0.12894 - 6)2.4923
;:2" 0.16233
0.19811
Et 0.23529
2:8 0.27270
3.0 0.30906 0.13203
3.2 0.34307 0.15972
0.37339 0.18920
xi 0.39876 0.21980
3:8 0.41803 0.25074
0.43017 0.28113 0.13209
t:2" 0.43439 0.31003 0.15614
0.43013 0.33645 0.18160
t-46 0.41707 0.35941 0.20799
418 0.39521 0.37796 0.23473 I - 43 I 1.3952
9.3860
2.0275
4.0270
2.8852
0.36483 0.39123 0.26114 0.13105
E! 0.32652 0.39847 0.28651 0.15252
514 0.28113 0.39906 0.31007 0.17515
0.22978 0.39257 0.33103 0.19856
55:: 0.17382 0.37877 0.34862 0.22230
6.0 0.11477 0.35764 0.36209 0.24584 0.12959
6.2 +0.05428 0.32941 0.37077 0.26860 0.14910
-0.00591 0.29453 0.37408 0.28996 0.16960
246 -0.06406 0.25368 0.37155 0.30928 0.19077
6:8 -0.11847 0.20774 0.36288 0.32590 0.21224
7.0 -0.16756 0.15780 0.34790 0.33920 0.23358 0.12797
7.2 -0.20987 0.10509 0.32663 0.34857 0.25432 0.14594
-0.24420 +0.05097 0.29930 0.35349 0.27393 0.16476
2 -0.26958 -0.00313 0.26629 0.35351 0.29188 0.18417
718 -0.28535 -0.05572 0.22820 0.34828 0.30762 0.20385
-0.29113 -0.10536 0.18577 0.33758 0.32059 0.22345 0.12632
-0.28692 -0.15065 0.13994 0.32131 0.33027 0.24257 0.14303
-0.27302 -0.19033 0.09175 0.29956 0.33619 0.26075 0.16049
-0.25005 -0.22326 +0.04237 0.27253 0.33790 0.27755 0.17847
-0.21896 -0.24854 -0.00699 0.24060 0.33508 0.29248 0.19670
9.0 -0.18094 -0.26547 -0.05504 0.20432 0.32746 0.30507 0.21488
9.2 -0.13740 -0.27362 -0.10053 0.16435 0.31490 0.31484 0.23266
-0.08997 -0.27284 -0.14224 0.12152 0.29737 0.24965
E! -0.04034 -0.26326 -0.17904 0.07676 0.27499 8'ZY 0.26546
9:8 +0.00970 -0.24528 -0.20993 +0.03107 0.24797 0:32318 0.27967
10.0 0.05838 -0.21960 -0.23406 -0.01446 0.21671 0.31785 0.29186
Compiled from British Association for the Advancement of Science, Bessel functions, Part II. Func-
tions of positive integer order, Mathematical Tables, vol. X (Cambridge Univ. Press, Cambridge, Eng-
land, 1952) and Mathematical Tables Project, Table of .f,(x) = n!(G)-nJ,(:x). J. Math. Phys. 23, 45-60
(1944) (with permission).
BESSEL FUNCTIONS OF INTEGER ORDER ‘399
BESSEL FUNCTIONS-ORDERS 3-9 Table 9.2
4 :1;162
-5.8215
-3.5899
II
3
2
1
-1.2097
-2.4302
-7.8751
-2.4420 -9.4432
II
-1.7897 -5.8564
-1.3896 -3.9059
2.0 -1.1278 -2.7659 -9.9360 1 -4.6914
-0.94591 -2.0603 -6.5462 1 -2.7695
2:
2:6
-0.81161
-0.70596
-1.6024
-1.2927
-4.5296
-372716
1
1
-1.7271
-1.1290
2.8 -0.61736 -1.0752 -2.4548 -7.6918
3.0 -0.53854 -0.91668 -1.9059 -5.4365
3.2 -0.46491 -0.79635 -1.5260 -3.9723
-0.39363 -0.70092 -1.2556 -2.9920 -9.3044
z-46 -0.32310 -0.62156 -1.0581 -2.3177 -6.6677
3:8 -0.25259 -0.55227 -0.91009 -1.8427 -4.9090
4.0 -0.18202 -0.48894 -0.79585 -1.5007 -3.7062 ( l)-1.1471
4.2 -0.11183 -0.42875 -0.70484 -1.2494 -2.8650 -8.3005
-0.04278 -0.36985 -0.62967 -1.0612 -2.2645 -6.1442
2: +0.02406 -0.31109 -0.56509 -0.91737 -1.8281 -4.6463
4:8 0.08751 -0.25190 -0.50735 -0.80507 -1.5053 -3.5855
5.0 0.14627 -0.19214 -0.45369 -0.71525 -1.2629 -2.8209 -7.7639
0.19905 -0.13204 -0.40218 -0.64139 -1.0780 -2.2608 -5.8783
:5 0.24463 -0.07211 -0.35146 -0.57874 -0.93462 -1.8444 -4.5302
516 0.28192 -0.01310 -0.30063 -0.52375 -0.82168 -1.5304 -3.5510
5.8 0.31001 +0.04407 -0.24922 -0.47377 -0.73099 -1.2907 -2.829l5
0.09839 -0.19706 -0.42683 -0.65659 -1.1052 y;;
2; oo%E 0.14877 -0.14426 -0138145 -0.59403 -0.95990
6:4 0:333s3 0.19413 -0.09117 -0.33658 -0.53992 -0.84450 -1:5713
6.6 0.32128 0.23344 -0.03833 -0.29151 -0.49169 -0.75147 -1.33011
6.8 0.29909 0.26576 +0.01357 -0.24581 -0.44735 -0.67521 -1.1414
0.26808 0.29031 0.06370 -0.19931 -0.40537 -0.61144 -0.992:20
;-2o 0.22934 0.30647 0.11119 -0.15204 -0.36459 -0.55689 -0.87293
714 0.18420 0.15509 -0.10426 -0.32416 -0.50902 -0.77643
0.13421 00%2B~ 0.19450 -0.05635 -0.28348 -0.46585 -0.69726
;:i 0.08106 0:30186 0.22854 -0.00886 -0.24217 -0.42581 -0.63128
8.0 +0.02654 0.28294 0.25640 +0.03756 -0.20006 -0.38767 -0.57528
-0.02753 0.25613 0.27741 0.08218 -0.15716 -0.35049 -0.52673
E -0.07935 0.22228 0.29104 0.12420 -0.11361 -0.31355 -0.48363
8:6 -0.12723 0.18244 0.29694 0.16284 -0.06973 -0.27635 -0.444140
8.8 -0.16959 0.13789 0.29495 0.19728 -0.02593 -0.23853 -0.40777
-0.20509 0.09003 0.28512 0.22677 +0.01724 -0.19995 -0.37271
%*20 -0.23262 +0.04037 0.26773 0.25064 0.05920 -0.16056 -0.33843
9:4 -0.25136 -0.00951 0.24326 0.26830 0.09925 -0.12048 -0.30433
-0.26079 -0.05804 0.21243 0.27932 0.13672 -0.07994 -0.26995
;:i -0.26074 -0.10366 0.17612 0.28338 0.17087 -0.03928 -0.23499
10.0 -0.25136 -0.14495 0.13540 0.28035 0.20102 +0.00108 -0.19930
400 BESSEL FUNCTIONS OF INTEGER ORDER
Ydx) Ys(4
lo”0
10:2
-0.25136
-0.23314
y4cd
-0.14495
-0.18061
y5w
0.13540
0.09148
yS(x)
0.28035
0.27030
y7cd
0.20102
0.22652
0.00108 -0.19930
Y9@)
0.04061 -0.16282
10.4 -0.20686 -0.20954 +0.04567 0.25346 0.24678 0.07874 -0.12563
10.6 -0.17359 -0.23087 -0.00065 0.23025 0.26131 0.11488 -0.08791
10.8 -0.13463 -0.24397 -0.04609 0.20130 0.26975 0.14838 -0.04793
11.0 -0.09148 -0.24851 -0.08925 0.16737 0.27184 0.17861 -(LO1205
11.2 -0.04577 -0.24445 -0.12884 0.12941 0.26750 0.20496 +0.02530
11.4 +0.00082 -0.23203 -0.16365 0.08848 0.25678 0.22687 0.06163
11.6 0.04657 -0.21178 -0.19262 0.04573 0.24384 0.09640
11.8 0.08981 -0.18450 -0.21489 +0.00238 od
. zsx 0.25545 0.12906
12.0 0.12901 -0.15122 -0.22982 -0.04030 0.18952 0.26140 0.15902
12.2 -0.11317 -0.23698 -0.08107 0.15724 0.26151 0.18573
12.4 EKi92971 -0.07175 -0.23623 -0.11875 0.12130 0.25571 0.20865
12.6 0:20959 -0.02845 -0.22766 -0.15223 0.08268 0.24409 0.22728
12.8 0.22112 +0.01518 -0.21163 -0.18052 0.04240 0.22689 0.24122
13.0 0.22420 0.05759 -0.1887d -0.20279 +0.00157 0.20448 0.25010
13.2 0.21883 0.09729 -0.15987 -0.21840 -0.03868 0.17738 0.25369
13.4 0.20534 0.13289 -0.12600 -0.22692 -0.07722 0.14625 0.25184
13.6 0.18432 0.16318 -0.22813 -0.11296 0.11185 0.24454
13.8 0.15666 0.18712 100'
. 00:;;; -0.22204 -0.14489 0.07505 0.23190
14.0 0.12350 0.20393 -0.00697 -0.20891 -0.17209 +0.03682 0.21417
14.2 0.08615 0.21308 +0.03390 -0.18921 -0.19380 -0.00186 0.19170
14.4 0.04605 0.21434 0.07303 -0.16363 -0.20939 -0.03994 0.16501
14.6 +0.00477 0.20775 q.10907 -0.13305 -0.21842 -0.07640 0.13470
14.8 -0.03613 0.19364 0.14080 -0.09850 -0.22067 -0.11024 0.10149
15. 0 -0.07511 0.17261 -0.06116 -0.21610 -0.14053 0.06620
15.2 -0.11072 0.14550 k 11:::70 -0.02228 -0.20489 -0.16644 +0.02969
15.4 -0.14165 0.11339 0120055 +0.01684 -0.18743 -0.18723 -0.00710
15.6 -0.16678 0.07750 0.20652 0.05489 -0.16430 -0.20234 -0.04322
15.8 -0.18523 +0.03920 0.20507 0.09059 -0.13627 -0.21134 -0.07775
16.0 -0.19637 -0.00007 0.19633 0.12278 -0.10425 -0.21399 -0.10975
16.2 -0.19986 -0.03885 0.18067 0.15038 -0.06928 -0.21025 -0.13838
16.4 -0.19566 -0.07571 0.15873 0.17250 -0.03251 -0.20025 -0.16286
16.6 -0.18402 -0.10930 0.13135 0.18843 +0.00487 -0.18432 -0.18253
16.8 -0.16547 -0.13841 0.09956 0.19767 0.04164 -0.16297 -0.19685
17. 0 -0.14078 -0.16200 0.06455 0.19996 0.07660 -0.13688 -0.20543
17.2 -0.11098 -0.17924 +0.02761 0.19529 0.10864 -0.10686 -0.20805
17.4 -0.07725 -0.18956 -0.00990 0.18387 0.13671 -0.07387 -0.20464
17.6 -0.04094 -0.19265 -0.04663 0.16616 0.15991 -0.03895 -0.19533
17.8 -0.00347 -0.18846 -0.08123 0.14282 0.17752 -0.00320 -0.18039
18.0 +0.03372 -0.17722 -0.11249 0.11472 0.18897 +0.03225 -0.16030
18.2 0.06920 -0.15942 -0.13928 0.08289 0.19393 0.06629 -0.13566
18.4 0.10163 -0.13580 -0.16067 0.04848 0.19229 0.09782 -0.10722
18.6 0.12977 -0.10731 -0.17593 +0.01272 0.18414 0.12587 -0.07586
18.8 0.15261 -0.07506 -0.18455 -0.02310 0.16980 0.14955 -0.04252
19.0 0.16930 -0.04031 -0.18628 -0.05773 0.14982 0.16812 -0.00824
19.2 0.17927 -0.00440 -0.18111 -0.08993 0.12490 0.18100 +0.02593
19.4 0.18221 +0.03131 -0.16930 -0.11857 0.09595 0.18782 0.05895
19.6 0.17805 0.06546 -0.15134 -0.14267 0.06399 0.18838 0.08979
19.8 0.16705 0.09678 -0.12794 -0.16139 +0.03013 0.18270 0.11750
0.14967 0.12409 -0.10004 -0.17411 -0.00443 0.17101 0.14124
[ 1 [ 1 [ 1 [ 1 [ 1 [ 1 II‘-$18 c-p
20.0
(-j+)l (-;I1 C-$)1 (-;)9 (-;I8 1
BESSEL FUNCTIONS OF INTEGER ORDER
z 10'ki-'oJlo(cz)
0. 0 2.69114 446 1.22324 748 -0.11828 049 3.91990 9.33311 -0.406017
0.1 2.69053 290 1.22299 266 -0.11831 335 3.91944 9.33205 -0.406071
2.68869 898 1.22222 850 -0.11841 200 3.91804 9.32886 -0.406231
Fl*: 2.68564 500 1.22095 588 -0.11857 661 3.91571 -0.406499
0: 4 2.68137 477 1.21917 626 -0.11880 750 3.91244 99.
. '3:::: -0.406873
0. 5 2.67589 362 1.21689 169 -0.11910 510 3.90825 9.30663 -0.407355
2.66920 838 1.21410 481 -0.11946 998 3.90314 9.29500 -0.407945
E 2.66132
2.65226
738
043
1.21081
1.20703
883
750
-0.11990
-0.12040
282
444
3.89710
3.89015
9.28128
9.26546
-0.408644
-0.409452
::9” 2.64201 878 1.20276 518 -0.12097 581 3.88228 9.24758 -0.410369
2.63061 512 1.19800 675 -0.12161 801 3.87350 9.22162 -0.411397
::1” 2.60437
2.61806 358
963
1.19276
1.18705
764
385
-0.12233
-0.12312
229
002
3.86383
3.85325
9.20562
9.18157
-0.412536
-0.413788
:-:
1:4
2.58958
2.51368
012
323
1.18087
1.17422
185
867
-0.12398
-0.12492
273
212
3.84179
3.82945
9.15550
9.12743
co.415153
-0.416632
1.5 2.55670 842 1.16713 182 -0.12594 004 3.81624 9.09737 -0.418228
1.6 2.53867 639 1.15958 931 -0.12703 852 3.80216 9.06534 -0.419940
2.51960 907 1.15160 961 -0.12821 977 3.78723 9.03137 -0.421771
:-ii 2.49952 955 1.14320 168 -0.12948 616 3.77146 8.99546 -0.423122
1:9 2.47846 207 1.13437 488 -0.13084 030 3.75485 8.95766 -0.425795
2.45643 192 1.12513 904 -0.13228 497 3.73742 8.91797 -0.427992
2: 2.43346
2.40959
545
000
1.11550
1.10548
438
152
-0.13382
-0.13545
319
821
3.71918
3.70015
8.87643
8.83306
-0.430315
-0.432764
$3
2:4
2.38483
2.35922
384
612
1.09508
1.08431
144
551
-0.13719
-0.13903
351
284
3.68032
3.65973
8.78790
8.74096
-0.435344
-0.438056
2.5 2.33279 682 1.07319 540 -0.14098 022 3.63831 8.69228 -0.440902
2.30557 613 1.06173 312 -0.14303 997 3.61627 8.64189 -0.443885
Z 2.21759 732
074
1.04994
1.03783
098
155
-0.14521
-0.14751
672
543
3.59344
3.56989
8.58981
8.53609
-0.447007
-0.450272
::t 2.24889
2.21948 976 1.02541 767 -0.14994 141 3.54564 8.48076 -0.453682
3. 0 2.18942 770 1.01271 242 -0.15250 037 3.52071 8.42385 -0.457241
3.1 2.15873 836 0.99972 906 -0.15519 840 3.49510 8.36539 -0.460951
2.12745 598 0.98648 108 -0.15804 206 3.46885 8.30542 -0.464816
:-: 2.09561 517 0.97298 213 -0.16103 836 3.44195 8.24397 -0.468840
3: 4 2.06325 085 0.95924 599 -0.16419 482 3.41444 8.18110 -0.473027
2.03039 820 0.94528 659 -0.16751 951 3.38633 8.11682 -0.477379
::2 1.99709
1.96336
260
956
0.93111
0.91675
794
415
-0.17102
-0.17470
110
889 '3*3':;;:
8.05119
7.98424
-0.481902
-0.486600
;:I. 1.92926
1.89481
467
352
0.90220
0.88749
939
785
-0.17859
-0.18268
286
376
3129855
3.26821
7.91600
7.84653
-0.491476
-0.496537
1.86005 168 0.87263 375 -0.18699 314 3.23736 1.77586 -0.501786
1.82501 462 0.85763 130 -0.19153 346 3.20601 7.70403 -0.507229
1.78973 765 0.84250 469 -0.19631 812 3.17419 7.63108 -0.512872
1.75425 588 0.82726 806 -0.20136 159 3.14192 7.55707 -0.518719
1.71860 416 0.81193 548 -0.20667 950 3.10921 1.48202 -0.524777
1.68281 701 0.79652 093 -0.21228 873 3.07608 7.40598 -0.531051
1.64692 860 0.78103 829 -0.21820 757 3.04256 7.32900 -0.537549
1.61097 267 0.76550 130 -0.22445 582 3.00866 7.25112 -0.544276
1.57498 249 0.74992 351 -0.23105 498 7.17238 -0.551240
1.53899 084 0.73431 852 -0.23802 840 5.. Eltl" 7.09282 -0.558448
1.50302 991 0.71869 942 -0.24540 147 2.90490 7.01250 -0.565907
[ 1 C-55)6
[ 1
(-55)5
[ 1 c 1
(93 (6;)3
Y,+1(@ Yn(z)-Yn-l(X)
E
Compiledfrom British Associationfor the Advancement of Science,Besselfunctions, Part II. Functions
of positive integer order, Mathematical Tables, vol. X (CambridgeUniv. Press,Cambridge, England,
1952),L. Fox, A short table for Besselfunctions of integer ordersand large arguments. Royal Society
Shorter Mathematical Tables No. 3 (CambridgeUniv. Press,Cambridge,England, 1954), and Mathe-
matical Tables Project, Table of fm(z)=n!(%-c-V,(a). J. Math. Phys. 23, 45-60 (1944) (with per-
mission).
BESSEL FUNCTIONS OF INTEGER ORDER 403
BESSEL FUNCTIONS-ORDERS 10, 11, 20 AND 21 Table 9.3
JIOG) 1027~-215~~(~)
ls”0 Jll(Z) YlO(X) wh-“oJzo(~) lo-?iGoY~~(z)
-0.09007 181 0.09995 048 0.21997 141 0.22134 33 0.61224 04 - 11.0024
15:1 -0.10575 330 0.08344 886 0.21134 904 0.21230 71 0.58873 25 - 11.51807
15.2 -0.12073 964 0.06666 618 0.20160 159 0.20356 16 0.56593 06 - 12.1974
15.3 -0.13494 535 0.04967 738 0.19077 902 0.19510 08 0.54382 12 - 12.8555
15.4 -0.14828 828 0.03256 035 0.17893 834 0.18691 87 0.52239 14 - 13.5585
15.5 -0.16069 032 +0.01539 539 0.16614 338 0.17900 91 0.50162 76 - 14.3098
15.6 -0.17207 791 -0.00173 513 0.15246 453 0.17136 62 0.48151 66 - 15.1136
15.7 -0.18238 269 -0.01874 731 0.13797 838 0.16398 38 0.46204 52 - 15.9742
15.8 -0.19154 204 -0.03555 621 0.12276 733 0.15685 60 0.44319 99 - 16.8962
15.9 -0.19949 958 -0.05207 632 0.10691 918 0.14997 67 0.42496 74 - 17.8849
16.0 -0.20620 569 -0.06822 215 0.09052 660 0.14334 00 0.40733 43 - 18.9460
16.1 -0.21161 797 -0.08390 874 0.07368 666 0.13694 00 0.39028 75 - 20.0855
16.2 -0.21570 160 -0.09905 224 0.05650 016 0.13077 08 0.37381 35 - 21.3104
16.3 -0.21842 977 -0.11357 046 0.03907 110 0.12482 65 0.35789 93 - 22.6279
16.4 -0.21978 394 -0.12738 344 0.02150 600 0.11910 14 0.34253 16 - 24.0462
16.5 -0.21975 411 -0.14041 403 +0.00391 319 0.11358 96 0.32769 75 - 25.5740
16. 6 -0.21833 905 -0.15258 841 -0.01359 786 0.10828 55 0.31338 39 - 27.2209
16.7 -0.21554 637 -0.16383 668 -0.03091 729 0.10318 34 0.29957 78 - 28.9975
16.8 -0.21139 267 -0.17409 338 -0.04793 557 0.09827 77 0.28626 66 - 30.9150
16.9 -0.20590 350 -0.18329 797 -0.06454 431 0.09356 30 0.27343 76 - 32.9859
17.0 -0.19911 332 -0.19139 539 -0.08063 696 0.08903 37 0.26107 81 - 35.2237
17.1 -0.19106 538 -0.19833 646 -0.09610 960 0.08468 45 0.24917 57 - 37.6429
17.2 -0.18181 155 -0.20407 831 -0.11086 170 0.08051 02 0.23771 82 - 40.25'94
17.3 -0.17141 203 -0.20858 485 -0.12479 683 0.07650 53 0.22669 32 - 43.0904
17.4 -0.15993 505 -0.21182 701 -0.13782 343 0.07266 49 0.21608 89 - 46.1543
17.5 -0.14745 649 -0.21378 318 -0.14985 544 0.06898 37 0.20589 33 - 49.4711
17.6 -0.13405 943 -0.21443 935 -0.16081 304 0.06545 69 0.19609 48 - 53.06822
17.7 -0.11983 363 -0.21378 944 -0.17062 321 0.06207 96 0.18668 17 - 56.9506
17.8 -0.10487 499 -0.21183 538 -0.17922 038 0.05884 68 0.17764 27 - 61.1611
17.9 -0.08928 492 -0.20858 727 -0.18654 691 0.05575 39 0.16896 66 - 65.7197
18.0 -0.07316 966 -0.20406 341 -0.19255 365 0.05279 63 0.16064 24 - 70.6543
18.1 -0.05663 961 -0.19829 032 -0.19720 030 0.04996 93 0.15265 91 - 75.9946
18.2 -0.03980 852 -0.19130 265 -0.20045 582 0.04726 85 0.14500 62 - 81.7717
18.3 -0.02279 278 -0.18314 307 -0.20229 875 0.04468 96 0.13767 32 - 88.0182
18.4 -0.00571 052 -0.17386 213 -0.20271 742 0.04222 83 0.13064 97 - 94.7683
18.5 +0.01131 917 -0.16351 793 -0.20171 011 0.03988 04 0.12392 57 -102.0574
18.6 0.02817 711 -Oil5217 591 -0.19928 520 0.03764 17 0.11749 14 -109.9219
18.7 0.04474 490 -0.13990 845 -0.19546 113 0.03550 84 0.11133 69 -118.3992
18. 8 0.06090 579 -0.12679 446 -0.19026 637 0.03347 64 0.10545 28 -127.5270
18.9 0.07654 556 -0.11291 893 -0.18373 930 0.03154 21 0.09982 98 -137.3432
19.0 0.09155 333 -0.09837 240 -0.17592 797 0.02970 16 0.09445 89 -147.8850
19.1 0.10582 247 -0.08325 039 -0.16688 985 0.02795 15 0.08933 lo -159.1885
19.2 0.11925 134 -0.06765 283 -0.15669 143 0.02628 80 0.08443 76 -171.2882
19.3 0.13174 416 -0.05168 334 -0.14540 785 0.02470 79 0.07977 01 -184.2155
19.4 0.14321 168 -0.03544 863 -0.13312 231 0.02320 78 0.07532 03 -197.9980
19.5 0.15357 193 -0.01905 771 -0.11992 560 0.02178 44 0.07108 01 -212.6582
19.6 0.16275 089 -0.00262 120 -0.10591 538 0.02043 46 0.06704 16 -228.2122
19.7 0.17068 305 +0.01374 948 -0.09119 555 0.01915 54 0.06319 71 -244.6678
19.8 0.17731 198 0.02994 285 -0.07587 548 0.01794 37 0.05953 92 -262.0226
19.9 0.18259 079 0.04584 818 -0.06006 922 0.01679 67 0.05606 06 -280.2622
20.0 0.18648 256 0.06135 630 -0.04389 465 0.01571 16 0.05275 42 -299.3574
[ 542
1 [ C-212
1 [ 1
(542
[
C-j)4
1 [ 1 [c-p1
(--j’9
Y
406 BESSEL FUNCTIONS OF INTEGER ORDER
Table 9.3
BESSEL FUNCTIONS-MODULUS AND PHASE OF ORDERS 10, 11, 20 AND 21
Y&j =Nn(z) sin &(.r)
!C-l .r*Mlo(z) ho@+-r GM, 1(z) 811 (z)-.c <s>
0.050 0.85676 701 -13.94798 864 0.87222 790 -14.96758 686 20
0.048 0.85136 682 -14.05389 581 0.86513 271 -15.09771 672 21
0.046 0.84633 336 -14.15926 984 0.85857 314 -15.22701 466 22
0.044 0.84164 245 -14.26413 968 0.85250 587 -15.35552 901 23
0.042 0.83727 251 -14.36853 333 0.84689 281 -15.48330 635 24
0.040 0.83320 419 -14.47247 807 0.84170 044 -15.61039 144 25
0.038 0.82942 012 -14.57600 035 0.83689 917 -15.73682 771 26
0.036 0.82590 472 -14.67912 589 0.83246 283 -15.86265 679 28
0.034 0.82264 403 -14.78187 967 0.82836 826 -15.98791 896 29
0.032 0.81962 546 -14.88428 611 0.82459 496 -16.11265 291 31
0.030 0.81683 775 -14.98636 880 0.82112 469 -16.23689 620 33
0.028 0.81427 076 -15.08815 085 0.81794 133 -16.36068 504 36
0.026 0.81191 546 -15.18965 477 0.81503 056 -16.48405 469 38
0.024 0.80976 370 -15.29090 253 0.81237 970 -16.60703 912 42
0.022 0.80780 825 -15.39191 569 0.80997 751 -16.72967 149 45
0.020 0.80604 267 -15.49271 527 0.80781 410 -16.85198 406 50
0.018 0.80446 127 -15.59332 192 0.80588 079 -16.97400 835 56
0.016 0.80305 902 -15.69375 598 0.80416 997 -17.09577 505 63
0.014 0.80183 156 -15.79403 741 0.80267 505 -17.21731 438 71
0.012 0.80077 512 -15.89418 589 0.80139 036 -17.33865 590 83
0.010 0.79988 647 -15.99422 093 0.80031 114 -17.45982 880 100
0.008 0.79916 297 -16.09416 168 0.79943 341 -17.58086 166 125
0.006 0.79860 244 -16.19402 726 0.79875 398 -17.70178 301 167
0.004 0.79820 323 -16.29383 652 0.79827 039 -17.82262 084 250
0.002 0.79796 417 -16.39360 832 0.79798 093 -17.94340 316 500
0.000 0.79788 456 -16.49336 143 0.79788 456 -18.06415 776 00
.r- 1
0.050
.2X20(~)
1.474083
02”(Z) -.l’
-21.047407
:,?
iI (.I.)
1.791133
.921(x)--R.
-21.290925
<r>
20
0.048 1.320938 -21.606130 1.525581 -21.927545 21
0.046 1.211667 -22.149524 1.347435 -22.550082
0.044 1.131459 -22.676802 1.224460 -23.154248 22:
0.042 1.070845 -23.188535 1.136653 -23.738936 24
0.040 1.023762 -23.685951 1.071741 -24.304948
0.038 -24.170500 1.022171 -24.853951 ;56
0.036 t 9985%; -24.643620 0.983229 -25.387848
0.034 0:930635 -25.106640 0.951902 -25.908478 2
0.032 0.909513 -25.560748 0.926211 -26.417500 31
0.030 0.891605 -26.006988 0.904821 -26.916369 33
0.028 0.876293 -26.446280 0.886799 -27.406346
0.026 0.863121 -26.879433 0.871483 -27.088527 2
0.024 0.851743 -27.307159 0.858385 -28.363869 5~ 42
0.022 0.841895 -27.7300+8 0.847145 -28.833211 45
0.020 0.833375 -28.148822 0.837487 -29.297299 5"
0.018 0.826019 -28.563847 0.829198 -29.756800 51,
0.016 0.819702 -28.975650 0.822114 -30.212318
0.014 0.814321 -29.384666 0.816105 -30.664405 76:
0.012 0.809796 -29.791303 0.811069 -31.113569 a3
0.010 0.806062 -30.195941 0.806925 -31.560285 1""
0.008 0.803071 -30.598942 0.803612 -32.005000 iis
0.006 0.800781 -31.000652 0.801081 -32.448139 167
0.004 0.799165 -31.401404 0.799297 -32.890109 250
0.002 0.798204 -31.801522 0.798237 -33.331307 500
0.000 0.797885 -32.201325 0.797885 -33.772121 0~
[ 1
C-73)5
[<2>:nearest
1
C-f)2
Integer to 5.
[(-P1
Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments.
Royal Society Shorter Mathematical Tables No. 3(Cambridge Univ. Press, Cambridge, England,
1954) (with permission).
BESSEL FUNCTIONS OF INTEGER ORDER 407
BESSEL FUNCTIONS-VARIOUS ORDERS Table 9.4
- 3)2.47663 8964
I- 3)7.03962 9756
I
1-
(-
(-
l)-1.77596
2
1 -3.27579
+4.65651
3.64831 7713
2306
1628
1376
1 3.91232 3605
1) 2.61140 5461
1.19800 6746
16)6.88540 8200 I - 10
11
7)2.51538
9
8I 2.30428
1.07294 5149
1.49494
1.93269 6448
4758
6283
2010
II
- 25
60 2.90600
80
42 3.48286 4948
1.10791
3.87350 5851
9794
3009
II
-
-
-
-
19 3.91897
33 3.65025
48 1.19607
65 3.22409
2805
6266
7458
5839
II - 45
33 2.77033
21
11 2.67117 0052
2.29424
8.70224 7616
1617
7278
- 2)+7.08409 7728
I- 1
2I +2.16710
-2.34061
-1.44588
3.17854 5282
1268
9177
4208
(- 1) 2.91855 6853
II- 1
2 +1.04058
+6.04912
-8.71210 4770
-8.14002 0126
5632
2682
(- 2)-2.71924 6104
-2 -7.41957
jl
-2 -3.35253
-2 +7.01726
-2 +4.33495
-2)-6.32367
3696
8314
9099
5988
6141
(- 1) 2.07486 1066 (- l)-1.13847 8491
I -7 1
2I 2.89720
6.33702 5497
1.23116
1.19571 6324
8393
5280 - 2)-6.98335 2016
I - 43
5i 1.52442
5.05646 4853
4.50797
4.31462
1.56675 7752
3144
6697
6192
I -30
-215I 1.78451
--12 6.03089 6078
1.15133
1.55109 3608
5312
6925
I - 2
1I +4.84342
+1.21409
-1.16704 2812
-1.38176 5725
3528
0219
II
-2 +8.14601
-3.86983 5850
+7.27017
+6.22174 3973
5482
2958
(-89) 6.59731 6064 (-21)+1.11592 7368 ( -2) +9.63666 7330
408 BESSEL FUNCTIONS OF INTEGER ORDER
II
20 8 -5.93396 5297
18 -4.02856 8418
200 29 -9.21681 6571
50 62)-1.97615 0576 42 -2.78883 7017
100 (185)-3.77528 7810 (155)-3.00082 6049 (115)-5.08486 3915
Table 9.5
ZEROS AND ASSOCIATED VALUES OF BESSEL FUNCTIONS AND THEIR DERIVATIVES
s YO. s Y’o(?/o,J Yl, d Y’l (VI, ,“) 1/z, s Y’Z(Y2, s)
O.iSi57 697 +0.87942 080 2.19714 +0.52079 3.38424
: 3.95767 842 -0.40254 267 5.42968 -0.34032 6.79381 :: ;;;;:
7.08605 lob +0.30009 761 8.59601 +0.27146 10.02348 +0:24967
i 10.22234 504 -0.24970 124 11.74915 -0.23246 13.20999 -0.21835
5 13.36109 747 +0.21835 830 14.89744 +0.20655 lb.37897 +0.19646
lb. 50092 244 -0.19646 494 18.04340 -0.18773 19.53904 -0.18006
76 19.64130 970 +O. 18006 318 21.18807 to.17327 22.69396 +0.16718
22.78202 805 -0.16718 450 24.33194 -0.16170 25.84561 -0.15672
: 25.92295 765 +0.15672 493 27.47529 +0.15218 28.99508 +0.14801
10 29.06403 025 -0.14801 108 30.61829 -0.14417 32.14300 -0.14061
32.20520 412 +0.14060 578 33.76102 +0.13730 35.28979 +0.13421
:: 35.34645 231 -0.13421 123 36.90356 -0.13132 38.43573 -0.12862
38.48775 bb5 +0.12861 6bl 40.04594 +O. 12607 41.58101
:43 4l.b2910 447 -0.12366 795 43.18822 -0.12140 44.72578 '-; ::;;:
15 44.77048 661 +0.11924 981 46.33040 +0.11721 47.87012 +0:11527
41.91189 633 -0.11527 369 49.47251 -0.11343 51.01413 -0.11167
:; 51.05332 855 +0.11167 049 52.61455 +0.10999 54.15785
54.19477 936 -0.10838 535 55.75654 -0.10685 .57.30135 tfl- :g:;
:t 57.33624 570 +0.10537 405 58.89850 +0.10396 60.44464 +0:10260
20 60.47772 516 -0.10260 057 62.04041 -0.10129 63.58777 -0.10004
.,
s f:,, s .JI( j’:;, ,) j'4, \ .J,(j’,, <) J i, \ .fr,(j',, x)
1 4.20119 to.43439 5.31755 +0.39965 6.41562 +0.37409
2 8.01524 -0.29116 9.28240 -0.27438 10.51986 -0.26109
3 11.34592 +0.24074 12.68191 +0.22959 13.98719 +0.22039
4 14.58585 -0.21097 15.96411 -0.20276 17.31284 -0.19580
5 17.78875 co.19042 19.19603 to.18403 20.57551 +0.17849
s J
.I
6, N Jdi'B. 8) .1 i, * ,Ji(j’7. 4 .I s. * .JY(~‘K,s)
7.50127 +0.35414 8.57784 9.64742 to.32438
11.73494 -0.25017 12.93239 I;*:;;;; 14.11552 -0.23303
15.26818 +0.21261 16.52937 +0:20588 17.77401 +0.19998
18.63744 -0.18978 19.94185 -0.18449 21.22906 -0.17979
21.93172 +0.17363 23.26505 +0.16929 24.58720 +0.16539
6 25.18393 -0.16127 26.54503 -0.15762 27.88927 -0.15431
7 28.40978 +0.15137 29.79075 +0.14823 31.15533 to.14537
31.61788 -0.14317 33.01518 -0.14044 34.39663 -0.13792
i 34.81339 +0.13623 36.22438 +0.13381 37.62008 +0.13158
10 37.99964 -0.13024 39.42227 -0.12808 40.83018 -0.12608
11 41.17885 +0.12499 42.61152 +0.12305 44.03001 +0.12124
44.35258 -0.12035 45.79400 -0.11859 47.22176 -0.11695
:: 47.52196 +0..11620 48.97107 50.40702 +0.11309
14 50.68782 -0J1246 52.14375 '_"o-::g; 53.58700 -0.10960
15 53.85079 +0.10906 55.31282 +0:10771 56.76260 +0.10643
57.01138 -0.10596 58.47887 -0.10471 59.93454 -0.10352
:t 60.16995 +0..10311 61.64239 co.10195 63.10340 +0.10084
63.32681 -0.10049 64.80374 -0.09940 66.26961 -0.09837
:t 66.48221 +0.09805 67.96324 +0.09704 69.43356 +0.09607
20 69.63635 -0-09579 71.12113 -0.09484 72.59554 -0.09393
412 BESSEL FUNCTIONS OF INTEGER ORDER
Table 9.5
ZEROS AND ASSOCIATED VALUES OF BESSEL FUNCTIONS AND THEIR DERIVATIVES
[(-fjP
0.00000
[ C-i)1
1 1
0.00000
[C-y]
.
['$'"I
-
[ (-;I5
1
_ _
From E. T. Goodwin and J. Staton, Table of .Io(,&,r), Quart. J. Mech. Appl.
Math. 1, 220-224 (1948) (with pernksion).
414 BESSEL FUNCTIONS OF INTEGER ORDER
A- ‘\s 1 2
1. 00 1.2558 4.0795 7.15358 lo.??710 13.33984 <A>
1
0. 80 1.3659 4.1361 7.1898 10.2950 13.4169
0. 60 1.5095 4.2249 7.2453 10.3346 13.4476 21
0. 40 1.7060 4.3818
0.20 1.9898 4.7131 7.3508
7.6177 10.4118
10.6223 13.6786
13.5079 ;
0.10 2.1795 5.0332 7.9569 10.9363 13.9580
0.08 2.2218 5.1172 8.0624 11.0477 14.0666 :;
0.06 2.2656 5.2085 8.1852 11.1864 14.2100
0. 04 2.3108 5.3068 8.3262 11.3575 14.3996 :;
0. 02 2.3572 5.4112 8.4840 11.5621 14.6433 50
0. 00 2.4048 5.5201 8.6537 11.7915 14.9309 01
Xk’\S 1 2 3 4 <A>
1. 00 1.8412 5.3314 8.5363 11.7060 14.8:36 1
0. 80 1.9844 5.3702 8.5600 11.7232 14.8771
0.60 2.1092 5.4085 8.5836 11.7404 14.8906 21
0.40 2.2192 5.4463 8.6072 11.7575 14I9041 I
0.20 2.3171 5.4835 8.6305 11.7745 14.9175 5
0.10 2.3621 5.5019 8.6421 11.7830 14.9242 10
0.08 2.3709 5.5055 8.6445 11.7847 14.9256 13
0. 06 2.3795 5.5092 8.6468 11.7864 14.9269 17
0. 04 2.3880 5.5128 8.6491 11.7881 14.9282 25
0.02 2.3965 5.5165 8.6514 11.7898 14.9296 50
0. 00 2.4048 5.5201 8.6537 11.7915 14.9309 co
<h> =nearestintegerto A.
A-‘\S 1 2 4 5
* 0. 80 12.59004 151 25.14465 37.7:706 50.27145 62.83662
0.60 4.75805 426 9.44837 14.15300 18.86146 23.57148
0.40 2.15647 249 4.22309 6.30658 8.39528 10.48619
0.20 0.84714 961 1.61108 2.38532 3.16421 3.94541
0.10 0.39409 416 0.73306 1.07483 1.41886 1.76433
0. 08 0.31223 576 0.57816 0.84552 1.11441 1.38440 *
0. 06 0.23235 256 0.42843 0.62483 0.82207 1.02001
0.04 0.15400 729 0.28296 0.41157 0.54044 0.66961
0.02 0.07672 788 0.14062 0.20409 0.26752 0.33097
0. 00 0.00000 000 0.00000 0.00000 0.00000 0.00000 00
A-‘\s 1 2 3 4 5 <A>
* 0. 80 6.56973 310 18.94971 31.47626 44.02544 56.58224 1
* 0. 60 2.60328 1% 7.16213 11.83783 16.53413 21.23751
0.40 li24266 626 3.22655 5.28885 7.36856 9.45462 3
0.20 0.51472 663 1.24657 2.00959 2.78326 3.56157 5
0.10 0.24481 004 0.57258 0.90956 1.25099 1.59489
0.08 0.19461 772 0.45251 0.71635 0.98327 1125203 :30 *
0. 06 0.14523 798 0.33597 0.53005 0.72594 0.92301
0.04 0.09647 602 0.22226 0.34957 0.47768 0.60634 :sT
0.02 0.04813 209 0.11059 0.17353 t* 203088: 0.29991 50
0. 00 0.00000 000 0.00000 0.00000 0.00000 00
<X> =nearest integer to A. ’
Compiled from British Association for the Advancement of Science, Bessel func-
tions, Part I. Functions of orders zero and unity, Mathematical Tables, vol.VI
(Cambridge Univ. Press, Cambridge, England, 1950) (with permission).
*see page II.
416 BESSEL FUNCTIONS OF INTEGER ORDER
[.
c-y
I [ 1 (6$3
[I 1c-y
[
c-t)3
1 [ C-t)6
1
420 BESSEL FUNCTIONS OF INTEGER ORDER
[ 1
c-p5
[
C-l'4
1
t-y
[ 1
BESSEL FUNCTIONS OF INTEGER ORDER 421
MODIFIED BESSEL FUNCTIONS-ORDERS 0, 1 AND 2 Table 9.8
[(-p1
20.0 0.27854 48766 0.28542 54970 0.30708 743
I:(-y 1 c 1
C-$)3
422 BESSEL FUNCTIONS OF INTEGER ORDER
2-l xfe-zZo(x) xf PZ1 (2) xfd2(x) T--lde~Ko(x) a-lx*e~Kl (x) R- ~xbKiz(x) <x>
0.050 0.40150 9761 0.39133 9722 0.36237 579 0.39651 5620 0.40631 0355 0.43714 666
0.048 0.40140 4058 0.39164 8743 0.36380 578 0.39661 0241 0.40601 9771 0.43558 814 :1”
0.046 0.40129 8619 0.39195 7336 0.36523 854 0.39670 5057 0.40572 8854 0.43403 211
0.044 0.40119 3443 0.39226 5502 0.36667 408 0.39680 0069 0.40543 7604 0.43247 858 ::
0.042 0.40108 8526 0.39257 3245 0.36811 237 0.39689 5278 0.40514 6017 0.43092 754 24
0.040 0.40098 3868 0.39288 0567 0.36955 342 0.39699 0686 0.40485 4094 0.42937 901 25
0.038 0.40087 9466 0.39318 7470 0.37099 722 0.39708 6293 0.40456 1832 0.42783 299
0.036 0.40077 5319 0.39349 3958 0.37244 375 0.39718 2101 0.40426 9230 0.42628 949 g
0.034 0.40067 1424 0.39380 0032 0.37389 302 0.39727 8110 0.40397 6286 0.42474 850
0.032 0.40056 7781 0.39410 5695 0.37534 502 0.39737 4322 0.40368 2998 0.42321 003 31
0.030 0.40046 4387 0.39441 0950 0.37679 973 0.39747 0738 0.40338 9365 0.42167 410 33
0.028 0.40036 1241 0.39471 5798 0.37825 716 0.39756 7359 0.40309 5386 0.42014 070
0.026 0.40025 8340 0.39502 0243 0.37971 729 0.39766 4186 0.40280 1058 0.41860 984 a;
0.024 0.40015 5684 0.39532 4286 0.38118 012 0.39776 1221 0.40250 6380 0.41708 153
0.022 0.40005 3270 0.39562 7929 0.38264 564 0.39785 8465 0.40221 1349 0.41555 576 45
0.020 0.39995 1098 0.39593 1176 0.38411 385 0.39795 5918 0.40191 5965 0.41403 256
0.018 0.39984 9164 0.39623 4028 0.38558 474 0.39805 3583 0.40162 0226 0.41251 191
0.016 0.39974 7469 0.39653 6487 0.38705 830 0.39815 1460 0.40132 4130 0.41099 383
0.014 0.39964 6009 0.39683 8556 0.38853 453 0.39824 9551 0.40102 7674 0.40947 833
0.012 0.39954 4785 0.39714 0236 0.39001 342 0.39834 7857 0.40073 0858 0.40796 540
0.010 0.39944 3793 0.39744 1530 0.39149 496 0.39844 6379 0.40043 3679 0.40645 505 1’00
0.008 0.39934 3033 0.39774 2440 0.39297 91.5 0.39854 5119 0.40013 6136 0.40494 730 125
0.006 0.39924 2503 0.39804 2968 0.39446 599 0.39864 4077 0.39983 8226 0.40344 214 167
0.004 0.39914 2202 0.39834 3116 0.39595 546 0.39874 3256 0.39953 9949 0.40193 958
0.002 0.39904 2128 0.39864 2886 0.39744 756 0.39884 2657 0.39924 1300 0.40043 962 ~~00
0.000 0.39894 2280 0.39894 2280 0.39894 228 0.39894 2280 0.39894 2280 0.39894 228 00
[ 1
q3)3 ['-;I51 [c-;'"] [q"] ['-;'"I ['-;I"]
For interpolating near x--1 =0 note that if f&-l) =zk~Z,(z) thenf,( -2-l) =r-de~K~(z),
Compiled from L. Fox, A short table for Bessel functions of integer orders and large arguments. Royal
Society Shorter Mathematical Tables No. 3 (Cambridge Univ. F’ress, Cambridge, England, 1954) (with
permission).
MODIFIED BESSEL FUNCTIONS-AUXILIARY TABLE FOR SMALL ARGUMENTS
::1” 0.11872
0.11593 152
387 0.99691
1.00000 180
000 1:1
lx0 0.49199
0.42102 896
444 0.49390
0.60190 723
093
z-3’ 0.12713
0.14124 128
511 0.98754
0.97158 819
448 :3 0.66373
0.57261 444
364 0.21236
0.36514 944
381
0: 4 0.16121 862 0.94852 090 1: 4 0.76632 938 +0.03176 677
1.0 0.42102 444 0.60190 723 2.0 1.69398 200 -1.92535 914
(- 5)4.0512
(- 5)5.7482
(- 5)7.9208 (- 5)1.4507
;; (-2)5.6454 (-3)7.2431 (- 4 1.0638 (- 5)2.0556
. (-2)5.8893 (-3)8.2288 (-3)2.4106 (- 4)6.1640 (- 41 1.3965 (- 5)2.8380
4.0 -3)9.2443 (-3)2.8291 (- 4)1.7968
4.2 -2 1 1.0283 (-3)3.2785 (- 4)2.2703
-2 1.1337 (- 4)2.8224
(-2)3.1221 -2 1 1.2402 (- 4)3.4578 (- 5)8.3667
(-2)3.2854 -2 1.3471 (- 4)4.1806 (- 4)1.0508
-2'1.4540
-2 i 1.5605
-2)1.6662
-2)1.7707
-2)1.8738
-2)1.9752
II
(- 3)3.1156
-2 1 2.0747
-2 2.1723 - 3 1.0484
1.3351
1.1870
(- 3)1.4924
(-2)4.5567 (-2)1.0849 (- 3)1.6587
22”
8.4 (-2 I 1.5854
-
-
-
3)6.7449
3 7.1440
3 I 7.5464
(- 3)2.8292 (- 331.0866
Ii 45 1.3160
1.2610
2.0785
3.6055
6.6436
0 1.6317
2: 0 1.5303
0 1.4414
1-z 0 1.3629
4:8 II 0 1.2931
0)2.2646 1)1.1973
::2"
:*: II 0 2.1186
1.8746
1.9895
1.7720 II 01 1.0645
7.7717
9.5285
8.5813
5:8
6.0
2:
II 0 1.3902
1.4528
1.5213
1.5967
1.6798 II 1i 1.5547
1.1467
1.3978
1.2630
1.7387
28"
1osx-‘ozlo(~) lO".r"Zll(X)
0.26911 445 1.22324 748 1.85794 idd 0.391990 0.933311 6.37771
0.26935 920 1.22426 724 1.85588 251 0.392177 0.933736 6.37435
0.27009 468 1.22733 125 1.84970 867 0.392738 0.935008 6.36429
0.27132 457 1.23245 366 1.83947 021 0.393674 0.937136 6.34757
0.27305 504 1.23965 820 1.82524 326 0.394988 0.940123 6.32424
0.27529 480 1.24897 831 1.80713 290 0.396684 0.943974 6.29437
0.27805 517 1.26045 740 1.78527 169 0.398766 0.948703 6.25807
0.28135 012 1.27414 918 1.75981 781 0.401239 0.954321 6.21545
0.28519 648 1.29011 798 1.73095 297 0.404112 0.960843 6.16665
0.28961 396 1.30843 932 1.69887 992 0.407392 0.968285 6.11184
0.29462 538 1.32920 036 1.66381 982 0.411087 0.976669 6.05118
0.30025 682 1.35250 061 1.62600 944 0.415209 0.986016 5.98488
0.30653 784 1.37845 262 1.58569 822 0.419768 0.996351 5.91314
0.31350 170 1.40718 285 1.54314 529 0.424778 1.007703 5.83620
0.32118 565 1.43883 260 1.49861 645 0.430253 1.020101 5.75428
0.32963 121 1.47355 907 1.45238 126 0.436209 1.033581 5.66764
0.33888 455 1.51153 657 1.40471 020 0.442662 1.048178 5.57655
0.34899 681 1.55295 782 1.35587 192 0;449632 1.063935 5.48128
0.36002 459 1.59803 551 1.30613 075 0.457139 1.080893 5.38210
0.37203 039 1.64700 388 1.25574 432 0.465205 1.099102 5.27932
0.38508 316 1.70012 064 1.20496 150 0.473853 1.118613 5.17321
0.39925 889 1.75766 896 1.15402 052 0.483111 1.139481 5.06408
0.41464 125 1.81995 978 1.10314 736 0.493006 1.161768 4.95224
0.43132 237 1.88733 435 1.05255 442 0.503569 1.185538 4.83197
0.44940 362 1.96016 700 1.00243 944 0.514832 1.210861 4.72159
0.46899 655 2.03886 82 0.95298 465 0.526830 1.237813 4.60339
0.49022 387 2.12388 83 0.90435 626 0.539601 1.266475 4.48367
0.51322 061 2.21572 08 0.85670 405 0.553186 1.296933 4.36272
0.53813 536 2.31490 71 0.81016 129 0.567630 1.329281 4.24084
0.56513 169 2.42204 09 0.76484 483 0.582979 1.363622 4.11830
0.59438 965 2.53777 36 0.72085 532 0.599284 1.400061 3.99537
0.62610 759 2.66282 00 0.61827 767 0.616599 1.438715 3.87234
0.66050 400 2.79796 48 0.63718 161 0.634984 1.479709 3.74945
0.69781 972 2.94406 93 0.59762 235 0.654501 1.523176 3.62695
0.73832 033 3.10208 00 0.55964 137 0.675219 1.569259 3.50507
0.78229 881 3.27303 69 0.52326 729 0.697210 1.618113 3.38405
0.83007 854 3.45808 34 0.48851 672 0.720554 1.669904 3.26411
0.88201 663 3.65847 74 0.45539 529 0.745333 1.724808 3.14543
0.93850 764 3.87560 29 0.42389 854 0.771639 1.783016 3.02821
0.99998 773 4.11098 38 0.39401 295 0.799570 1.844734 2.91264
1.06693 936 4.36629 90 0.36571 690 0.829231 1.910180 2.79887
1.13989 641 4.64339 88 0.33898 159 0.860735 1.979593 2.68705
1.21945 007 4.94432 35 0.31377 202 0.894204 2.053225 2.57733
1.30625 534 5.27132 42 0.29004 783 0.929769 2.131351 2.46983
1.40103 829 5.62688 64 0.26776 418 0.967571 2.214264 2.36466
1.50460 429 6.01375 48 0.24687 251 1.007764 2.302281 2.26193
1.61784 713 6.43496 31 0.22732 134 1.050510 2.395741 2.16172
1.74175 933 6.89386 57 0.20905 690 1.095988 2.495011 2.06411
1.87744 369 7.39417 36 0.19202 382 1.144389 2.600488 1.96916
2.02612 620 7.93999 51 0.17616 568 1.195919 2.712593 1.87692
[c-y1
2.18917 062 8.53588 02 0.16142 553 1.250800 2.831786 1.78744
[ 1
(-;)2
[ 1
C-l)6
[c-p1 [I 1
(-;)4
2n
Kn+1(2)=~Kn.(z)+Kn-l(2)
Compiled from British Associationfor the Advancement of Science,Besselfunctions, Part II. Func-
tions of positive integer order, Mathematical Tables, vol. X (CambridgeUniv. Press,Cambridge,Eng-
land, 1952)and L. Fox, A short table for Besselfunctions of integer ordersand largearguments. Royal
Society Shorter Mathematical Tables No. 3 (CambridgeUniv. Press,Cambridge,England, 1954) (with
permission).
426 BESSEL FUNCTIONS OF INTEGER ORDER
eZKlo(z) 1024r20120(2)
lo”0
10:2
e-zllo(z)
0.00099
0.00107
38819
29935
e-21l1(2)
0.00038
0.00042
75284
45861
35.55633
32.60759
91
68
1.25080
1.30927
2.83179
2.95856
1.787443
1.700753
10.4 0.00115 52835 0.00046 37417 29.98423 91 1.37160 3.09345 1.616873
10.6 0.00124 06973 0.00050 50080 27.64297 29 1.43806 3.23703 1.535814
10.8 0.00132 91744 0.00054 83934 25.54714 23 1.50895 3.38992 1.457578
Z?%(l) Zn(5)
II
I 011.26606 5878 I 112.72398 7182
- 1I Sib5159 1040
- 1 1.35747 6698
- 2 2.21684 2492 10 1.75056
5.10823 4214
1.03311
2.43356 4764
1497
5017
II
- 3 2.73712 0221
II
f oj2.15797 4547
- 5346 2.76993
1.60017 9323
3.04418
2.24639
9.82567 1420
3364
5903
6951
- 12 2.56488
7.41166 6690
7192285
1.93157 3216
1882
9417
f- lOj2.75294 8040
II - 11
14 1.24897
13
16 1.99563
7.11879 8308
5.19576 1678
1153
0054
20
ioo
50
-
-
-
-
25 3.96683
42 3.53950
60 1.12150
80i 2.93463
5986
0588
9741
5309
- .9 4.31056 0576
II
- 33 3.89351 9664
- 48 1.25586 9192
- 65 3.35304 2830
II - 11
21 5.02423
45
32 2.93146 9358
1.18042
3.99784 6980
9647
4971
II 3 2.67098
1.22649 6628
2.28151
2.81571
1.75838 0538
8968
0717
8304 ( 2012.49509 894
10 f4116.49897 552
::
:34 II
41 4.01657
4.59832 209
ii84924
5.21214 700
794
227
11:
17
I 191912.25869
3.07376 455
581
18
19
20 4 1.25079 9736 18 5.44206 840 41 1.44834 613
30
40
50
100
112 7.78756
-20 2.04212
-30 I 4.75689
(-88)1.08234
9783
3274
4561
4202
16
13
+lO I
4.27499
6.00717
1.76508
(-16)2.72788
365
897
024
795
II
40
38
36
1.20615
3.84170
4.82195
(21)4.64153
487
550
809
494
BESSEL FUNCTIONS OF INTEGER ORDER 429
MODIFIED BESSEL FUNCTIONS-VARIOUS ORDERS Table 9.11
II-10 6.47385
2.19591 8728
2.53759
1.39865
1.13893 3909
8818
5927
7546
I 1.00050 8312
52347I 6.22552
4.42070
3.65383
3.60960 2033
4099
5896
1230
II 2130 5.21498
9.75856 2921
9.65850
1.92563
4.15465 3277
2913
2829
4995
20 II 2.97998 5755
6853
9886
1740
II
22 6.29436 9360 45
30
16
62 4.27112
5.77085
9.94083
39 4.70614 5527
Ip 58 1.11422 0651
77 3.40689 6854
100 (185)5.90033 3184 (155)4.61941 5978 (115)7.03986 0193
2
87
9
10
::
13
14
15 -1
-1
2.65656
8.81629
3849
2510 II
-21 4.21679
-22 1.16980
8.17096
5.81495
3.11621 235
398
523
828
117
I II
-44 7.65542
-45 1.23283 963
1.07829
9.52475
8.49696
I -44 1.42348
2.32445
1.95464
1.65987
148
044
206
797
2.79144 325
645
763
531
371
1176 0 3.08686 9988
1 1.13769 8721
ii II 1 4.40440 2395
20 -21 1.70614 838 -44 3.38520 541
El
50
100 (85)4.59667 4084
II
-19
-16
-13
2.00581
1.29986
4.00601
(+13)1.63940
681
971
347
352
II
-43
-41
-40
3.97060
1.20842
9.27452
(-25)7.61712
205
080
265
963
430 BESSEL FUNCTIONS OF INTEGER ORDER
KELVIN FUNCTIONS-AUXILIARY
[(-;I6 1 TABLE
[(-;I6 1
FOR SMALL
[(92 1
ARGUMENTS
2: ker x+ber .z’In .g kei .I,+bei .I’In .I’ ,r(kerl .c+berl .CIn .r) .@eil .r+beil .XIn .G)
0. 0 0.11593 1516 -0.78539 8163 -0.70710 6781 -0.70710 6781
0.1 0.11789 2485 -0.78260 7108 -0.70651 7131 -0.70215 4903
0.12374 5076 -0.77421 9267 -0.70486 2164 -0.68733 0339
t: 0.13339 8210 -0.76019 0919 -0.70248 3157 -0.66272 8003
0:4 0.14669 9682 -0.74045 0212 -0.69994 6658 -0.62851 1738
0. 5 0.16343 5574 -0.71489
[ 1
I (78
8693
[ C-$)1 -0.69804
1
I [C-f)1
1049
I
-0.58492 2770
Compiled from National Bureau of Standards, Tables of the Bessel functions JO(Z) and J,(z) for complex
arguments, 2d ed. (Columbia Univ. Press, New York, N.Y., 1947) and National Bureau of Standards,
Tables of the Bessel functions I’s(z) and YI(z) for complex arguments (Columbia Univ. Press, New York,
N.Y., 1950) (with permission).
BESSEL FUNCTIONS OF INTEGER ORDER 431
KELVIN FUNCTIONS-ORDERS 0 AND 1 Table 9.12
kern: kei z ker, 2 kei, .X
-0.78539 8163
2.4200047 3980 -0.77685 0646 -7.14628 1711 16.94074 2153
1.73314 2752 -0.75812 4933 -3.63868 3342 -3.32341 7218
1.333721 8637 -0.73310 1912 -2.47074 2357 -2.08283 4751
1.06262 3902 -0.70380 0212 -1.88202 4050 -1.44430 5150
0.85590 5872 -0.67158 1695 -1.52240 3406 -1.05118 2085
0.69312 0695 -0.63744 9494 -1.27611 7712 -0.78373 8860
0.56137 8274 -0.60217 5451 -1.09407 2943 -0.59017 5251
0.45288 2093 -0.56636 7650 -0.95203 2751 -0.44426 9985
0.36251 4812 -0.53051 1122 -0.83672 7829 -0.33122 6820
0.28670 6208 -0.49499 4636 -0.74032 2276 -0.24199 5966
0.22284 4513 -0.46012 9528 -0.65791 0729 -0.17068 4462
0.16894 5592 -0.42616 3604 -0.58627 4386 -0.11325 6800
0.12345 5395 -0.39329 1826 -0.52321 5989 -0.06683 2622
0.08512 6048 -0.36166 4781 -0.46718 3076 -0.02928 3749
0.05293 4915 -0.33139 5562 -0.41704 4285 +0.00100 8681
0.02602 9861 -0.30256 5474 -0.37195 1238 0.02530 6776
+0.00369 1104 -0.27522 8834 -0.33125 0485 0.04461 5190
-0.01469 6087 -0.24941 7069 -0.29442 5803 0.05974 7779
-0.02966 1407 -0.22514 2235 -0.26105 9495 0.07137 3592
-0.04166 4514 -0.20240 0068 -0.23080 5929 0.08004 9398
-0.05110 6500 -0.18117 2644 -0.20337 3135 0.08624 3202
-0.05833 8834 -0.16143 0701 -0.17850 9812 0.09035 1619
-0.06367 0454 -0.14313 5677 -0.15599 6054 0.09271 2940
-0.06737 3493 -0.12624 1488 -0.13563 6638 0.09361 7161
-0.06968 7972 -0.11069 6099 -0.11725 6136 0.09331 3788
-0.07082 5700 -0.09644 2891 -0.10069 5314 0.09201 8037
-0.07097 3560 -0.08342 1858 -0.08580 8451 0.08991 5810
-0.07029 6321 -0.07157 0648 -0.07246 1339 0.08716 7762
-0.06893 9052 -0.06082 5473 -0.06052 9755 0.08391 2666
-0.06702 9233 -0.05112 1884 -0.04989 8308 0.08027 0223
-0.06467 8610 -0.04239 5446 -0.04045 9533 0.07634 3451
-0.06198 4833 -0.03458 2313 -0.03211 3183 0.07222 0724
-0.05903 2916 -0.02761 9697 -0.02476 5662 0.06797 7529
-0.05589 6550 -0.02144 6287 -0.01832 9556 0.06367 7999
-0.05263 9277 -0.01600 2568 -0.01272 3249 0.05937 6256
-0.04931 5556 -0.01123 1096 -0.00787 0585 0.05511 7592
-0.04597 1723 -0.00707 6704 -0.00370 0576 0.05093 9514
-0.04264 6864 -0.00348 6665 -0.00014 7138 0.04687 2681
-0.03937 3608 -0.00041 0809 +0.00285 1155 0.04294 1728
-0.03617 8848 +0.00219 8399 0.00535 1296 0.03916 6011
-0.03308 4395 0.00438 5818 0.00740 6063 0.03556 0272
-0.03010 7574 0.00619 3613 0.00906 4226 0.03213 5235
-0.02726 1764 0.00766 1269 0.01037 0752 0.02889 8142
-0.02455 6892 0.00882 5624 0.01136 6998 0.02585 3229
-0.02199 9875 0.00972 0918 0.01209 0904 0.02300 2160
-0.01959 5024 0.01037 8865 0.01257 7182 0.02034 4409
-0.01734 4409 0.01082 8725 0.01285 7498 0.01787 7607
-0.01524 8188 0.01109 7399 0.01296 0651 0.01559 7847
-0.01330 4899 0.01120 9526 0.01291 2753 0.01349 9960
-0.01151 1727 0.01118 7587 0.01273 7390 0.01157 7754
[(-;)I1
432 BESSEL FUNCTIONS OF INTEGER ORDER
a>=nearest integer to z.
BESSEL FUNCTIONS OF INTEGER ORDER 433
KELVIN FUNCTIONS-MODULUS AND PHASE Table 9.12
ker z=No (2) cos do (z) kerl :c =NI (.c)cos 41(z)
kei ~=No(x) sin do(x) kei 1 z =NI (cc)sin $1(x)
x No(4 $0(4 Wl (x) @l(X)
0.0 0.000000 -2.356194
1.8G702 -0.412350 4.927993 -2.401447
0”:: 1.274560 -0.584989 2.372347 -2.487035
0.941678 -0.743582 1.497572 -2.590827
0”:; 0.725172 -0.896284 1.050591 -2.704976
0.572032 -1.045803 0.778870 -2.825662
:$I 0.458430 -1.193368 0.597114 -2.950763
1:4 -1.339631 0.468100 -3.078993
is ;E!i -1.484977 0.372811 -3.209526
::: 0:249850 -1.629650 0.300427 -3.341804
2. 0 0.206644 -1.773813 0.244293 -3.475437
2.2 0.171649 -1.917579 0.200073 -3.610143
0.143095 -2.061029 0.164807 -3.745715
:-z 0.119656 -2.204225 0.136407 -3.881994
2: a 0.100319 -2.347212 0.113353 -4.018860
0.084299 -2.490025 0.094515 -4.156217
::2” 0.070979 -2.632692 0.079039 -4.293990
0.059870 -2.775236 0.066264 -4.432118
::“6 0.050578 -2.917672 0.055677 -4.570551
3. 8 0.042789 -3.060017 0.046873 -4.709250
-3.202283 0.039530 -4.848179
2: 8. %!i -3.344478 -4.987312
0:026095 -3.486612 ii*E:; -5.126623
:"b 0.022174 -3.628692 0:023918 -5.266093
4: 8 0.018859 -3.770724 0.020280 -5.405705
5.0 0.016052 -3.912712 0.017213 -5.545443
0.013674 -4.054662 0.014624 -5.685295
::: 0.011656 -4.196576 0.012435 -5.825250
0.009942 -4.338460 0.010583 -5.965298
::i 0.008485 -4.480314 0.009013 -6.105430
0.007246 -4.622142 -6.245638
i-20 0.006191 -4.763947 :* EE: -6.385917
6: 4 0.005292 -4.905730 0:005590 -6.526260
0.004526 -5.047493 0.004773 -6.666662
286 0.003872 -5.189238 0.004077 -6.807119
7.0 0.003315 -5.330966 0.003485 -6.947625
<a>=nearest integer to 2.
10. Bessel Functions of Fractional Order
H. A. ANTOSIEWICZ~
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 437
10.1. Spherical Bessel Functions . . . . . . . . . . . . . . . 437
10.2. Modified Spherical Bessel Functions . . . . . . . . . . . 443
10.3. Riccati-Bessel Functions . . . . . . . . . . . . . . . . 445
10.4. Airy Functions . . . . . . . . . . . . . . . . . . . . 446
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 452
10.5. Use and Extension of the Tables . . . . . . . . . . . . 452
References . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Table 10.1. Spherical Bessel Functions-Orders 0, 1, and 2 (0 _<x110) . 457
sd4, Y&d
n=O, 1, 2; %=0(.1)5, 6--S, x=5(.1)10, 5s
Table 10.2. Spherical Bessel Functions--Orders 3-10 (0 Iz 5 10) . . . 459
j&), YnO)
n=3(1)8; 2=0(.1)10, 5s
2-njn(z), z”+‘Y”(4
n=9, 10; x=0(.1)10, 7-8s
Table 10.3. Spherical Bessel Functions--Orders 20 and 21 (0 Iz 225) 463
2-B exp (2/(4n+2)lj,(~)
z”+l exp (--2/(4+2))y,(z)
n=20, 21; x=0(.5)25, 6-8s
Table 10.4. Spherical Bessel Functions-Modulus and Phase--Orders
9,10,20 and 21 . . . . . . . . . . . . . . . . . . . . . . . 464
ai%,++ 09, en++(4 -z where
Ad4 = am&+&> CO8k+t (4
~44 =-\13+%++ (4 sin en++(4
n=9, 10; z-‘=.l(-.005)0, 8D
n=20, 21; z-*=.04(-.002)0, 8D
Table Spherical Bessel Functions-Various
10.5. Orders (0 In 5 100) . 465
Ah), Y&d
n=0(1)20, 30, 40, 50, 100
2=1, 2, 5, 10, 50, 100, 10s
Table 10.6. Zeros of Bessel Functions of Half-Integer Order (0 -<n 5 19) . 467
Zeros j “, *, yy.Bof J&>, Y&z> and Values of JX.& & YL(y,.,)
v=n+$, n=0(1)19, 6-7D
Table 10.7. Zeros of the Derivative of Bessel Functions of Half-Integer
Order (OIn519). . . . . . . . . . . . . . . . . . . . . . . 468
Zeros jL, ,, yi. t of J:(z), Y:(z) and Values of J&j:, d), Y,(yl, ,)
v=n+$, n=0(1)19, 6D
Page
T&Ie 10.8. ModiGed Spherical Bessel Functions-Orders 0, 1 and 2
(0<2<5)........................... 469
Awn+* (4 , lmK+t 0)
n=o, I, 2;2=0(.1)5, 4-9D
Table 10.9. Modified Spherical Bessel Fuuctions-Crders 9 and 10
(O<z<co)........................... 470
z-"&&k+t (4, ~n+'J*~/~K+t (4
n=9, 10; 2=0(.1)5, 7-8s
e-‘l,+t (4, @/9WJL+t (4
n=9, lO;s=5(.1)10, 6s
4% exp I-z+W+ 1YCWl~~tC4
4% exp [2-n(n+l>l(2z)lK,+t(2)
n=9, 10;z-1=.1(-.005)0, 7-S
Table 10.10. Modified Spherical Bessel Functions-Various Orders
(O~n~lOO) . . . . . . . . . . . . . . . . . . . . . . . . . 473,
l&d&+1 cd, ma+* (4
n=0(1)20, 30, 40, 50, 100
z=l, 2, 5, 10, 50, 100, 10s
Table 10.11. Airy Functions (OlzS w) . . . . . . . . . . . . . . 475
Ai( Ai’( Bi(z), Bi’(z)
z=o(.ol)l, sD
Ai(--2), Ai’( Bi(-z), Bi’(-z)
~=O(.Ol)l(.l)lO, 8D
Auxiliary Functions for Large Positive Arguments
Ai (2) = jz-1/4e-y( - i) ; Bi (2) =z-1/4ey([)
Ai’(z&~“~e-~g(-[); Bi’(z)=z1/4etg(t)
f(h[), g(kl); i=$i’2, [-‘=1.5(-.1).5(-.05)0, 6D
Auxiliary Functions for Large Negative Arguments
Ai(--s)=z-1/41f,(t) cos t+fi(t) sin ,$I
Bi(--z)=z-1’41fi([) cos c-fi([) sin .$I
Ai’(-z)=s”4[g1([) sin E-g&$) cos []
Bi’(-z>=z1/41g2(t) sin t+gl(t) cos [I
flc9,AW, m(E>, 9269; t=iw2
t-'=.05(-.Ol)O, 6-7D
Table 10.12. Integrals of Airy Functions (OIz<lO). . . . . . . . . 478
437
438 BESSEL FUNCTIONS OF FRACTIONAL ORDER
10.1.10
j,(z)=fn(z) sin z+(-l)n+lf-n-~(z) cos 2
f&9 =2-l, f~(z)=z-2
fn-,(~)+fn+l(~)=~2~+w-‘fn(4
(n=O, fl, f2, . . .)
The Functionsj,(z), y,(z) for n=O, 1, 2
10.1.11 j,(z)==s’nz
sin 2 cos 2
j,(Zl=~ -7
10.1.12
y&)=-j-&)=-~
J,(x)
‘\
\ Y, lx)
\
\
\
\
10.1.14
10.1.17
(See 9.2.28.)
hyz)=i”+‘z-‘e-‘$ (n+t, k) (2iz)-k *
10.1.28 (~/z)M:,,(z)=j;(z)+y;(z)=z-2
10.1.18
10.1.29
h~~_,(z)=i(-l)“h~)(z)
(3?r/z)~~,,(z)=j:(z)+y~(z)=z-~+z-'
h~~-,(z)=--i(-l)nh~)(z) (n=O, 1, 2, . . .)
10.1.30
Elementary Properties
Recurrence Relations
(%r/z)M~,2(2)=j:(z)+y:(z)=2-2+32-'+92-~
f.(z) :j,c7!, Y&f>, h?(z), h?(z) cross Products
(n=O, fl, f2, . . .) 10.1.31 j,(z>y,~~(z)-j,-~~z)y,(z)=2--2
10.1.19 L(z) +fn+*(~)=(2~+l)~-1f,(~) 10.1.32
- (-t)"
i sin&&ZZ=~ 7 y,-l(z) @lKl4
10.1.26 0
,-0=
(3+J
IWW sin x-Si(2x) co8 2)
[$ j.bl],w-l=
(&f/x) { Ci(2x) co8 x+Si(2x) sin x}
+sin 33 I$ (-1W2n+3d341
10.1.43 10.1.54
-g5 (6949p4+474908p3+330638j.i2
m
+904678Op-5075147)(8/V)-‘- ...
a;.1-(n+~)+.8086165(n+#>Y3-.236680(n+~)-1’3 10.1.66
-.20736(n+~)-1+.0233(n+~)-5’3+ ... Y,@L, .> - -&WU <n+#-“’
10.1.60 h[(n+$)-VQ (Z[(n+$)-2’3b:])--‘2
a:,,-(n+~)+1.8210980(n+~p3
{I+& ~k[(~+3)-2’3b:l(,+3$-2kj
I
+.802728(n++)-‘la-.1174O(n+$)-’
A,(,$), z(t) are defined as in 9.5.26, 9.3.38, 9.3.39.
+.0249(n++)-5’3+ ... a:, b: s-th (negative) real zero of Ai’( Bi’(z)
10.1.61 (see 10.4.95, 10.4.99.)
Complex Zeros of h:‘(Z), hY(Z)
j,(u:J -.8458430(n+#-5’E { 1-.566032(~1++)-~‘~
f@ (2) and h$ (ze”“**), m any integer, have the
+.38081(n+~)-4’a-.2203(n+~)-2+ . . .} same zeros.
/&I) (z) has n zeros, symmetrically distributed with
10.1.62 respect to the imaginary axis and lying approxi-
mately on the finite arc joining z=-n and z=n
y,(b:,,)-.7183921(n+~)-“‘6{ 1-l.274769(n++)-2’3
shown in Figure 9.6. If n is odd, one zero lies on
+1.23038(n+))-4’3-1.0070(n+~)-2+ . . .) the imaginary axis.
hc)‘( z) has n+ 1 zeros lying approximately on the
See [lo.311 for corresponding expansions for same curve. If n is even, one zero lies on the
6=2, 3. imaginary axis.
442 BESSEL FUNCTIONS OF FRACTIONAL ORDER
sinh z cash z
dmn+t(4=l. 3.5 . 2”
. . &+I) &&,&)=-~+y
12
(32”) 2 ~I~,2(z) =(s+k) sinh s-3 cash z
{ 1+,!(:+3)+2!(27&+3)(2n+5)+ * * *)
10.2.14
10.2.6
J3?rlzI...l,2(z> -559
12 &qa-3,2(z) =q-=g
(322>2
‘+l!(l?Zn)+2!(1-2n)(3-2n)+ ’’’
&&I-~,~(z)=-$ sinh z+(-$+i) cash s
(n=O, 1,2, . . ,)
*See page 11.
444 BESSEL FUNCTIONS OF FRACTIONAL ORDER
10.2.15
&$~K,+t(z) =3aie(~+1)r~‘2h~1)(zel”)
(- 7r<arg 2 I+r)
=- 4 Irie-'"+l'"'/2h~2'(Ze-fri)
(h<arg 25 4
n
= (&r/z)e-z c (n+ 3, k) (2.4 --L
0
10.2.16
Elementary Properties
Recurrence Relations
FIGURE 10.4. &L++(x), & IL+&). n=0(1)3.
J&(4: d&L+&>, (--l)n+l&&&+tM
(n=O, fl, zk2,. . .)
10.2.18 fn-l(z)-fn+,(z)=(2n+l)z-‘f,(z)
10.2.19 nfn-l(z)+(n+l)f.+l(z)=(2nfl)$f.(z)
2-
10.2.21 -~~~cz,+$f.(z)=fn+l(z)
(See 10.2.23.)
DilTerentiation Formulae
I-
IO 4-
fn(4: vmL+&>, (-1)“+‘4mIL+,(4 s-
10.2.22 ; -$ ~[zl+tfn(z)l=z”-“+%-01(2)
( >
I 2 3 4 5 6 7
10.2.23 l[z-Iyn(z)l=z-.-mf.+m(z) -
FIQURE 10.5. f K,,+&). x=10.
(m=l, 2,3,. ..)
BESSEL FUNCTIONS OF FRACTIONAL ORDER 445
Formulas of Rayleigh’s Type Addition Theorems and Degenerate Forms
(; g >*C=p (n=O,
J~Ln-*(z)=2”
1,2, . . .) hmm”+t(~P)l~n(eos 0)
10.2.36
Formulas for IX++(Z) -IT,-+ (2)
eEcase=$2 @n+l) hmL+t~4lcdcos 0)
10.2.26
10.2.37
2$ -$ (-l)k+’ (2n-4) ! (2n--2k)! (22)a-2n e--y$ (-1)“(2n+l)[~~~+t(z)]P,(cos e>
k! [(n-k) !]’
(n=O, 1,2, . . .) Duplication Formula
10.2.38
10.2.27 &r/2)[1:*(2)-12-~,*(2)]=-2-* &+tcw=
Differential Equation
10.4.1 w”-zw=o
Pairs of linearly independent solutions are
Ai (z), Bi (z),
Ai (z), Ai (ze22*1’3), .a -
Ai (z), Ai (ze-2ff’3).
Ascending Series
10.4.2 Ai (z)=cJ(z)-c2g(z)
=q 3 (gk &
-.6 -
=$ 3”(gk&$ ”
-1.0 I-
10.4.4 2.0 -
10.4.8
Bi (z)+e2rifi Bi (~2~*fl)+e-2*f” Bi (~--2*~f/3)=0
(See 10.4.2.)'
2=
(
; 1 2J3
>
10.4.39
S 0
‘Ai (--t)dt=-c$(--z)+c&(-z)
Gi (z)=s-l~-sin(~P+zt)dt
w(0) =i Bi (O)=$ Ai (0)=.20497 55424 78
=gBi (z)+s’[Ai (z) Bi (t)-Ai (t) Bi (z)]dt
0
w’(O)=aBi’ (0)=-l Ai’ (0)=.14942 9452449
10.4.43 8
w(z)=Gi(z)
Gi’ (z)=iBi’ (z)+S’[Ai’(z) Bi (t)-Ai (t) Bi’(z)]dt
10.4.56 W” -zwq(-l
10.4.44
w(O)=: Bi (O)=-$ Ai (0)=.40995 10849 56
Hi(z)=7rS11 expi-i Pfzl) dt
w’(O) =f Bi’ (0) =--$ Ai’ (0) =.29885 89048 98
=iBi (z)+J’[Ai (t) Bi (z)-Ai Bi(t)]dt
0
w(z) =Hi (z)
10.4.45
Differential Equation for Products of Airy Functions
Hi’(z)=iBi’ (z)+S’[Ai (t) Bi’(z)-Ai’ Bi (t)]dt
0 10.4.57 W “‘--4~w~-~w=o
+sin(C+$ I$ (--Ukd2k+l
Fml] N(z)=J[Ai’* (--2)+Bi’2 (-s)],
$+)=arctan [Bi’ (--z)/Ai’ (-x)1
Differential Equations for Modulus and Phaw
10.4.77 e'2+3(e"'/e'>-g(e"/e')2=2
Asymptotic Expansions of Modulus and Phase for
Large 2
10.4.67
Bi’ (--z)vr-f2f $’ (-l)kd,r{-a *2?g (22)-o+'28,2,03;e525(22)-"- . . .]
+ 4+ (2s)-9-2065,~~(22)-'p+ . ..I
hl 2 $ (-l)“d,k+,r-“-l-j
>
Asymptotic Forms of ‘Ai ( f 1) dt, “Bi ( f f) dt tot Large 3:
(lw 4<3 4 s 0 s 0
2 10.4.104
10.4.84
S= 0
Bi (t)dt-s-1/2z--3/4 exp
(
3 x312
>
10.4.85
S' Bi 0
(- t)&-~-1&--3/4 sin (i x3/,,:)
10.4.105
AsymptoticFormsofGi(rtz),Gi’(*z),Hi(fz),Hi’(f~)
for Large 2
Bi (&)=(-l)a-l&e*~‘Egl
(21 sufficiently
$ (JS-~)+:
large
ln 2
1
10.4.86 Gi (2) &x-*x-’
f(z)-22’3 1+$ z-2-4 z-4+% z-6
10.4.87 Gi (-x) ~~~~~~~~~~~ cos (; .3,2+g
1080 56875 z-8
7 69 67296
10.4.88 Gi’ (z) mss x-1x12
16 23755 96875 z-Io
- ..*
+ 3344 30208 >
10.4.89 Gi’ (-x) ~1r-1f2x1/4 sin (; x3,2+;)
181223 --B
g(z)-22’3 (1-i z-2+g8 c4-- 207360 2
10.4.90 Hi (x) ~~~~~~~~~~~ exp (zx”‘“)
10.4.91 Hi (-z) WFW~ 186 83371 --8
+ 12 44160 ’
10.4.92 Hi’ (x) ~~~~~~~~~~ exp ($x3/2)
9 11458 84361
z-*0+ . . .)
1911 02976
10.4.93 Hi' (-5) A, -5 ~-1x-2
5 2-2-46o8
1525 2 -4
fI(Z) wr-“%z1’6 1 +s
Zeros and Their Asymptotic Expansions
23 97875 z-e-ee.
Ai (z), Ai’ (z) have zeros on the negative real +
6 63552
axis only. Bi (z), Bi’ (z) have zeros on the nega-
-1/22-1/6 I-- 7 -2 1673 m4 *
tive real axis and in the sector $r<] arg z]<+x. g&IN*
( 96' +6144’
a,, a:; &, b: s-th (real) negative zero of Ai (z),
Ai’ (2); Bi (z), Bi’ (z), respectively. & ~3:; s,, 5: 843 94709
s-th complex zero of Bi (z), Bi’ (z) in the sectors -26542080 '--'+ ' - *
&r<arg z<+r, -$r<arg z<-f?r, respectively. Formal and Asymptotic Solutions of Ordinary Differ-
ential Equations of Second Order With Turning
10.4.94 a*=--j[3*(4s-1)/8] Points
10.4.11s t (g2=P(z) t
(n-0, 1, 2, . . .) 10.4.116 *
Uniform Asymptotic Expansions of Solutions
The recurrence relation 10.2.18 yields successively y’(x), respectively, the following formulas may be
used, in which d, d’ denote approximations to c, c’
-&?r/3.6&2(3.6)=-.Oll92 222 and u=y(d)/y’(d), v=y’(d’)/d”y(d’).
= .04942,4480
-Jj$i3Kg,2(3.6)=-.~246i 718
-(105+76d’3+24d’8);
-A (.04942 4480)
=-.12072 034 -(945+756d’“+272d$-. . .}
[lO.l] H. Bateman and R. C. Archibald, A guide to tables [lo.171 H. K. Crowder and G. C. Francis, Tables of
of Bessel functions, Math. Tables Aids Comp. 1, spherical Bessel functions and ordinary Bessel
205-308 (1944), in particular, pp. 229-240. functions of order half and odd integer of the
[10.2] T. M. Cherry, Uniform asymptotic formulae for first and second kind, Ballistic Research Labora-
functions with transition points, Trans. Amer. tory Memorandum Report No. 1027, Aberdeen
Math. Sot. 68, 224257 (1950). Proving Ground, Md. (1956).
(10.31 A. Erdelyi et al., Higher transcendental functions, [lO.lS] A. T. Doodson, Bessel functions of half int,egral
vol. 1, 2 (McGraw-Hill Book Co., Inc., New order [Riccati-Bessel functions], British Assoc.
York, N.Y., 1953). Adv. Sci. Report, 87-102 (1914).
[10.4] A. Erdelyi, Asymptotic expansions, Caliiornia [10.19] A. T. Doodson, Riccati-Bessel functions, British
Institute of Technology, Dept. of Math., Assoc. Adv. Sci. Report, 97-107 (1916).
Technical Report No. 3, Pasadena, Calif. (1955). [19.20] A. T. Doodson, Riccati-Bessel functions, British
[IO.51 A. Erdelyi, Asymptotic solutions of differential Assoc. Adv. Sci. Report, 263-270 (1922).
equations with transition points or singularities, [10.21] Harvard University, Tables of the modified Hankel
J. Mathematical Physics 1, 16-26 (1960). functions of order one-third and of their deriva-
[10.6] H. Jeffreys, On certain approximate solutions of tives (Harvard Univ. Press, Cambridge, Mass.,
linear differential equations of the second order, 1945).
Proc. London Math. SOC. 23, 428-436 (1925). [10.22] E. Jahnke and F. Emde, Tables of functions, 4th
110.71 H. Jeffreys, The effect on Love waves of hetero-
ed. (Dover Publications, Inc., New York, N.Y.,
geneity in the lower layer, Monthly Nat. Roy. 1945).
A&r. Sot., Geophys. Suppl. 2, 101-111 (1928).
[10.8] H. Jeffreys, On the use of asymptotic approxima- [IO.231 C. W. Jones, A short table for the Bessel functions
In+i(z), (2/a)&++(z) (Cambridge Univ. Press,
tions of Green’s type when the coefficient has
zeros, Proc. Cambridge Philos. Sot. 52, 61-66 Cambridge, England, 1952).
(1956). [10.24] J. C. P. Miller, The Airy integral, British Assoc.
[10.9] R. E. Langer, On the asymptotic solutions of Adv. Sci. Mathematical Tables, Part-vol. B
differential equations with an application to the (Cambridge Univ. Press, Cambridge, England,
Bessel functions of large complex order, Trans. 1946).
Amer. Math. Sot. 34, 447-480 (1932). [10.25] National Bureau of Standards, Tables of spherical
[lO.lO] R. E. Langer, The asymptotic solutions of ordinary Bessel functions, ~01s. I, II (Columbia Univ.
linear differential equations of the second order, Press, New York, N.Y., 1947).
with special reference to a turning point, Trans. [10.26] National Bureau of Standards, Tables of Bessel
Amer. Math. Sot. 67, 461-490 (1949). functions of fractional order, ~01s. I, II (Co-
[lO.ll] W. Magnus and F. Oberhettinger, Formeln und lumbia Univ. Press, New York, N.Y., 1948-49).
Siitze fiir die speziellen Funktionen der mathe- [10.27] National Bureau of Standards, Integrals of Airy
matischen Physik, 2d ed. (Springer-Verlag, functions, Applied Math. Series 52 (U.S. Gov-
Berlin, Germany, 1948). ernment Printing Office, Washington, D.C.,
[10.12] F. W. J. Olver, The asymptotic solution of linear 1958).
differential equations of the second order for [10.28] J. Proudman, A. T. Doodson and G. Kennedy,
large values of a parameter, Philos. Trans. Roy. Numerical results of the theory of the diffraction
Sot. London [A] 247, 307-327 (1954-55). of a plane electromagnetic wave by a conducting
[10.13] F. W. J. Olver, The asymptotic expansion of Bessel sphere, Philos. Trans. Roy. Sot. London [A]
functions of large order, Philos. Trans. Roy. 217, 279-314 (1916-18), in particular pp. 284-
SOC. London [A] 247, 328-368 (1954). 288.
[10.14] F. W. J. Olver, Uniform asymptotic expansions of [10.29] M. Rothman, The problem of an infinite plate
solutions of linear second-order differential under an inclined loading, with tables of the
equations for large values of a parameter, Philos. integrals of Ai (+z), Bi (&.z), Quart. J. Mech.
Trans. Roy. Sot. London [A] 250,479-517 (1958). Appl. Math. 7, l-7 (1954).
[10.15] W. R. Wasow, Turning point problems for systems [10.30] M. Rothman, Tables of the integrals and differ-
of linear differential equations. Part I: The
ential coefficients of Gi (+z), Hi (--z), Quart. J.
formal theory; Part II: The analytic theory.
Comm. Pure Appl. Math. 14, 657-673 (1961) ; Mech. Appl. Math. 7, 379-384 (1954).
15, 173-187 (1962). [10.31] Royal Society Mathematical Tables, vol. 7,
[10.16] G. N. Watson, A treatise on the theory of Bessel Bessel functions, Part III. Zeros and associated
functions, 2d ed. (Cambridge Univ. Press, values (Cambridge Univ. Press, Cambridge,
Cambridge, England, 1958). England, 1960).
456 BESSEL FUNCTIONS OF FRACTIONAL ORDER
[10.32] R. S. Scorer, Numerical evaluation of integrals of [10.33] A. D. Smirnov, Tables of Aiiy functions (and
the form special confluent hypergeometric functions).
Translated from the Russian by D. G. Fry
(Pergamon Press, New York, N.Y., 1960).
[10.34] I. M. Vinogradov and N. G. Cetaev, Tables of
Bessel functions of imaginary argument (Izdat.
and the tabulation of the function Akad. Nauk SSSR., Moscow, U.S.S.R., 1950).
[10.35] P. M. Woodward, A. M. Woodward, R. Hensman,
Gi (z) = (l/r)h- sin (uz+ 1/3ua)du, H. H. Davies and N. Gamble, Four-figure tables
of the Airy functions in the complex plane,
Quart. J. Mech. Appl. Math. 3, 107-112 (1950). Phil. Mag. (7) 37, 236-261 (1946).
BESSEL FTJNCTIONS OF FRACTIONAL ORDER 457
SPHERICAL BESSEL FUNCTIONS-ORDERS 0.1 AND 2 Table 10.1
x
&(4 j,(x) & (4 Y,(X) ?/I (4 Y,(X)
1.00000 000 0.00000 0000
i:! 0.99833 417 0.033300012 0.00000 619061 -9.95;04 17
0.00066000000 -,0,,",875 - 30;5:0125
0.99334 665 0.066400381 0.0026590561 -4.90033 29 -25.495011 -377.52483
EE 0.98506 736 0.099102888 0.0059615249 -3.18445 50 -11.599917 -112.81472
0:4 0.97354 586 0.13121215 0.010545302 -2.30265 25 --6.73017 71 -48.173676
0.95885 108 0.16253703 0.016371107 -1.75516 51 -4.46918 13 -25.059923
0.94107 079 0.19289196 0.023388995 -1.37555 94 -3.23366 97 -14.792789
0.92031 098 0.22209828 0.031538780 -1.09263 17 -2.48121 34 -9.54114 00
0.89669 511 0.24998551 0.040750531 -0.87088 339 -1.98529 93 -6.57398 92
0.87036 323 0.27639252 0.050945155 -0.69067 774 -1.63778 29 -4.76859 87
1.0 0.84147 098 0.30116868 0.062035052 -0.54030 231 -1.38177 33 -3.60501 76
0.81018 851 0.32417490 0.073924849 -0.41236 011 -1.18506 13 -2.81962 54
::: 0.77669 924 0.34528457 0.086512186 -0.30196 480 -1.02833 66 -2.26887 66
0.74119 860 0.36438444 0.099688571 -0.20576 833 -0.89948 193 -1.86995 92
::.i 0.70389 266 0.38137537 0.11334028 -0.12140 510 -0.79061 059 -1.57276 05
1.5 0.66499 666 0.39617297 0.12734928 -0.04715 8134 -0.69643 541 -1.34571 27
0.62473 350 0.40870814 0.14159426 +0.018249701 -0.61332 744 -1.16823 87
i-7" 0.58333 224 0.41892749 0.15595157 0.075790879 -0.53874 937 -1.02652 51
1:8 0.54102 646 0.42679364 0.17029628 0.12622339 -0.47090 236 -0.91106 065
1.9 0.49805 268 0.43228539 0.18450320 0.17015240 -0.40849 878 -0.81515 048
2.0 0.45464 871 0.43539778 0.19844795 0.20807342 -0.35061 200 -0.73399 142
0.41105 208 0.43614199 0.21200791 0.24040291 -0.29657 450 -0.66408 077
2: 0.36749 837 0.43454522 0.22506330 0.26750051 -0.24590 723 -0.60282 854
0.32421 966 0.43065030 0.23749812 0.28968523 -0.19826 956 -0.54829 769
::: 0.28144 299 0.42451529 0.24920113 0.30724738 -0.15342 325 -0.49902 644
2.5 0.23938 886 0.41621299 0.26006673 0.32045745 -0.11120 588 -0.45390 450
0.19826 976 0.40583020 0.26999585 0.32957260 -0.07151 1067 -0.41208 537
::; 0.15828 884 0.39346703 0.27889675 0.33484153 -0.03427 3462 -0.37292 316
0.11963 863 0.37923606 0.28668572 0.33650798 +0.00054 2796 -0.33592 641
2: 0.08249 9769 0.36326136 0.29328784 0.33481316 0.032953045 -0.30072 380
0.04704 0003 0.34567750 0.29863750 0.32999750 0.062959164 -0.26703 834
z-1" +0.01341 3117 0.32662847 0.30267895 0.32230166 0.090555161 -0.23466 763
3:2 -0.01824 1920 0.30626652 0.30536678 0.31196712 0.11573164 -0.20346 870
-0.04780 1726 0.28475092 0.30666620 0.29923629 0.13847939 -0.17334 594
33:: -0.07515 9148 0.26224678 0.30655336 0.28435241 0.15879221 -0.14424 164
-0.10022 378 0.23892369 0.30501551 0.26755905 0.17666922 -0.11612 829
::6' -0.12292 235 0.21495446 0.30205107 0.24909956 0.19211667 -0.08900 2337
-0.14319 896 0.19051380 0.29766961 0.22921622 0.20514929 -0.06287 8964
:-; -0.16101 523 0.16577697 0.29189179 0.20814940 0.21579139 -0.03778 7773
3:9 -0.17635 030 0.14091846 0.28474912 0.18613649 0.22407760 -0.01376 91'l2
4.0 -0.18920 062 0.11611075 0.27628369 0.16341091 0.23005335 +0.009129107
4.1 -0.19957 978 0.091522967 0.26654781 0.14020096 0.23377514 0.030854018
-0.20751 804 0.067319710 0.25560355 0.11672877 0.23531060 0.051350236
:-: -0.21306 185 0.043659843 0.24352220 0.093209110 0.23473838 0.070561855
4:4 -0.21627 320 +0.020695380 0.23038368 0.069848380 0.23214783 0.088434232
4.5 -0.21722 892 -0.00142 95812 0.21627586 0.046843511 0.22763858 0.10491554
-0.21601 978 -0.02257 9838 0.20129380 0.024380984 0.22132000 0.11995814
t:; -0.21274 963 -0.04262 9993 0.18553900 +0.002635886 0.21331046 0.13351972
-0.20753 429 -0.06146 5266 0.16911850 -0.01822 8955 0.20373659 0.14556433
29" -0.20050 053 -0.07898 2225 0.15214407 -0.03806 3749 0.19273242 0.15606319
5.0 -0.19178 485 -0.09508 9408 0.13473121 -0.05673 2437 0.18043837 0.16499546
c 1
(-$)4
[ /-- 1
(-$)3
-1 -1.2885
II-1
-1
-1
-1
-1.3849
-1.4644
-1.5268
-1.5720
-2 -5.6210
-1 -1.0770
II
-2
-2
-2
-1
-6.9018
-8.0795
-9.1466
-1.0097
-2 6.0883
-2
II
-2.9542
II
-2
-2
-2
-2
-9.6415
-a.4493
-7.2065
-5.9263
-1 -1.3123
-2 4.7295
-2 3.3674
-2 2.0132
-3 +6.7812
-3 -6.2736
-2
-3
-3
Ii-2 II II
-1.6303
-3.2520
+9.4953
2.1829
-2 1.8188
-1
-1
-1
-1
-1.3171
-1.3092
-1.2891
-1.2571
-2
-2
-2
-2
-2
-1.8929
-3.1089
-4.2662
-5.3561
-6.3711
II II
-2
II
a.1877
-2 -1.1105
-1 -a.5177
-9.4473
-1.0313
-7.5334
-2
-2
-2
-2
3.0067
4.1360
5.1973
6.1820
-2
-2
-2
-2
-1
-7.3040
-8.1487
-8.8997
-9.5527
-1.0104
-2
-2
-2
-1
II
Ii-1 II
a.8851
9.4810
9.9723
1.0357
1.0632 -1 1.0124 -2 -3.4542
-1
-1
-1
-1
-1.0551
-1.0892
-1.1126
-1.1253
II-2
-3 +2.6357
1.3382
2.4227
3.5066
4.5791
II
r:
-1
-1 II II
:* oOts0:
ioa13
1.0663
1; +F
-2 -1.0349
211498
3.1395
yfl -1 1.0415
-1 1.0596
-1 1.0669
-1 1.0635
-1 1.0497
-2
-2
-3
-3
-2.3621
-1.2710
-1.9101
+8.6782
(-2)-5.4402 (-2)
II 7.8467
I-
(-2) 7.7942
-1
-2
-2
-2
1.0257
9.9213
9.4941
8.9817
(-2) 8.3907 (-2) 6.2793
I--
(-2)-6.5069
Y&)= +/xyn+* (x)=(-‘)“+l-\l~“ix~-~~+*~(x)
d
BESSEL FUNCTIONS OF FRACTIONAL ORDER 459
SPHERICAL BESSEL FUNCTIONS-ORDERS 3-10 Table 10.2
j7 (4
0.0000 i.0000 1.52734 93 7.27309-i9
-14)4.9319 2.9012 1.52698 56 7.27151 10
-12)6.3072 7.4212 1.52589 53 7.26677 00
1.8995 1.52407 96 7.25887 47
1.8938 1.52154 09 7.24783 46
1.1261 1.51828 26 7.23366 29
4.8282 1.51430 88 7.21637 65
1.6515 1.50962 48 7.19599 61
4.7873 1.50423 66 7.17254 61
1.2228 1.49815 12 7.14605 44
1.0 II-3 9.0066 2.8265 1.49137 65 7.11655 26
-2 1.1847 6.0254 1.48392 11 7.08407 57
-2 1.5183 1.2013 1.47579 48 7.04866 21
2.2640 1.46700 80 7.01035 39
::i -2 1.9033
2.3411 II- 54 1.4786
3.3461
2.2643
9.2561
4.7963 I 4.0669 1.45757 18 6.96919 61
‘3.:
3:2
1.4983
1.9160
2.4283
1.21214 41
1.23023 38 5.97131
5.89135 85
26
1.19371 48 5.80979 75
;:: 3.8056
3.0520 1.15592
1.17496 82
54 5.72674
5.64226 82
00
'3.:
. 1.0325
8.5665 1.05702 31
1.07722 33 5.20181 62
5.29201 05
4.0 1.2372 1.03665 63 5.11072 78
4.1 1.4743 1.01614 44 5.01885 80
:s 2.0603
1.7473 0.99550
0.97477 88
06 4.83311 07
4.92629 51
414 2.4174 0.95395 10 4.73942 00
::2 3.2814
2.8229 0.91215 06
0.93307 01 4.55082
4.64529 34
25
i-i 3.7976
4.3763 0.89120
0.87026 94
97 4.45609
4.36119 35
18
419 5.0226 0.84934 88 4.26620 13
5.0 (-1)2.2982 (-1)1.8702 (- 1)1.0681 (- 2)4.7967 (- 2)1.7903 (- 3)5.7414 0.82846 70 4.17120 50
[ (-yJ] [‘-y]
&Cd =&IxJ~+~ (4
Compiledfrom National Bureau of Standards,Tablesof sphericalBesselfunctions, ~01s.I, II. Columbia
Univ. Press,New York, N.Y., 1947 (with permission).
460 BESSEL FUNCTIONS OF FRACTIONAL ORDER
1.5
4I
3
3
3
3
-1.0881
-5.6378
-3.0988
-1.7901
-1.0790
/I I
543 -1.4045
-1.7686
-9.8790
-3.3227
-6.6058
6
5
5
5
5
4
-2.0959
-8.9515
-4.1224
-2.0227
-1.0477
-5.6859
-0.35490
-0.35710
-0.35954
-0.36221
-0.36512
-0.36827
04
89
56
57
46
87
-0.67221
-0.67595
-0.68007
-0.68458
-0.68949
-0.69481
50
30
37
47
42
14
4 -3.2143 -0.37168 46 -0.70054 60
i-76 4 -1.8835 -0.37534 96 -0.70670 90
1:s 4 -1.1395 -0.37928 17 -0.71331 20
1.9 3I -7.0931 -0.38348 96 -0.72036 75
1 -9.7792 3 -4.5301 -0.38798 26 -0.72788 93
::1" 1 -7.0870 3 -2.9613 -0.39277 08 -0.73589 19
1I -5.2238 3 1-1.9771 -0.39786 50 -0.74439 11
I
;*: 1 -3.9108 3 -1.3458 -0.40327 71 -0.75340 38
2:4 1 -2.9702 2 1-9.3247 -0.40901 97 -0.76294 81
2.5 1 -2.2859 2 -6.5676 -0.41510 62 -0.77304 34
2.6 1 -1.7812 -0.42155 14 -0.78371 06
1 -1.4041 g I 1:;;;; -0.42837 10 -0.79497 18
z-i 1 -1.1189 2 -2:5025 -0.43558 18 -0.80685 08
2:9 0 -9.0069 2 1 -1.8615 -0.44320 20 -0.81937 31
-0.45125 11 -0.83256 59
-0.45975 01 -0.84645 82
I
-0.46872 14 -0.86108 11
-0.47818 95 -0.87646 78
-0.48818 03 -0.89265 39
II I
0 -2.9528 1 -3.9249 -0.49872 20 -0.90967 72
0 -2.5201 1 -3.1246 -0.50984 49 -0.92757 84
0 -2.1660 1 -2.5070 -0.52158 17 -0.94640 10
0 -1.8743 1 -2.0265 -0.53396 75 -0.96619 15
0 -1.6325 0 -9.8471
-4.7139
-5.6086
-6.7182
-8.1040 1 -1.6498 -0.54704 05 -0.98699 97
-0.64247 43 -1.13650 10
22
6:1
2.8417
3.0876
3.3461
3.6168
0.53155
0.51352
0.49577
0.47831
94
10
04
68
2.79677
2.71153
2.62739
2.54443
98
12
98
09
3.8996 0.46116 89 2.46266 76
4.1940 0.44433 45 2.38215 03
4.4994 0.42782 11 2.30291 70
4.8154 0.41163 52 2.22500 27
5.1412 0.39578 30 2.14844 05
5.4759 0.38026 97 2.07326 03
-1 I 1.5077 -2 I 5.8188 0.36510 02 1.99948 99
-1 1.5217 -2 6.1686 0.35027 86 1.92715 45
-1 1.5312 -2 6.5244 0.33580 85 1.85627 66
-1 1.5360 -2 6.8849 0.32169 28 1.78687 63
-1)1.5361 -2 7.2486 0.30793 39 1.71897 14
I
-2 7.6143
-2 7.9804
-2 8.3451
-2 8.7069
0.29453
0.28149
0.26881
0.25649
36
30
29
33
1.65257
1.58770
1.52436
1.46257
72
64
97
53
-2)9.0640 0.24453 39 1.40232 92
9'::
z-:
9:4
I II -1 1.3795
1.3746
-1 1.3655
1.3520
-1 1.3341
-1 1.1000
1.1270 0.18025
-1 1.1747
0.17076 78
1.1520 0.16161
84 1.07343
0.15280 93
1.02401 42
62 0.92973
72
0.97612 24
83
-1 1.1949 0.14432 46 0.88485 16
;:z
Ki
9:9
(-2)-5.5535
-2
-2
-2
'-2
-2
I II II
8.5853
7.8016
6.9921
6.1608
5.3120
(-2)4.4501
-1
-1
-1
-1
-1
1.3117
1.2849
1.2536
1.2180
1.1780
(-1)1.1339
-1
-1
-1
-1
-1
1.2126
1.2275
1.2394
1.2482
1.2537
(-1)1.2558
0.13616
0.12833
0.12081
0.11360
0.10670
93
53
68
83
35
0.84144
0.79950
0.75902
0.71996
0.68231
0.10009 64 0.64605 15
75
99
10
20
26
[(-:)5] [(-;)“I
j,(x) =
462 BESSEL FUNCTIONS OF FRACTIONAL ORDER
y,(+ -cn+&) [ 1
(73
BESSEL FUNCTIONS OF FRACTIONAL ORDER 463
SPHERICAL BESSEL FUNCTIONS-ORDERS 20 AND 21 Table 10.3
X lo26.f20(4 1027.f21
(4 lo-24920(x~ lo-25921(4
7.62597 90 1.77348 35 -0.31983 10 -1.31130 70
8:: -7.62705 91 1.77371 23 -0.31988 11 -1.31149 33
7.63028 29 1.77439 56 -0.32003 25 -1.31205 61
:-i 7.63560 15 1.77552 32 -0.32028 86 -1.31300 70
2:o 7.64293 25 1.77707 85 -0.32065 49 -1.31436 61
7.65215 99 1.77903 78 -0.32113 96 -1.31616 11
9.: 7.66313 22 1.78137 03 -0.32175 30 -1.31842 87
3:5 7.67566 19 1.78403 80 -0.32250 82 -1.32121 43
7.68952 28 1.78699 49 -0.32342 08 -1.32457 29
44:: 7.70444 90 1.79018 73 -0.32450 98 -1.32856 95
7.72013 23 1.79355 29 -0.32579 69 -1.33328 02
::; 7.73621 95 1.79702 05 -0.32730 79 -1.33879 33
7.75231 00 1.80050 95 -0.32907 24 -1.34521 03
66’! 7.76795 28 1.80392 94 -0.33112 44 -1.35264 77
7:o 7.78264 38 1.80717 91 -0.33350 34 -1.36123 89
7.79582 23 1.81014 64 -0.33625 47 -1.37113 69
7.80686 80 1.81270 77 -0.33943 07 -1.38251 67
7.81509 84 1.81472 70 -0.34309 23 -1.39557 96
7.81976 53 1.81605 56 -0.34731 02 -1.41055 73
7.82005 32 1.81653 14 -0.35216 70 -1.42771 82
10.0 7.815076 1.815979 -0.35776 04 -1.447374
10.5 7.803876 1.814208 -0.36420 59 -1.469891
11.0 7.785428 1.811016 -0.37164 20 -1.495697
11.5 7.758627 1.806185 -0.38023 59 -1.525305
12.0 7.722309 1.799482 -0.39019 23 -1.559325
12.5 7.675238 1.790664 -0.40176 53 -1.5b8497
13.0 7.616116 1.779472 -0.41527 46 -1.643728
13.5 7.543601 1.765639 -0.43113 22 -1.696143
14.0 7.456316 1.748885 -0.44987 76 -1.757166
14.5 7.352841 1.728929 -0.47223 40 -1.828625
15.0 7.231764 1.705481 -0.49918 70 -1.912922
15.5 7.091689 1.678251 -0.53209 15 -2.013273
16.0 6.931265 1.646956 -0.57279 98 -2.134049
16.5 6.749220 1.611324 -0.62378 79 -2.281228
17.0 6.544411 1.571096 -0.68821 72 -2.462936
17.5 6.315851 1.526041 -0.76981 49 -2.689957
18.0 6.062784 1.475960 -0.87240 01 -2.975953
18.5 5.784739 1.420698 -0.99883 14 -3.336925
19.0 5.481584 1.360155 -1.149171 -3.789188
19.5 5.153621 1.294299 -1.317987 -4.344958
20.0 4.801647 1.223178 -1.490982 -5.004711
20.5 4.427041 1.146936 -1.641599 -5.745922
21.0 4.031843 1.065826 -1.728777 -6.508927
21.5 3.618830 0.98022 63 -1.697442 -7.182333
22.0 3.191590 0.89065 46 -1.483467 -7.592679
22.5 2.754567 0.79777 92 -1.024223 -7.504782
23.0 2.313103 0.70243 25 -0.274630 -6.640003
23.5 1.873442 0.60561 45 co.773430 -4.717888
24.0 1.442686 0.50849 80 2.072631 -1.52185
24.5 1.028721 0.41242 27 3.508629 +3.01816
25.0 0.640055 0.31888 30 4.901591 +8.74251
exp (~2/4n+2)
Compiled from National Bureau of Standards, Tables of spherical Bessel func-
tions, ~01s. I, II.ColumbiaUniv. Press, New York, N.Y., 1947(with permission).
464 BESSEL FUNCTIONS OF FRACTIONAL ORDER
Table 10.4
SPHERICAL BESSEL FUNCTIONS-MODULUS AND PHASE-ORDERS 9, 10, 20 AND 21
1 1(5312
[ 6$6
1 II 1
(-;I6
c
'#9
I
Compiled from National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II.
Columbia Univ. Press,NewYork,N.Y.,1947 (with permission).
BESSEL FUNCTIONS OF FRACTIONAL ORDER 465
SPHERICAL BESSEL FUNCTIONS-VARIOUS ORDERS Table 10.5
I-
x=1
123I 9.00658
1.01101
6.20350 5808
3.01168
8.41470 5201
1117
6789
9848
II
f-
-
x=2
114.54648 7134
2 4.35397
1 6.07220 9276
1.40793
1.98447 9766
9491
7750
2.29820 6182
1.87017 6553
ii!
::
II
f- 816.82530 0865
- 10
12
11
9 4.81014
3.58145 8901
2.48104
5.97687 1612
9119
1402
14
I - 22
21I,.1.55670
24
19
18 3.08874 2364
1.20385
4.45117
5.13268 5742
7504
8271
6115 1.08428 0182
1.67993 9976
II- 20
66 7.63264
49
34 4.01157 1101
1.66097
5.83661 5290
8779
7888 I - 12
33I 2.85747
46
22 1.21034 7583
4.28273
5.42772 9350
0217
6761
x=50 x=100
(- 3)-5.24749 7074 -3)-5.06365 6411
- l)-1.05589 2851
f- 21-2.00483 0056 -3)-9.29014 8935
2
7
98
I -3 +3.15754
-3 +9.70062
-3 i -1.70245
(-3)-9.99004
5454
9844
0977
6510
10 I- 21 6.46051 5449 (- 2 -1.50392 2146
(- 2 3155744 1489 - 3 +9.90845 4236
11: - 2 +1.95971 1041
- 4 -1.09899 0300
ii I - 23I 7.46558
1.72159 4477
2.94107 8342
9974 I- 2I -1.96564 5589
- 2 -1.12908 4539
:z - 2 +1.26561 3175
- 2 +1.96438 9234
1187 3.23884 7439 - 3 +1.09459 2888
19 8.89662 7269 I- 2I -1.88338 9360
20 2 -1.57850 2990
30 - 3 -1.49467 3454
40 - 2 -2.60633 6952
50 I-- 2I +1.88291 0737
100 (-90) 5.83204 0182 (-22)+1.01901 2263 (-2)+1.08804 7701
466 BESSEL FUNCTIONS OF FRACTIONAL ORDER
( -l)-3.20465 0467
-1 I -5.18407 5714
0 -1.02739 4639
0 -2.56377 6345
0 -7.68944 4934
10 5 -3.55414 7201
11 6 -3.69396 5631
12 ll)-3.23191 3629 7 -4.21251 9003
8 -5.22870 9098
1143 I 9I -7.01663 2092
15 -6.29800 7233 11 -1.01218 2944
:65 17 -1.95020 7734 I 12 -1.56186 6932
17 18 -6.42938 7516 13 -2.56695 8608
20 -2.24833 5423 14 -4.47655 8894
1198 I 21 I -8.31241 1677 I 15I -8.25596 4368
23 -3.23959 2219
$i 40 -2.94642 8547
58 -8.02845 0851
50 I 78 I ~2.73919 2285 63)-1.23502 1944
100 (186)-6.68307 9463 (156)-2.65595 5830 (116)-1.79971 3983
-3 +3.72067 8486
2 -3 +9.49950 2019
7 -3 -2.48574 3224
-3 I -9.87236 3502
98 -4)+8.07441 4285
:10
12
13
14
-3 t6.25864 1510
112 -3 1-6.78002 3635
-3 -8.49604 9309
:; -3 t3.80640 6377
II
19 -3 1c9.90441 9669
20 -5 t5.63172 9379
-2 -4.19000
-5 -2.24122
Cl.37595
t4.97879 7221
6812
0150
3130 -3 -5.41292 9349
zoo -4 -7.04842 0407
50 -2 I tl.07478 2297
100 (85)-8.57322 6309 (+18)-1.12569 2891 (-2)-2.29838 5049
BESSEL FUNCTIONS OF FRACTIONAL ORDER 467
ZEROS OF BESSEL FUNCTIONS OF HALF-INTEGER ORDER Table 10.6
J&,)=0 Y”(V”, ,I=0
Y s Jib, s) y, ,~ (- l)“+‘T(!/“, s)
J., 8 ?h,, (-l)“+lY~~w,.,~) y s h,, .JX.i, ,)
l/2 1 3.141593 -0.45015 82 1.570796 -0.63661 98 15/2 1 11.657032 -0.20550 46 9.457882 +0.20754 83
2 15.431289 +0.19008 87 13.600629 -0.19801 01
32 9.424778
6.283185 +0.31830
-0.25989 89
99 4.712389
7.853982 +0.36755
-0.28470 26
50 3 18.922999 -0.17582 99 17.197777 +0.18264 01
4 12.566370 +0.22507 91 10.995574 +0.24061 97 4 22.295348 +0.16402 38 20.619612 -0.16964 44
5 15.707963 -0.20131 68 14.137167 -0.21220 66 5 23.955267 +0.15890 14
6 18.849556 +0.18377 63 17.278760 +0.19194 81
7 21.991149 -0.17014 38 20.420352 -0.17656 66
8 23.561945 +0.16437 45
17/2 1 12.790782 -0.19382 82 10.529989 -0.19361 38
2 16.641003 +0.18155 15 14.777175 +0.18810 92
3 20.182471 -0.16922 10 18.434529 -0.17517 27
4 23.591275 +0.15870 04 21.898570 +0.16373 75
3/2 1 4.493409 -0.36741 35 2.798386 +0.44914 84
: 10'904122
7 725252 +0 28469
-0'24061 69
20 6.121250
9.317866 +0.25989
-0.31827 33
37
4 14:066194 +0:21220 57 12.486454 -0.22507 76
5 17.220755 -0.19194 77 15.644128 +0.20131 63 19/2 1 13.915823 -0.18376 12 11.597038 +0.18186 42
6 20.371303 +0.17656 64 18.796404 -0.18377 61 2 17.838643 +0.17398 80 15.942945 -0.17944 10
7 23.519452 -0.16437 44 21.945613 +0.17014 37 3 21.428487 -0.16326 17 19.658369 +0.16849 33
4 24.873214 +0.15383 84 23.163734 -0.15837 45
512 ; 5.763459 -0.31710 58 3.959528 -0.36184 68 21/2 1 15.033469 -0.17496 82 12.659840 -0.17179 22
9.095011 +0.25973 30 7.451610 +0.28430 75 2 19.025854 +0.16722 59 17.099480 +0.17176 97
3 12.322941 -0.22503 59 10.715647 -0.24053 93 3 22.662721 -0.15785 09 20.870973 -0.16247 13
4 15.514603 +0.20130 14 13.921686 +0.21218 15 4 24.416749 +0.15347 56
5 18.689036 -0.18376 96 17.103359 -0.19193 81
6 21.853874 +0.17014 05 20.272369 +0.17656 19
7 23.433926 -0.16437 21
23/2 1 16.144743 -0.16720 39 13.719013 +0.16304 06
2 20.203943 +0.16113 25 18.247994 -0.16491 86
3 23.886531 -0.15290 87 22.073692 +0.15700 50
13/2 1 10.512835 -0.21926 48 8.379626 -0.22441 70 37/2 1 23.797849 -0.13037 81 21.062860 -0.12321 13
2 14.207392 +0.19983 04 12.411301 +0.20946 65
3 17.647975 -0.18321 82 15.945983 -0.19106 59
4 20.983463 +0.16988 82 19.324820 +0.17619 60
5 24.262768 -0.15902 21 22.628417 -0.16419 26 39/2 1 24.878005 -0.12669 81 22.104735 +0.11937 34
Values to greater accuracy and over a wider range are given in [10.31].
From National Bureau of Standards, Tables of spherical Bessel functions, ~01s. I, II. Columbia Univ.
Press, New York, N.Y., 1947 (with permission).
468 BESSEL FUNCTIONS OF FRACTIONAL ORDER
13/2 1 8.040535 +0.345649 10.391621 -0.277420 37/2 1 20.663347 to.249423 23.668335 -0.205855
2 12.335631 -0.245384 14.151399 +0.224513
3 15.901023 +0.209127 17.610124 -0.197009
20.954335 +0.178651
54 22:602185
19 291967 ;:::%;: 24.238863 -0.165043 39/2 1 21.700865 to.245275 24.747606 to.202629
-Values to greater accuracy and over a wider range are given in [10.31].
From National Bureau of Standards, Tables of spherical Bessel functiomqvols. I, II. Columbia Univ.
Press, New York, N.Y., 1947 (with permission).
BESSEL FUNCTIONS OF FRACTIONAL ORDER 469
MODIFIED SPHERICAL BESSEL FUNCTIONS--ORDERS 0, 1 AND 2 Table 10.8
[
(-;)3
I c 1
(-;)4
[(-y 1
in(x)= d ; ,/XIn+*(x)
Compiled from C. W. Jones, A short table for the Bessel functions Zn++(z), (~/T)K~+P),
Cambridge Univ. Press, Cambridge, England, 1952 (with permission).
BESSEL FUNCTIONS OF FRACTIONAL ORDER 471
MODIFIED SPHERICAL BESSEL FUNCTIONS-ORDERS 9 AND 10 Table 10.9
II(-4
-4 1.56545
1.43285
1.30831
1.19157
1.08240
II(-4
-4 2.35684
1.85569
1.70632
2.18075
2.01376
Table 10.9
MODIFIED SPHERICAL BESSEL FUNCTIONS-ORDERS 9 AND 10
1
fQ@) flO(@ gQ @> glow
o.;io 1.10630 573 1.21411 149 0.65502 364 0.56777 303
0.095 1.08238 951 1.17260 877 0.68557 030 0.60351 931
0.090 1.06167 683 1.13650 462 0.71563 676 0.63926 956
0.085 1.04394 741 1.10534 464 0.74502 124 0.67473 612
0.080 1.02899 406 1.07872 041 0.77352 114 0.70961 813
0.075 1.01661 895 1.05626 085 0.80093 667 0.74360 745 13
0.070 1.00662 998 1.03762 412 0.82707 483 0.77639 538
0.065 0.99883 728 1.02248 982 0.85175 354 0.80768 018
0.060 0.99304 985 1.01055 159 0.87480 587 0.83717 510
0.055 0.98907 251 1.00151 009 0.89608 425 0.86461 675
0.050 0.98670 320 0.99506 643 0.91546 455 0.88977 340
0.045 0.98573 080 0.99091 634 0.93284 978 0.91245 301
0.040 0.98593 357 0.98874 519 0.94817 344 0.93251 041
0.035 0.98707 842 0.98822 421 0.96140 216 0.94985 358
0.030 0.98892 100 0.98900 824 0.97253 769 0.96444 830
0.025 0.99120 680 0.99073 519 0.98161 804 0.97632 121
0.020
0.015
0.99367
0.99605
323
259
0.99302
0.99549
746
538
0.98871
0.99394
764
654
0.98556
0.99231
077
623
s”o”
0.010
0.005
0.99807
0.99947
595
760
0.99774
0.99937
259
316
0.99744
0.99939
863
894
0.99679
0.99925
434
415
1El
200
0.000 1.00000 000 1.00000 000 1.00000 000 1.00000 000 Co
[ (-;I4
1 [
C-$3
I c 1
'y3
&%119(x) =fQ(x)ez-4k-1
-5
@z121(x) =flo(x)e+55~-1
Y
@&&9(x) = gQ(x)e-z+452-1
-5
I- 43
7I 3.58484
6
5 8.59805 8301
7.09794
8.24936
5.40595 3854
9394
4523
2086 I- 23I 2.56465
1 5.33186 8690
7.83315
2.14704
7.45140 3294
9422
4364
1251
II II
(- 3j1.20941 3702
- 46
5 2.52325
8.74937 7454
1.46862
4.87152 7470
7330
8858
tI
- 10
12
11
9 8.12182
8 2.82275 9483
5.57826
4.11114
7.01394 9636
2138
8275
3211
- 18
24 1.60182
21
22
19 3.16500 2725
4.57312
5.29060
1.23512 3796
2995
7153
0086
I-I
- 26
61 7.71514
81
43 3.65054 7565
1.55685
5.65589 5122
5412
8686
II
- 20
66 8.37672
34
49 6.21921 8478
1.74298
4.17042 9214
6176
4440
1I - 12
33 9.70826
22
46 1.63577 9664
3.64245
6.36889 1994
3001
6441
II
(
x=10
3)1.10132 3287
2 9.91190
3.91520 9985
5.89207
8.03965 9633
4237
9640
I 19I 4.23682
x=50
5.18470 4844
5.08101
4.59302
4.87984 1073
6934
1418
5529
5
7
:10
II
40 1.15601
41 8.55360 0470
9.36222
1.01451
1.08840 6574
3425
8456
0111
12
13
14
II II
- 2
3 2.23450
4 4.23421
1.66914
1.02488 9437
6.26543 6979
3574
7720
8379
II 18 3.40719
1.17158
1.70426
2.43274 1747
4.68149 8938
7856
6870
3423
20
30
5400
-125 2.37154
-31
--21 2.81471
5.88991 3577
1.22928 6154
5830
4325
II +17
12
15
9 2.00489
7.90430
5.67659 4104
7.34905 8633
3929
8082
(40)1.64074 7551
Table 10.10
II
- 1
x=1
i02 5.77863
4.04504 6749
1.53711
2.13809
1.15572 7350
5724
7375
5597
’
I-
x=2
111.06292 0829
x=5
Ii - 3i 2.11678
2.54014
6.18102 a479
3.64087 6184
2359
6175
(- 211.22943 0749
3I 1.40478 6594
4 1.56063 6427
5 2.04287 5221
6 3.07991 9195
7 5.25629 1384
II II
1 1.11621 7817
::: 1 4.96235 0604
12 2 2.39430 3059
3 1.24677 5036
:i 14
13
11
10
9 1.00177
1.21727
4.86068
2.10898
3.29151 4384
9443
1836
5282
5179 9 4.79605
a
7
6
5 4.52287
9.50401 1652
6.99881
5.56068 2999
9354
0749
7078 I 3I 6.97201 5499
I
15I 9.55756 6814 4 4.16844 6493
20
30
17 2.96613
la 9.79781
20 3.43219
22 1.27089
7227
0417
9783
3701
23)4.95991 7633
12
16I 2.15637
14
13
11 6.27234 9105
1.16395
3.57187
1.38508
( 17)2.27598 6819
6139
7368
6330
0704 I 5 2.65415
6 1.79342
7 1.28194
7I 9.66570
8)7.66744 6235
18)7.97979 3303
6981
a072
1220
7838
40
50
100 ( a7)1.04451 3645 (156)4.08894 4237 (116)2.49323 8041
x=10 x=50
-6 7.13140 4291
-6 7.84454 4720
-6 9.48476 7707
-5 1.25869 2857
I-5 I 1.82956 1771 I -24 I 6.05934
6.18053
6.82355
7.38547 6353
6.43017 1115
5506
8350
3280
-46 6.78387 0523
9.90762 2914
II -46
-46
-46
-46
-45
7.20097
7.71999
a.35897
9.14102
1.00957
0973
6750
0485
1732
6461
::
12
13
14
15
16
2.50109
6.74327
1.93592
5.90133
1.90497
6.49556
2290
4558
7868
2701
9270
9007
3.63628 4300
II -45
-45
-45
-45
1.12611
1.26858
1.44325
1.65826
3230
2504
8856
2396
2.33403 5699
:; a.81868 1848
19 3.49631 5854
t-22)3.67748 3017 (-45)4.68935 4218
fi
it
100 (a5)8.14750 7624 (+12)5.97531 1344 (-25)1.48279 6529
BESSEL FUNCTIONS OF FRACTIONAL ORDER 475
AIRY FUNCTIONS Table 10.11
x Ai Ai’ Bi(x) Bi’(x) IX Ai Ai’ Bi(x) Bi’ (x)
0.00 0.35502 805 -0.25881 940 0.61492 663 0.44828 836 0.50 0.23169 361 -0.22491 053 0.85427 704 0.54457 256
0.01 0.35243 992 -0.25880 174 0.61940 962 0.44831 926 0.51 0.22945 031 -0.22374 617 0.85974 431 0.54890 049
0.02 0.34985 214 -0.25874 909 0.62389 322 0.44841 254 0.52 0.22721 872 -0.22257 027 0.86525 543 0.55334 239
0.03 0.34726 505 -0.25866 197 0.62837 808 0.44856 911 0.53 0.22499 894 -0.22138 322 0.87081 154 0.55789 959
0.04 0.34467 901 -0.25854 090 0.63286 482 0.44878 987 0.54 0.22279 109 -0.22018 541 0.87641 381 0.56257 345
0.05 0.34209 435 -0.25838 640 0.63735 409 0.44907 570 0.55 0.22059 527 -0.21897 720 0.88206 341 0.56736 532
0.06 0.33951 139 -0.25819 898 0.64184 655 0.44942 752 0.56 0.21841 158 -0.21775 898 0.88776 152 0.57227 662
0.07 0.33693 047 -0.25797 916 0.64634 286 0.44984 622 0.57 0.21624 012 -0.21653 112 0.89350 934 0.57730 873
0.08 0.33435 191 -0.25772 745 0.65084 370 0.45033 270 0.58 0.21408 099 -0.21529 397 0.89930 810 0.58246 311
0.09 0.33177 603 -0.25744 437 0.65534 975 0.45088 787 0.59 0.21193 427 -0.21404 790 0.90515 902 0.58774 120
0.10 0.32920 313 -0.25713 042 0.65986 169 0.45151 263 0.60 0.20980 006 -0.21279 326 0.91106 334 0.59314 448
0.11 0.32663 352 -0.25678 613 0.66438 023 0.45220 789 0.61 0.20767 844 -0.21153 041 0.91702 233 0.59867 447
0.12 0.32406 751 -0.25641 200 0.66890 609 0.45297 457 0.62 0.20556 948 -0.21025 970 0.92303 726 0.60433 267
0.13 0.32150 538 -0.25600 854 0.67343 997 0.45381 357 0.63 0.20347 327 -0.20898 146 0.92910 941 0.61012 064
0.14 0.31894 743 -0.25557 625 0.67798 260 0.45472 582 0.64 0.20138 987 -0.20769 605 0.93524 011 0.61603 997
0.15 0.31639 395 -0.25511 565 0.68253 473 0.45571 223 0.65 0.19931 937 -0.20640 378 0.94143 066 0.62209 226
0.16 0.31384 521 -0.25462 724 0.68709 709 0.45677 373 0.66 0.19726 182 -0.20510 500 0.94768 241 0.62827 912
0.17 0.31130 150 -0.25411 151 0.69167 046 0.45791 125 0.67 0.19521 729 -0.20380 004 0.95399 670 0.63460 222
0.18 0.30876 307 -0.25356 898 0.69625 558 0.45912 572 0.68 0.19318 584 -0.20248 920 0.96037 491 0.64106 324
0.19 0.30623 020 -0.25300 013 0.70085 323 0.46041 808 0.69 0.19116 752 -0.20117 281 0.96681 843 0.64766 389
0.20 0.30370 315 -0.25240 547 0.70546 420 0.46178 928 0.70 0.18916 240 -0.19985 119 0.97332 866 0.65440 592
0.21 0.30118 218 -0.25178 548 0.71008 928 0.46324 026 0.71 0.18717 052 -0.19852 464 0.97990 703 0.66129 109
0.22 0.29866 753 -0.25114 067 0.71472 927 0.46477 197 0.72 0.18519 192 -0.19719 347 0.98655 496 0.66832 121
0.23 0.29615 945 -0.25047 151 0.71938 499 0.46638 539 0.73 0.18322 666 -0.19585 798 0.99327 394 0.67549 810
0.24 0.29365 818 -0.24977 850 0.72405 726 0.46808 147 0.74 0.18127 478 -0.19451 846 1.00006 542 0.68282 363
0.25 0.29116 395 -0.24906 211 0.72874 690 0.46986 119 0.75 0.17933 631 -0.19317 521 1.00693 091 0.69029 970
0.26 0.28867 701 -0.24832 284 0.73345 477 0.47172 554 0.76 0.17741 128 -0.19182 851 1.01387 192 0.69792 824
0.27 0.28619 757 -0.24756 115 0.73818 170 0.47367 549 0.77 0.17549 975 -0.19047 865 1.02088 999 0.70571 121
3.28 0.28372 586 -0.24677 753 0.74292 857 0.47571 205 0.78 0.17360 172 -0.18912 591 1.02798 667 0.71365 062
0.29 0.28126 209 -0.24597 244 0.74769 624 0.47783 623 0.79 0.17171 724 -0.18777 055 1.03516 353 0.72174 849
0.30 0.27880 648 -0.24514 636 0.75248 559 0.48004 903 0.80 -0.16984 632 -0.18641 286 1.04242 217 0.73000 690
0.31 0.27635 923 -0.24429 976 0.75729 752 0.48235 148 0.81 0.16798 899 -0.18505 310 1.04976 421 0.73842 795
0.32 0.27392 055 -0.24343 309 0.76213 292 0.48474 462 0.82 0.16614 526 -0.18369 153 1.05719 128 0.74701 380
0.33 0.27149 064 -0.24254 682 0.76699 272 0.48722 948 0.83 0.16431 516 -0.18232 840 1.06470 504 0.75576 663
0.34 0.26906 968 -0.24164 140 0.77187 782 0.48980 713 0.84 0.16249 870 -0.18096 398 1.07230 717 0.76468 865
0.35 0.26665 787 -0.24071 730 0.77678 917 0.49247 861 0.85 0.16069 588 -0.17959 851 1.07999 939 0.77378 215
0.36 0.26425 540 -0.23977 495 0.78172 770 0.49524 501 0.86 0.15890 673 -0.17823 223 1.08778 340 0.78304 942
0.37 0.26186 243 -0.23881 481 0.78669 439 0.49810 741 0.87 0.15713 124 -0.17686 539 1.09566 096 0.79249 282
0.38 0.25947 916 -0.23783 731 0.79169 018 0.50106 692 0.88 0.15536 942 -0.17549 823 1.10363 385 0.80211 473
0.39 0.25710 574 -0.23684 291 0.79671 605 0.50412 463 0.89 0.15362 128 -0.17413 097 1.11170 386 0.81191 759
0.40 0.25474 235 -0.23583 203 0.80177 300 0.50728 168 0.90 0.15188 680 -0.17276 384 1.11987 281 0.82190 389
0.41 0.25238 916 -0.23480 512 0.80686 202 0.51053 920 0.91 0.15016 600 -0.17139 708 1.12814 255 0.83207 615
0.42 0.25004 630 -0.23376 259 0.81198 412 0.51389 833 0.92 0.14845 886 -0.17003 090 1.13651 496 0.84243 695
0.43 0.24771 395 -0.23270 487 0.81714 033 0.51736 025 0.93 0.14676 538 -0.16866 551 1.14499 193 0.85298 891
0.44 0.24539 226 -0.23163 239 0.82233 167 0.52092 614 0.94 0.14508 555 -0.16730 113 1.15357 539 0.86373 470
0.45 0.24308 135 -0.23054 556 0.82755 920 0.52459 717 0.95 0.14341 935 -0.16593 797 1.16226 728 0.87467 704
0.46 0.24078 139 -0.22944 479 0.83282 397 0.52837 457 0.96 0.14176 678 -0.16457 623 1.17106 959 0.88581 871
0.47 0.23849 250 -0.22833 050 0.83812 705 0.53225 956 0.97 0.14012 782 -0.16321 611 1.17998 433 0.89716 253
0.48 0.23621 482 -0.22720 310 0.84346 952 0.53625 338 0.98 0.13850 245 -0.16185 781 1.18901 352 0.90871 137
0.49 0.23394 848 -0.22606 297 0.84885 248 0.54035 729 0.99 0.13689 066 -0.16050 153 1.19815 925 0.92046 818
0.50 0.23169 361 -0.22491 053 0.85427 704 0.54457 256 1.00 0.13529 242 -0.15914 744 1.20742 359 0.93243 593
[‘-y] [‘-f)4] [‘-;‘“I [c-y] [‘y72] [‘-pl] [‘-y-j [‘a’“]
1.0 1.310371 0.536489 0.616764 0.605068 ;.;;;qO;: 0.25 3.301927 0.555456 0.575908 0.576635 0.548255
0.9 1.405721 0.538618 0.614022 0.601782 0.20 3.831547 0.557058 0.573135 0.574320 0.551930
1.520550 0.540844 0.610309 0.598372 0:501859 0.15 4.641589 0.558724 0.570636 0.571927 0.555296
0.10 6.082202 0.560462 0.568343 0.569448 0.558428
0:6
E 1.662119
1.842016 0.545636
0.543180 0.599723
0.605543 0.594823
0.591120 ;.:1";;;;
. 0.05 9.654894 0.562280 0.566204 0.566873 0.561382
Om5[(-i)7]
2.080084 [(-45)2]
0.548230 [(-911
0.593015 [(-45)2]
0.587245 F;‘j
. 0*0°
From J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical
Tables, Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 (with
permission).
476 BESSEL FUNCTIONS OF FRACTIONAL ORDER
0.10 0.38084 867 -0.25695 811 0.56999 904 0.45121 336 0.60 0.49484 953 -0.17736 260 0.32879 184 0.52540 115
0.11 0.38341 628 -0.25655 685 0.56548 397 0.45180 945 0.61 0.49660 821 -0.17436 341 0.32352 796 0.52737 438
Oil2 0;38597 961 -0;25611 443 0.56096 268 0.45245 712 0.62 0.49833 b59 -0.17130 392 0.31824 435 0.52934 780
0.13 0.38853 843 -0.25563 033 0.55643 466 0.45315 546 0.63 0.50003 408 -0.16818 399 0.31294 101 0.53132 022
0.14 0.39109 213 -0.25510 406 0.55189 940 0.45390 355 0.64 0.50170 007 -0.16500 345 0.30761 795 0.53329 046
0.15 0.39364 037 -0.25453 511 0.54735 642 0.45470 047 0.65 0.50333 395 -0.16176 218 0.30227 521 0.53525 733
O.lb 0.39618 269 -0;25392 297 0.54280 523 0.45554 530 0.66 0.50493 511 -0.15846 007 0.29691 282 0.53721 964
0.17 0.39871 868 -0.25326 716 0.53824 536 0.45643 713 0.67 0.50650 295 -0.15509 701 0.29153 084 0.53917 618
0.18 0.40124 789 -0.25256 716 0.53367 b34 0.45737 503 0.68 0.50803 685 -0.15167 290 0.28612 932 0.54112 575
0.19 0.40376 987 -0.25182 250 0.52909 771 0.45835 806 0.69 0.50953 620 -0.14818 768 0.28070 835 0.54306 714
0.20 0.40628 419 -0.25103 267 0.52450 903 0.45938 529 0.70 0.51100 040 -0.14464 129 0.27526 801 0.54499 912
0.21 0.40879 038 -0.25019 720 0.51990 986 0.46045 578 0.71 0.51242 882 -0.14103 366 0.26980 840 0.54692 048
0.22 0.41128 798 -0.24931 559 0.51529 977 0.46156 860 0.72 0.51382 087 -0.13736 479 0.26432 964 0.54883 000
0.23 0.41377 653 -0.24838 737 0.51067 835 0.46272 279 0.73 0.51517 591 -0.13363 464 0.25883 185 0.55072 642
0.24 0.41625 557 -0.24741 206 0.50604 518 0.46391 740 0.74 0.51649 336 -0.12984 322 0.25331 516 0.55260 852
0.25 0.41872 461 -0.24638 919 0.50139 987 0.46515 148 0.75 0.51777 258 -0.12599 055 0.24777 973 0.55447 506
0.26 0.42118 319 -0.24531 828 0.49674 203 0.46642 408 0.76 0.51901 296 -0.12207 665 0.24222 571 0.55632 480
0.27 0.42363 082 -0.24419 888 0.49207 127 0.46773 423 0.77 0.52021 390 -0.11810 157 0.23665 329 0.55815 647
0.28 0.42606 701 -0.24303 053 0.48738 722 0.46908 095 0.78 0.52137 479 -0.11406 538 0.2310b 265 0.55996 884
0.29 0.42849 126 -0.24181 276 0.48268 953 0.47046 327 0.79 0.52249 501 -0.10996 815 0.22545 398 0.56176 063
0.30 0.43090 310 -0.24054 513 0.47797 784 0.47188 022 0.80 0.52357 395 -0.10580 999 0.21982 751 0.56353 059
0.31 0.43330 200 -0.23922 719 0.47325 181 0.47333 081 0.81 0.52461 101 -0.10159 101 0.21418 345 0.56527 745
0.32 0.43568 747 -0.23785 851 0.46851 112 0.47481 405 0.82 0.52560 557 -0.09731 134 0.20852 204 0.56699 994
0.33 0.43805 900 -0.23643 865 0.46375 543 0.47632 895 0.83 0.52655 703 -0.09297 U3 0.20284 354 0.56869 679
0.34 0.44041 607 -0.23496 718 0.45898 443 0.47787 450 0.84 0.52746 479 -0.08857 055 0.19714 820 0.57036 671
0.35 0.44275 817 -0.23344 368 0.45419 784 0.47944 970 0.85 0.52832 824 -0.08410 979 0.19143 630 0.57200 845
0.36 0.44508 477 -0.23186 773 0.44939 534 0.48105 354 0.86 0.52914 678 -0.07958 904 0.18570 813 0.57362 071
0.37 0.44739 535 -0.23023 893 0.44457 667 0.48268 500 0.87 0.52991 982 -0.07500 854 0.17996 399 0.57520 720
0;38 0;44968 937 -0;22855 687 0.43974 156 0.48434 307 0;88 0.53064 676 -0.07036 852 0.17420 419 0.57675 lb5
0.39 0.45196 631 -0.22682 llb 0.43488 973 0.48602 670 0.89 0.53132 700 -0.06566 925 0.16842 906 0.57826 777
0.40 0.45422 561 -0.22503 141 0.43002 094 0.48773 486 0.90 0.53195 995 -0.06091 100 0.16263 895 0.57974 926
0.41 0.45646 675 -0.22318 723 0.42513 495 0.48946 652 0.91 0.53254 502 -0.05609 407 0.15683 420 0.58119 484
0.42 0.45868 918 -0.22128 826 0.42023 153 0.49122 062 0.92 0.53308 163 -0.05121 879 0.15101 518 0.5R7hn 171
0.43 i46089 233 -0;21933 412 0.41531 047 0;49299 611 0.93 0.53356 920 -0.04628 549 0.14518 226 0.58397 309
0.44 0.46307 567 -0.21732 447 0.41037 154 0.49479 193 0.94 0.53400 715 -0.04129 452 0.13933 585 0.58530 317
0.45 0.46523 864 -0.21525 894 0.40541 457 0.49660 702 0.95 0.53439 490 -0.03624 628 0.13347 634 0.58659 217
0.46 0.46738 066 -0.21313 721 0.40043 934 0.49844 031 0.96 0.53473 189 -0.03114 116 0.12760 415 0.58783 879
0.47 0.46950 119 -0.21095 893 0.39544 570 0.50029 070 0.97 0.53501 754 -0.02597 957 0.12171 971 0.5R904 174
0;48 0;47159 965 -0;20872 379 0;39043 348 0.50215 713 i;ii 0.53525 129 -0.02076 197 0.11582 346 0.59019 973
0.49 0.47367 548 -0.20643 147 0.38540 251 0.50403 850 0.99 0.53543 259 -0.01548 880 0.10991 587 0.59131 145
0.50 0.47572 809 -0.20408 lb7 0.38035 266 0.50593 371 1.00 0.53556 088 -0.01016 057 0.10399 739 0.59237 563
[ C-f’31 [t-j”] [ C-j)7 [t-i’“] [C-:)7] [‘-j’“] [C-t’“] [C-j+]
BESSEL FUNCTIONS OF FRACTIONAL ORDER 477
AIRY FUNCTIONS Table 10.11
x Ai Ai’( -x) Bi( -x) Bi’(-x) x Ai( -x) Ai’( -n) Bi( -x) Bi’( -x)
1.0 0.53556 088 -0.01016 057 +0.10399 739 0.59237 563 5.5 +0.01778 154 0.86419 722 -0.36781 345 +0.02511 158
1.1 0.53381 051 +0.04602 915 +0.04432 659 0.60011 970 5.6 -0.06833 070 0.85003 256 -0.36017 223 -0.17783 760
1.2 0.52619 437 0.10703 157 -0.01582 137 0.60171 016 5.7 -0.15062 016 0.78781 722 -0.33245 825 -0.37440 903
1.3 0.51227 201 0.17199 181 -0.07576 964 0.59592 975 5.8 -0.22435 192 0.67943 152 -0.28589 021 -0.55300 203
1.4 0.49170 018 0.23981 912 -0.13472 406 0.58165 624 5.9 -0.28512 278 0.52962 857 -0.22282 969 -0.70247 952
1.5 0.46425 658 0.30918 697 -0.19178 486 0.55790 810 6.0 -0.32914 517 0.34593 549 -0.14669 838 -0.81289 879
1.6 0.42986 298 0.37854 219 -0.24596 320 0.52389 354 6.1 -0.35351 168 +0.13836 394 -0.06182 255 -0.87622 530
1;7 0;38860 704 0.44612 455 -0.29620 266 0.47906 134 6.2 -0.35642 107 -0.08106 856 +0.02679 081 -0.88697 896
1.8 0.34076 156 0.50999 763 -0.34140 583 0.42315 137 6.3 -0.33734 765 -0.29899 161 0.11373 701 -0.84276 110
1.9 0.28680 006 0.56809 172 -0.38046 588 0.35624 251 6.4 -0.29713 762 -0.5014: 985 0.19354 136 -0.74461 387
2.0 0.22740 743 0.61825 902 -0.41230 259 0.27879 517 6.5 -0.23802 030 -0.67495 249 0.26101 266 -0.59717 067
2.1 0.16348 451 0.65834 069 -0.43590 235 0.19168 563 6.6 -0.16352 646 -0.80711 925 0.31159 995 -0.40856 734
2.2 0.09614 538 0.68624 482 -0.45036 098 +0.09622 919 6.7 -0.07831 247. -0.88790 797 0.34172 774 -0.19009 878
2.3 +0.02670 633 0.70003 366 -0.45492 823 -0.00581 106 6.8 +0.01210 452 -0.91030 401 0.34908 418 +0.04437 678
2.4 -0.04333 414 0.69801 760 -0.44905 228 -0.11223 237 6.9 0.10168 800 -0.87103 106 0.33283 784 0.27926 391
2.5 -0.11232 507 0.67885 273 -0.43242 247 -0.22042 015 7.0 0.18428 084 -0.77100 817 0.29376 207 0.49824 459
2.6 -0.17850 243 0.64163 799 -0.40500 828 -0.32739 717 7.1 0.25403 633 -0.61552 879 0.23425 088 0.68542 058
2.7 -0.24003 811 0.58600 720 -0.36709 211 -0.42989 534 7.2 0.30585 152 -0.41412 428 0.15821 739 0.82650 634
2.8 -0.29509 759 0.51221 098 -0.31929 389 -0.52445 040 7.3 0.33577 037 -0.18009 580 +0.07087 411 0.90998 427
2.9 -0.34190 510 0.42118 281 -0.26258 500 -0.60751 829 7.4 0.34132 375 +0.07027 632 -0.02159 652 0.92812 809
3.0 -0.37881 429 0.31458 377 -0.19828 963 -0.67561 122 7.5 0.32177 572 0.31880 951 -0.11246 349 0.87780 228
3.1 -0.40438 222 0.19482 045 -0.12807 165 -0.72544 957 7.6 0.27825 023 0.54671 882 -0.19493 376 0.76095 509
3.2 -0.41744 342 +0.06503 115 -0.05390 576 -0.75412 455 7.7 0.21372 037 0.73605 242 -0.26267 007 0.58474 045
3.3 -0;41718 094 -0.07096 362 +0.02196 800 -0.75926 518 7.8 0.13285 154 0.87115 540 -0.31030 057 0.36122 930
3.4 -0.40319 048 -0.20874 905 0.09710 619 -0.73920 163 7.9 +0.04170 188 0.94004 300 -0.33387 856 +0.10670 215
3.5 -0.37553 382 -0.34344 343 0.16893 984 -0.69311 628 8.0 -0.05270 505 0.93556 094 -0.33125 158 -0.15945 050
3.6 -0.33477 748 -0.46986 397 0.23486 631 -0.62117 283 8.1 -0.14290 815 0.85621 859 -0.30230 331 -0.41615 664
3.7 -0.28201 306 -0.58272 780 0.29235 261 -0.52461 361 8.2 -0.22159 945 0.70659 870 -0.24904 019 -0.64232 293
3.8 -0.21885 598 -0.67688 257 0.33904 647 -0.40581 592 8.3 -0.28223 176 0.49727 679 -0.17550 556 -0.81860 044
3.9 -0.14741 991 -0.74755 809 0.37289 058 -0.26829 836 8.4 -0.31959 219 +0.24422 089 -0.08751 798 -0.92910 958
4.0 -0.07026 553 -0.79062 858 0.39223 471 -0.11667 057 8.5 -0.33029 024 -0.03231 335 +0.00775 444 -0.96296 917
4.1 +0.00967 698 -0.80287 254 0.39593 974 +0.04347 872 8.6 -0.31311 245 -0.30933 027 0.10235 647 -0.91547 918
4.2 0.08921 076 -0.78221 561 0.38346 736 0.20575 691 8.7 -0.26920 454 -0.56297 685 0.18820 363 -0.78882 623
4.3 0.16499 781 -0.72794 081 0.35494 906 0.36320 468 8.8 -0.20205 445 -0.77061 301 0.25778 240 -0.59221 371
4.4 0.23370 326 -0.64085 018 0.31122 860 0.50858 932 8.9 -0.11726 631 -0.91289 276 0.30483 241 -0.34136 475
4.5 0.29215 278 I -0.52336 253 0.25387 266 0.63474 477 9.0 -0.02213 372 -0.97566 398 0.32494 732 -0.05740 051
4.6 0.33749 598 -0.37953 391 0.18514 576 0.73494 444 9.1 +0.07495 989 -0.95149 682 0.31603 471 +0.23484 379
4.7 0.36736 748 -0.21499 018 0.10794 695 0.80328 926 9.2 0.16526 800 -0.84067 107 0.27858 425 0.50894 402
4.8 0.38003 668 -0.03676 510 +0.02570 779 0.83508 976 9.3 0.24047 380 -0.65149 241 0.21570 835 0.73928 028
4.9 0.37453 635 +0.14695 743 -0.05774 655 0.82721 903 9.4 0.29347 756 -0.39986 237 0.13293 876 0.90348 537
5.0 0.35076 101 0.32719 282 -0.13836 913 0.77841 177 9.5 0.31910 325 -0.10809 532 +0.03778 543 0.98471 407
5.1 0.30952 600 0.49458 600 -0.21208 913 0.68948 513 9.6 0.31465 158 +0.19695 044 -0.06091 293 0.97349 918
5.2 0.25258 034 0.63990 517 -0.27502 704 0.56345 898 9.7 0.28023 750 0.48628 629 -0.15379 421 0.86898 388
5.3 0.18256 793 0.75457 542 -0.32371 608 0.40555 694 9.8 0.21886 743 0.73154 486 -0.23186 331 0.67936 774
5.4 0.10293 460 0.83122 307 -0.35531 708 0.22307 496 9.9 0.13623 503 0.90781 333 -0.28738 356 0.42147 209
5.5 0.01778 154 0.86419 722 -0.36781 345 0.02511 158 10.0 0.04024 124 0.99626 504 -0.31467 983 0.11941 411
[q’“] [q’“] [‘-a”] [y] [(-93)4] [qy] [(-;)“I [(-;Lo)1]
A;Y-x)-x’ [gl(r) sin r-g,(r) cos I] Bi’(--x)=x’ [g,(r) cos r+g,(r) sin r]
3
,=yxz <r>=nearest integer to t.
478 BESSEL FUNCTIONS OF FRACTIONAL ORDER
zs
i.1”
0:2
5 Ai (t)
0.03421
0.06585
dt Ji Ai(-t)dtJi
0.00000 00 0.00000 00
15
01
-0.03679
-0.07615
54
70
Bi(t)dt
0.00000 00
C.06373
0.13199
67
45
s i Bi(-t)dl
0.00000 00
-0.05924
-0.11398
87
10
-0.11802 51 0.20487 68 -0.16411 57
2: 0.09497
0.12164 06
09 -0.16229 44 0.28256 70 -0.20952 89
0.36533 85 -0.25006 28 0.33331 97 -0.82151 82 -0.01617 86
E 0.16801
0.14595 79
33 -0.25736
-0.20880 07
95 0.45356 50 -0.28553 62 0.33332 27 -0.81897 90 +0.02038 99
0.54773 36 -0.31575 56 0.33332 50 -0.80797 96 0.05518 54
;*i 0.18795
0.20589 45
52 -0.30768
-0.35944 05
15 0.64845 82 -0.34052 58 0.33332 69 -0.78914 06 0.08625 18
0:9 0.22196 97 -0.41225 56 0.75649 64 -0.35966 27 0.33332 83 -0.76354 19 0.11181 25
1.0 0.23631 73 -0.46567 40 0.87276 91 -0.37300 50 0,33332 95 -0.73267 53 0.13038 11
0.99838 41 -0.38042 77 21" 0.33333 03 -0.69836 93 0.14086 00
:-: 0.24907
0.26037 33
12 -0.51918
-0.57224 94
05 1.13466 38 -0.38185 43 0.33333 10 -0.66268 96 0.14262 05
1:3 0.27034 09 -0.62421 79 1.28318 00 -0.37726 99 t: 0.33333 16 -0.62781 93 0.13555 73
1.4 0.27910 66 -0.67447 31 1.44579 42 -0.36673 34 6:4 0.33333 20 -0.59592 62 0.12011 15
0.28678 67 -0.72232 88 1.62470 81 -0.35038 81 0.33333 23 -0.56902 35 0.09726 08
0.29349 24 -0.76709 26 1.82252 33 -0.32847 24 0.33333 25 -0.54883 59 0.06847 29
2.04231 52 -0.30132 67 0.33333 27 -0.53667 65 0.03562 42
:*i 0.29932
0.30438 75
82 -0.80807
-0.84459 24
41 2.28772 12 -0.26939 97 0.93333 29 -0.53334 74 +0.00088 80
1:9 0.30876 29 -0.87602 06 2.56304 90 -0.23325 04 0.33333 30 -0.53906 98 -0.03340 40
2.0 0.31253 28 -0.90177 28 2.87340 83 -0.19354 74 7.0 0.33333 31 -0.55345 17 -0.06491 67
0.31577 11 -0.92135 09 -0.15106 46 0.33333 31 -0.57549 72 -0.09147 36
2: 0.31854 43 -0.93435 56 -0.10667 18 ::: 0.33333 32 -0.60365 96 -0.11121 47
0.32091 19 -0.94050 97 -0.06132 23 0.33333 32 -0.63593 60 -0.12273 90
2: 0.32292 74 -0.93967 67 -0.01603 45 ::: 0.33333 33 -0.66999 96 -0.12521 80
2.5 0.32463 80 -0.93187 78 +0.02812 94 7.5 0.33333 33 -0.70336 19 -0.11847 31
0.07009 01 -0.73355 34 -0.10300 57
::; 0.32730
0.32608 57
74 -0.89633
-0.91730 20
54 0.10878 06 ::; -0.75830 99 -0.07997 85
0.14317 88 -0.77575 13 -0.05114 35
::; 0.32919
0.32833 55
83 -0.83758
-0.86951 77
37 0.17234 20 ::9" -0.78453 65 -0.01872 22
3.0 0.32992 04 -0.80146 29 0.19544 25 -0.78398 26 +0.01475 64
3.1 0.33052 31 -0.76220 32 0.21180 21 if -0.77413 57 0.04664 84
0.33102 49 -0.72100 37 0.22092 49 8:2 -0.75578 55 0.07440 43
0.33144 15 -0.67915 91 0.22252 61 8.3 -0.73041 93 0.09577 a7
0.33178 65 -0.63802 56 0.21655 57 8.4 -0.70011 70 0.10902 22
3.5 0.33207 15 -0.59897 71 0.20321 50 -0.66739 21 0.11303 86
0.18296 47 -0.63499 08 0.10749 35
:*! 0.33230
0.33249 63
93 -0.56335
-0.53242 61
25 0.15652 33 -0.60566 32 0.09285 98
3:s 0.33265 76 -0.50730 05 0.12485 43 -0.58192 70 0.07039 64
3.9 0.33278 70 -0.48892 77 0.08914 28 -0.56584 22 0.04205 63
4.0 0.33289 27 -0.47800 75 +0.05076 01 -0.55881 97 +0.01033 04
4.1 0.33297 86 -0.47496 79 +0.01121 78 -0.56148 12 -0.02196 26
4.2 0.33304 84 -0.47992 95 -0.02788 79 -0.57358 51 -0.05192 24
-0.06494 00 -0.59403 00 -0.07682 93
i:: 0.33315
0.33310 07
50 -0.51269
-0.49268 28
51 -0.09837 02 -0.62093 76 -0.09439 a7
4.5 0.33318 76 -0.53908 35 -0.12673 04 -0.65181 01 -0.10300 27
4.6 0.33321 73 -0.57068 59 -0.14876 50 -0.68375 25 -0.10183 70
-0.16347 66 -0.71373 85 -0.09101 44
2; 0.33324
0.33326 11
02 -0.60606
-0.64358 63
51 -0.17018 59 -0.73889 84 -0.07157 33
4:9 0.33327 54 -0.68146 70 -0.16857 74 -0.75680 07 -0.04539 57
5.0 cLL~3zz;~ 4..~;2j
From J. C. P. Miller, The Airy integral, British Assoc. Adv. Sci. Mathematical
Tables Part-vol. B. Cambridge Univ. Press, Cambridge, England, 1946 and
F. W. J. Olver. The asymptotic expansion of Bessel functions of large order.
Philoa. Trans. Roy. Sot. London [A] 247, 32&368,1954(with permission).
11. Integrals of Bessel Functions
YTJDELL L. LUKE~
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 480
11.1. Simple Integrals of Bessel Functions . . . . . . . . . . . 480
11.2. Repeated Integrals of Jn(z) and K,(z) . . . . . . . . . . 482
11.3. Reduction Formulas for Indefinite Integrals . . . . . . . 483
11.4. Definite Integrals . . . . . . . . . . . . . . . . . . . 485
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 488
11.5. Use and Extension of the Tables . . . . . . . . . . . . 488
References . . . . . . . . . . . . . . . . . . . . . . . . . . 490
Table 11.1 Integrals of Bessel Functions . . . . . . . . . . . . . 492
= [l-W)ldl, m Y,wt, 8D
s 0 t s t
2=0(.1)5
e --z = [r,(t)-lldt:,J, ,& - K&h? 6-,-,
;
s0 t sz t
The author acknowledges the assistance of Geraldine Coombs, Betty Kahn, Marilyn
Kemp, Betty Ruhlman, and Anna Lee Samuels for checking formulas and developing
numerical examples, only a portion of which could be accommodated here.
1 Midwest Research Institute. (Prepared under contract with the National Bureau
of Standards.)
479
11. Integrals of Bessel Functions
Mathema ! ical Properties
S 0
atkL(t)dt S 0
z.zo(t)dt=xzo(x) ++rxr -Lo(z)Z,(~)+L,(~)Zo(~~
ZIP(x) =AI,(z) +Be*YrKr(z),,v=O,l
I
S 0
,
w4 2k
+‘k& (k!)2(2k+1)2
(92(P+~+1)>0)
11.1.2
y (Euler’s constant)=.57721 56649 . . .
S 0
&)&=a go J”+2k+,(z>(9’v>-l)
.- In this and all other integrals of 11.1, x is real
and positive although all the results remain valid
11.1.3 jyJ2.(t)dkSo'J,(t)dt-2 ~J,,,,(z) for extended portions of the complex plane unless
stated to the contrary.
11.1.4 g~~,+l(t)dt=1-Jo(z)-2 g J&) 11.1.10
s0 B
-t-z
11.1.5
Recurrence Relations
S K,(t)dt=;
0SzJo(t)dt+i;
J’Yo@)ch 0 0
Asymptotic Expansions
S0
aJ,+l(t)dt=
s
ogJ.4(t)dt-2J.(z) b>O> 11.1.11
Jm [J,(t)+iY,(l)]dt-(~)~ e’(‘-*‘*)
11.1.6
S 0
kl(t)dt=l--Jo(z) z
x[go (--)‘a2k+lx -ak-i+i 5kIO (-)ka2kx-ak 1
11.1.12
,,-r(k+f) 5 us+*)
r(i) r-0 2d r (3)
11.1.7
Soz~o(t)dt=x~o(x)+~~x{Ho(~)~(~)--~~~)~(~)~
11.1.13
480
INTEGRALS OF BESSEL FUNCTIONS 481
11.1.18 75x<=
11.1.14 de-" Xlo(t)dt-(2~)-~ go a,x-"
s0
where the ak are defined as in 11.1.12.
de”
S
2
OD
K~(t)dt=$~
I
(-)“ek(x/7)-“+c(x)
Ic(x)l<2xlO-7
11.1.15 de" (DK,(t)&- f ' PO (-)bkxwk k
sz 0 - 0 1.2533: 414
where the ak are defined as in 11.1.12. 1 0.11190 289
2 . 02576 646
Polynomial Approximations * 3 .00933 994
4 . 00417 454
11.1.16 81X<= 5 .00163 271
.00033 934
im [Jo(t)+iYo(t)Mt
s
=xTje*(z-*l*) go (-)kak(x/8)-w-1
11.1.19
7 1
s0
[e(x)1 12x10-9
=22-l PO
I (2;+3)w+2) -al)1 J2k+2(~)
=1-2x-V,(x)
ak bk
sz
11.1.17 Slxl=
[e(x)1<2x10-6
k di
0 . 39894 23
1 . 03117 34
2 . 00591 91
3 . 00569 56
4 -. 01148 58
5 * 01774 40
{ $(k+l) *&-ln i}
6 -. 00739 95
11.1.23
2 Approximation 11.1.16
Polynomial approximations
ls from A. J. M. Hitchcock.
to Bessel functions of order K,(t)&
--ia i* ODJ,(t)& ?r - Yo(t)dt
zero and one and to related functions, Math. Tables Aids
Comp. 11, 86-88 (1957) (with permission). S--iz t=z S z T----Z S
z t
482 INTEGRALS OF BESSEL JTUNCTIONS
ODK,(t)&
2 Qez - t =& (-lkd, (;)-kt4xl
sz
k dk
0 1.25331 41
1 0.50913 39
2 .32191 84
3 .26214 46
11.1.25 go(x)=2$ 2 S 4
5
6
.20601 26
11103 96
: 02724 00
11.1.26 aY2e2 "~O(W~
where
z t S 0
; 4&)kckx-k
11.1.31 52x2 00
2(k+l)ck+l= ~(~+l~2+~~c~-,+,c~-~ k fk
11.1.28 33/ze-z + S 0
[I,(t);lldt-(~,&f$o ckz-k
0
1
2
0.39893
13320
-: 04938
14
55
43
3 1.47800 44
where cnis defined as in 11.1.27. 4 -8.65560 13
Polynomial Approximations 5 28.12214 78
6 -48.05241 15
11.1.29 55x1 aJ 7 40.39473 40
8 -11.90943 95
S2- ~~=2gl(~)~Pox))_go(x>~Px)
t X2 2
Q
10
-3.51950
2.19454
09
64
k ak b
- 11.2.2
k:5999 2815 ki i1998 5629 Then
.10161 9385 .30485 8155
.13081 1585 52324 6341 11.2.3
.20740 4022 1: 03702 0112
: 28330
27902
.17891
.06622
0508
9488
5710
8328
1.95320
1.69980
1.43132
0. 59605
6413
3050
5684
4956
s,(z-t)‘-‘Jn(t)dt
S.&)=+) (9-?Kp>O)
.01070 2234 .10702 2336
INTEGRALS OF BESSEL FUNCTIONS 483
Recurrence Relations 11.3.2 Z”W a b
11.2.5
J”(Z), Y”(Z), H?(z), W’(z) 1 1
r(r--l)j”,l, t&4=2@--1M. nw
I”(Z) -1 1
K(z) 1 -1
-[(1-+--n2+~21f,-l, T&M
K&(z)=
S I
mK,(t)&, . . ., K&(z)=
S z
m Ki,-,(t)dt +(p-v-l)e- Pzz~-lZ”(z)-pe-pzz~Z”(z)
+P(2r--l)g,-l,“(~)+[~2-~(cc--1)219r-2.”~~>
11.2.10 +WCc+vh.v--1(z)
- e-’ ooabf &a
Ki&) =
S 0 cash’ t
Case 1: p2+ab=0, v= f (r-l)
+2”-Y2A)r(v) (v+3) ‘.
11.3. Reduction Formulas for Indefinite
Integrals 11.3.11
E
Let
11.3.1 6&)=
S ” e-%!@Z”(t)dt S 0
ef~t”Y”(t)dt-2v+l-eirZY+1 [Y”(Z) -iYy+l(z)]
11.3.13 11.3.25
St~I,-,(t)dt=z%(z)
0
z Wv>O)
St-~I,+,(t)dt=z-~‘I,(z)-
’ e-lr,(t>dt=ze-‘[r,(z)+Il(z)l
s0 11.3.26 g l
0 2’Iyvfl)
+nIe-.lo(z)-11+2e-‘~~ (n--kV~(z)
11.3.27
11.3.14
S t-YK”+l(t)dt=z-YK”(z)
s0
1 11.3.28 -
b#3> E
F24(2V-l)r(v)
Indefinite Integrals of Products of Beeeel Functiona
11.3.15
Let %$(z) and .Qy(z) denote any two cylinder
’ e*‘t’K,(t)dt ==g [K”(Z) fKv+1(z)l functions of orders cc and v respectively.
s0
11.3.29
F2”r(v+l)
vi?v>--3)
2v+1
z (~-z’)t-~}~(kl)~(zt)dt
King’s integral (see [11.5]) s{
=z{k~~+l(kz)~“(z~)-z~(~~)~“+l(~~) 1 *
11.3.16 s zecK,(t)~t=ze”[K,(z~+~~(Z)l-~
0 - (/.J--v)%?~(kz)9”(zz)
11.3.17 11.3.30
m
sI
e’t-‘K,(t)&
S ap-“---l
%+*(~>~+1(W
=G [K,(z)+K-,@)I W>3)
Case 2: p=o, /A=fv
11.3.31
11.3.18 ag”,v-l(z)=~“~“(d
11.3.19 ag-,,,+,(z)=--z-“Z”(z) r z t’+p+‘~p(t)~(t)dt
J
StJ?-l(t)&=2 (~x-0
0 2vr(v+ij
11.3.22 * PO (vS2&c+&)
0 I
2n- ‘J2nwt=I 2 n
c -; z m- 1V*,-,(d 11.3.33
s0 t
11.3.23
=; ,$+l w--1)J2,-,b) @>O) Szt[J:-l(t>-~J:+l(t)ldt=2vJt(z)
0
(%>O>
(2nfl)l Jzn+l(Qdt=SD J,(t)&
11.3.34
St~~(t)dl=~[J:(z)+J:(z)] 0
2
11.3.35
-Jl(Z) -; &I kJ2k (2)
11.3.24
SzJn(tv.,l(t)dt=~
0
[I-WI-&:(t)
S0
z t’Y._,(t)dt=z.P.(z)+~ (gv>o)
*seepageII.
INTEGRALS OF BESSEL FUNCTIONS 485
11.3.36 2. There must exist numbers k, and k, (both
not zero) so that for all n
(P+v)szt-‘~(t)W(t)dt
11.4.4 kA,%C+l (La> --k,%@na) = 0
--(/h+v+w
S zt-%$+.(t)L%+.(t)dt
n-1 In connection with these formulae, see 11.3.29.
=~(z)~(z)+~+.(Z)~+,(2)+2~l~+R(2)~+n(2) If o=O, the above is valid provided B=O. This
case is covered by the following result.
Convolution Type Integrals
11.3.37 11.4.5
StJ,(cY,t)J,(a,t)dt=O
go(-)nJr+“+2k+l(2) 1
S 0
* J,w”(z--w=2
(9%>-1,9v>-l>
0
=3[J:hJ12
(m#n, v>-1)
S 0
zJ”(t)J~-~(2-t)dt=Jo(2)-cos 2 (-1<9?v<q
S 0
J,(t)J-,(z--t)dt=sin 2 w4<1>
al, %, . . . are the positive zeros of
aJV(z) +brJ~(~) = 0, where a and b are real con-
stants.
11.3.40
I:t-‘J,(t)J”
(z-t)d&$ 11.4.6
S 0
(9&>0,9?v>-1) S t-‘J0
m
“+*n+l(t)J”+Sm+l(t)~t=O Cm #n)
11.3.41 1
=2(2n+v+l)
c J,(t)J”(Z-t)dt=(~+v)J~+Y(2)
S 0 t(z-t> lzvz
(9&4>0, c-?v>O)
(m=n)(v+n+m>-1)
Definite Integrals Over a Finite Range
*
11.4. Definite Integrals
Orthogonality Properties of Bessel Functions
11.4.7
S ’ Jzn(22
0
sin t)dt=i J",(z)
Let $32)
In particular,
be a cylinder
let
function of order v. 11.4.8
S t)
0
* Jo(22 sin cos Zntdt=d~(z)
11.4.11
i
provided the following two conditions
1. A, is a real zero of
hold :
S 0
J,(z sin2 t)J,(z cos2 t) csc 2tdt
Infinite Integrals
11.4.23
Integrals of the Form me-wz”(t)dt
0
11.4.12
s
11.4.24
S -
-m
mew-‘J”(t)dt=e
t”b+d
r(p+v)
l-(+--p)
S0 r(3)2flr(v-P+l)
where T,(w) is the Chebyshev
first kind (see chapter 22).
polynomial of the
(S&r<;9
9&+4>0) 11.4.25
11.4.13
(D
S0
m
e-ttp-ll,(t)dt= w+4w-d
u3Pw-~+1)
S --m
t-‘em* orJ, (t)dt
=z (4)“(1-u~)w,-,(0)(~2<1)
(%<$ w(,+v)>o)
=O(w”>l)
11.4.14
where U,(W) is the Chebyshev polynomial of the
S0 - 00s bt K,(t) fit=&
(l+bY (lJbl<l) second kind (see chapter 22).
11.4.26
11.4.15
11.4.16
S0 - tu,(t)dt=
r(!I$!A)
where P,(W)
chapter 22).
is the Legendre polynomial (see
11.4.27
(m+4>-1, %<;)
11.4.17
S0
- J,(t)&=1 @v>-1)
where ~(a, z) is the incomplete gamma function
11.4.18 (see chapter 6).
r($)r(+)
S0 t’
- [l--Jowt=
2F{ r (ILL))2 o<gK3) 11.4.28
Integrals of the Form
s
o- e- cza*%PZ” (bt) a2
OD
S0 e-‘ata t+J,(bt)dt
11.4.19
(9 (P+v)>o, aa2>o>
11.4.20
S0 - Y,(t)&=-tan y (1~4<1> where the notation M(a, b, z) stands for the con-
fluent hypergeometric function (see chapter 13).
11.4.21
11.4622
S0 mYo(t)dt=O
m
11.4.29
S0mta&)dt=2p-l r (E!$!2)r(!!) S 0
e -a2%+1J~(bt)dt
b=
S 0
- J&t) cos bt dt=--p-----
--ar sin y
11.4.31 =(b2-u2)f[b+(b2-a2)f]F
b’J
OD @>a>@ WC----l)
S 0
e -“ll.(bt)dt=~ esI+, (z2)
11.4.38
SJ,(d)
(a>-1, a%z~>O)
11.4.32 m sin bt dt=
b= 0
OD
S 0
e -azt’Ko(bt)dt=~ es K. ($) M’a”>O) arcosy
=(b2-a2)f[b+(b2-a2)f]p
Weher-Schafheitlln Type Integrals
11.4.33 @>a>@ @‘cc>---2)
S-J,(ut)J.(bt)dt=
( bYr P+~--x+l
0 ta pa’-A+lr(v+l)r
2
r--vy+y
> 11.4.39
S 0
meratJo (at)dt = ’
(a”- b2)+ (05 b<a)
( i
=(b2_a2)1 (O<a<b)
X& ( p+u--x+1,
2
v-p--x+1.
, v+1;
b2
-
2 a2>
(D
11.4.40
b
11.4.34
W’(cc+v-X+1)>% 93>-l, O<b<a)
S 0
eib’Yo (at)dt= 2a
+2-bZ)f . ;
arc sm (05 b<4
S-J,,(at)J,(bt)dt=
0
arr lJ+v--x+1
( ta
2 >
-1
=(b2_a2)t
2i
+ ,+-&)t
=
bYr(v-p)
Special Cases of the Discontinuous Weber-Schafheitlin
Integral @>a>@
11.4.35
S J,,(d)
0
m sin bt dt
t
1
=;sin[rorcsini] (Olb<a)
ar sin 3
(b Za>O)
11.4.42
S 0
mJp(at)Jr-l(bt)dt=~ (O<b<a)
=r[b+(b2-a2>+]~ 1
(O<b=a)
(WP>---1) =zT;
11.4.36
=o @>a>01
S 0
- J,,(d) cos bt dt
t
=;
1
cos [parcsin ;] (O<b<a)
@P>O>
=p[b+(b2-a2)t]r
ar CO8=; 11.4.43
S 0
= ‘+ {l-J,(bt)}dt=O (O<bla)
=ln ; (b 2a)O)
488 INTEGRALS OF BESSEL FUNCTIONS
-v+‘J,(at)dt ODK(at)dt 7?
cos~ wa4 -Yhz)l
s0 (t2+22)r+l
aw-r
=2’r(p+l)
(a>o,
IL,(az)
Wz>o+au<aap+;)
s 0 tV(t2+22)=4ZY+1
11.4.48
Wa>O, Wz>O, %<3)
11.4.45
sa J,(at)dt
0 tqt2+2$
=z
2zy+l
[Iv(az) - L,(az)]
11.4.49
(a>O, Wz>O, B!v>-1)
Numerical Methods
11.5. Use and Extension of the Tables This value is readily checked using ~~3.1 and
h=--.05. Now /Jo(x) 121 for all x and IJn(x)>
1 Jo(t)dt,l Y,V)dt ,,l IoWt,jy= KOW <2-t, nk 1 for all 2. In Table 11.1, we can
always choose Ih( 5.05. Thus if all terms of O(h4)
and higher are neglected, then a bound for the
For moderatevalues of x, use 11.1.2 and 11.1.P absolute error is 2*h4/48<.2.10b6 for all x if (hl
11.1.10 as appropriate. For x sufficiently large, 5.05. Similarly, the absolute error for quadratic
use the asymptotic expansions or the polynomial interpolation does not exceed
approximations 11.1.11-11.1.18.
a.05 ha(2++2)/24<.2.10-‘.
Example 1. Compute to 5D. z
S
J,(t)dt
s Example 3. Interpolation of J,(t)dt using
0
Using 11.1.2 and interpolating in Tables 9.1 and Simpson’s rule. We have
9.2, we have
a.oa so=+* Jo(t)dt=so= Jo(t)dt+S’+* Jo(t)dt
2
&,(t)dt=2[.32019 09+.31783 69+.04611 52
s0
+. 00283 19+ .00009 72+ .OOOOO211 l+” JoWt=$ CJ,(x)+4Jo(x+;)+Jo(z+h)]+R
= 1.37415
3.0s
R= -&=, JP(e), x<t<x+h
Example 2. Compute J,(t)dt to 5D by Now
s 0
interpolation of Table 11.1 using Taylor’s formula. JP)(x)=; [J~(x)-~J&)+~Jo(~$I
We have
Z+A
S 0 S Jo(t)dt=
0
’ Jo(t)dt +hJo(z) -; Jl(x)
and with lhl5.05, it follows that
+; tJP(z)-Job)l+~ [3J1(~~-Jd~)l+ ... IRl<.s. 1o-‘o
Then with x=3.0 and h=.O5, Thus if s=3.0 and h=.05
a.os a.os
S 0
J,(t)&=1.387567+(.05)(--260052)
- (.00125) (.339059)
S 0
J,(t)dt=1.38756 72520+9 [-.26005
and the terms on this last line are tabulated. Thus If x/r is not large the formula can still be used
for x=2, provided that the starting values are sufficiently
accurate to offset the growth of rounding error.
j-1=- .57672 48,jo=.22389 08,j,=1.4257703 For tables of Ki,(x), see [ll.ll].
490 INTEGRALS OF BESSEL FUNCTIONS
Then
the latter following from 11.3.27 with v=l. In
11.3.5, put a=l, b= -1, p=O and v=O. Let “-““=(~-1)~(~-3)~ . . . (r-2k+1)2
p=m. Then
~,I[~K~(x)+x(T--~)Ko(~)I/(T--~)~
M4=[@- 1>%-2(d--;F2~1(d since for x fixed, fi(x) is positive and decreases as T
--Cm- 1)~0(41/~ (G-1) increases.
Using tabular values off0 and fi, one can compute Example 9. Computef,(x) to 5D for x=3 and
in succession fa, f3, . . . provided that m/x is not m=0(2)10. We have
large.
Example 8. Computef&) to 5D for x=5 and K,= .03473 95 K,=.04015 64
m=0(1)6. We have, retaining two additional
If r=16,
decimals
e1&.86* 1o-2 fiO<1.4’ 1o-6
K,= .00369 11 K,=. 00404 46
Taking g16=0, we compute the following values
Thus jo=l. 56738 74 f1=. 19595 54 of Q14, g1a, * * *, go by recurrence. Also recorded
are the required values off,,, to 5D.
ja=.O5791 27, j,=.O1458 93, ja=.00685 36
Similarly starting with fi, we can compute f3 and f5.
If m>x, employ the recurrence formula in
backward form and write
fm-~(x~=~~~~(x)+~~l(x)+~(m-ll)Ko(;c)ll(m-ll)a
In the latter expression, replacef, by g,,,. Fix x.
Take r>m and assume g,=O. Compute g,+
g,-4, etc. Then
- -
lim g+&x)=f,,,(x), m=r-2k
r-f- For tables off,,,(x), see [11.21].
References
Texts [11.7] Y. L. Luke, An associated Bessel function, J. Math.
[ll.l] H. Bateman and R. C. Archibald, A guide to tables Phys. 31, 131-138 (1952).
of Bessel functions, Math. Tables Aids Comp. [ll.S] F. Oberhettinger, On some expansions for Bessel
1, 247-252 (1943). See also Supplements I, II, integral functions, J. Research NBS 59, 197-201
IV, same journal, 1,403-404 (1943) ; 2,59 (1946) ; (1957) RP 2786.
2, 190 (1946), respectively. [11.9] G. Petiau, La thQrie des fonctions de Bessel
[11.2] A. Erdelyi et al., Higher transcendental functions, (Centre National de la Recherche Scientifique,
vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., New Paris, France, 1955).
York, N.Y., 1953). [ll.lO] G. N. Watson, A treatise on the theory of Bessel
[11.3] A. Erdelyi et al., Tables of integral transforms, ~01s. functions, 2d ed. (Cambridge Univ. Press,
1, 2 (McGraw-Hill Book Co., Inc., New York, Cambridge, England, 1958).
N.Y., 1954).
[11.4] W. Griibner and N. Hofreiter, Integraltafel, II Teil Tables
(Springer-Verlag, Wien and Innsbruck, Austria,
1949-1950). [ll.ll] W. G. Bickley and J. Nayler, A short table of the
Ill.51 L. V. King, On the convection of heat from small functions Kin(z), from n=l to n=16. Philos.
cylinders in a stream of fluid, Trans. Roy. Soo.
London 214A, 373-432 (1914). Mag. 7,20,343-347 (1935). Kir(z)=l-K&)&
[11.6] Y. L. Luke, Some notes on integrals involving
Ki.(z) = m K&,-l(t)&, n=1(1)16, ~=0(.05).2
Bessel functions, J. Math. Phys. 29, 27-30 J-
(1950). (.1)2, 3, “9D.
INTEGRALS OF BESSEL FUNCTIONS 491
[11.12] V. R. Bursian and V. Fock, Table of the functions [11.18] H. L. Knudsen, Bidrag til teorien for antenne-
systemer med he1 eller delvis rotations-symmetri.
I (Kommission Has Teknisk Forlag, Copenhagen,
Akad. Nauk, Leningrad, Inst. Fiz. Mat., Denmark, 1953). = J,(t)dt, n=0(1)8, z=
s0
Trudy (Travaux) 2, 6-10 (1931). jzmKo(t)dt,
O(.Ol)lO, 5D. Also s J,(t)e*dt, a=t, a=%-t.
s
2=0(.1)12, 7D; e* z =io(t)dt, 2=0(.1)16, 7D; [11.19] Y. L. Luke and D. Ufforod, Tables of the function
I
‘lo(t)&, 2=0(.1)6, 7D; e-= ’ zo(t)dt, x= i?o(z) = o=Ko(t)dt. Math. Tables Aids Comp.
s s0 s
O’(.l) 16, 7D. UMT 129. r&r)=-[[y+ln (z/2)lAl(z)+A&),
[11.13] E. A. Chistova, Tablitsy funktsii Besselya ot A,(z), A&). s=O(.O1).5(.05)1, SD.
deistvitel’ nogo argumenta i integralov ot [11.20] C. Mack and M. Castle, Tables of oaZo(z)dz and
nikh (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., s
mK&c)dz, Roy. Sot. Unpublished Math. Table
1958). J,,(X), Y,,(X), lrn + dt, 5” y & s
File No. 6. a=O(.O2)2(.1)4, 9D.
n=o, 1; s=o(.o01)15(.01)100, 7D. [11.21] G. M. Muller, Table of the function
Also tabulated are auxiliary expressions to facil-
itate interpolation near the origin. Kj.(z)
zu*Ko(u)du, =x-n
s
[11.14] A. J. M. Hitchcock, Polynomial approximations to Office of Technical Se&es, U.S. Department
Bessel functions of order zero and one and to of Commerce, Washington, D.C. (1954).
related functions, Math. Tables Aids Comp. n=0(1)31, z=O(.O1)2(.02)5, 8s.
11, 86-88 (1957). Polynomial approximations [11.22] National Bureau of Standards, Tables of functions
2 and zeros of functions, Applied Math. Series 37
for Jo(t)dt and -Ko(t)dt.
s sz (U.S. Government Printing Office, Washington,
[11.15] C. W. ‘Horton, A short table’ of Struve functions D.C., 1954). (1) pp. 21-31: ff Jo(t)dt, l Yo(t)dt,
and of some integrals involving Bessel and
Struve functions, J. Math. Phys. 29, 56-58 z=O(.ol)lo, 10D. (2) pp. 33-39: m Jo(t)dt/t,
sz
(1950). C.(z)=~t”J,(t)dt,n=l(l)4,z=O(.l)lO,
2=0(.1)10(1)22, 1oD; F(z) =sm Jo(t)dt/t
4D; D,(z) =l t”E,(t)dt, n=0(1)4, z=O(.l)lO, +in (z/2), 2=0(.1)3, 10D; F(“(i)/n!, z=
10(1)22, n=0(1)13, 12D.
4D, where E,(z) is Struve’s function; see [11.23] National Physical Laboratory, Integrals of Bessel
chapter 12. functions, Roy. Sot. Unpublished Math. Table
[11.16] J. C. Jaeger, Repeated integrals of Bessel functions File No. 17. s,’ Jo(t)& s, Yo(t)dt, 2=0(.5)50,
and the theory of transients in filter circuits,
J. Math. Phys. 27, 210-219 (1948). fi(z)= 1oD.
s” Jo(t)dt, f,(z) =lz f,-l(t)dk 2-‘f,(z), r=1(1)7, [11.24] M. Rothman, Table of OZ~(x)dz for 0(.1)20(1)25,
0 s
Quart. J. Mech. Appl.‘Math. 2, 212-21.7 (1949).
z=O(1)24,8D. Also o.(z)=~mJo[2(zt)“]J.(t)dt, 8S-9s.
a,,(x), a;(x), n=l(l)7, z=O(l;24, 4D. [11.25] P. W. Schmidt, Tables of ‘Jo(t)dt for large z,
s
[11.17] L. N. Karmazina and E. A. Chistova, Tablitsy J. Math. Phys. 34, 169-172 yl955). z=lO(.2)40,
funktsii Besselya ot mnimogo argumenta i 6D.
integralov ot nikh (Izdat. Akad. Nauk SSSR., [11.26] G. N. Watson, A treatise on the theory of Bessel
MOSCOW, U.S.S.R., 1958). e-zZ&), e-*11(z), functions, 2d ed. (Cambridge Univ. Press, Cam-
bridge, England, 1958). Table VIII, p. 752:
ezKo(z), eZZG(x), e=, e-2 = Zo(t)dt, es s - Ko(t)dt,
s0 1 =
5 Jo(t) dt, f j- =Y,(t)dt, x=0(.02)1, 7D, with
2=0(.001)5(.005)15(.Ol)lOO, 7D exlept for ea s
which is 75. Also tabulated are auxiliary expres- the” first 16 m:xima and minima of the integrals
sions to facilitate interpolation near the origin. to 7D.
492 INTEGRALS OF BESSEL FUNCTIONS
,I,Jo (t)dt
0.71531 19178
so=Yo
(t)dt
0.19971 93876
e-=
s
0.210459
’ l,,(t)dt
46
ez zp Ko(t)dt
s
0.50593 10
0.69920 74098 0.16818 49405 0.21154 58 0.50179 55
0.68647 10457 0.13551 34784 0.20860 68 0.49776 16
0.67716 40870 0.10205 01932 0.20577 28 0.49382 50
0.67131 39407 0.06814 12463 0.20303 89 0.48998 19
5. 5 0.66891 44989 0.03413 05806 0.20040 08 0.48622 86
0.66992 67724 +0.00035 67983 0.19785 40 0.48256 16
267 0.67427 98068 -0.03284 98697 0.19539 44 0.47897 75
0.68187 18713 -0.06517 04775 0.19301 81 0.47547 34
55:: 0.69257 19078 -0.09630 01348 0.19072 13 0.47204 60
0.70622 12236 -0.12595 06129 0.18850 02 0.46869 29
2; 0.72263 54100 -0.15385 27646 0.18635 16 0.46541 11
6:2 0.74160 64692 -0.17975 87372 0.18427 20 0.46219 83
0.76290 51256 -0.20344 39625 0.18225 84 0.45905 20
66:; 0.78628 33012 -0.22470 89068 0.18030 78 0.45596 99
0.81147 67291 -0.24338 05692 0.17841 74 0.45294 98
22 0.83820 76824 -0.25931 37161 0.17658 44 0.44998 97
6:7 0.86618 77897 -0.27239 18447 0.17480 64 0.44708 76
0.89512 09137 -0.28252 78684 0.17308 09 0.44424 15
66:: 0.92470 60635 -0.28966 45218 0.17140 55 0.44144 97
7. 0 0.95464 03155 -0.29377 44843 0.16977 82 0.43871 05
0.98462 17153 -0.29486 02239 0.16819 68 0.43602 22
3:: 1.01435 21344 -0.29295 35658 0.16665 93 0.43338 34
1.04354 00558 -0.28811 49927 0.16516 39 0.43079 23
::i 1.07190 32638 -0.28043 26862 0.16370 89 0.42824 76
1.09917 14142 -0.27002 13202 0.16229 24 0:42574 81
::2 1.12508 84628 -0.25702 06208 0.16091 30 0.42329 20
1.14941 49299 -0.24159 37080 0.15956 91 0.42087 86
7'8 1.17192 99830 -0.22392 52368 0.15825 93 0.41850 63
7: 9 1.19243 33198 -0.20421 93575 0.15698 21 0.41617 40
8. 0 1.21074 68348 -0.18269 75150 0.15573 64 0.41388 07
1.22671 60587 -0.15959 61109 0.15452 08 0.41162 52
8"-:. 1.24021 13565 -0.13516 40494 0.15333 42 0.40940 65
8: 3 1.25112 88778 -0.10966 01934 0.15217 55 0.40722 37
8.4 1.25939 12520 -0.08335 07540 0.15104 36 0.40507 56
1.26494 80240 -0.05650 66385 0.14993 74 0.40296 15
E 1.26777 58297 -0.02940 07834 0.14885 61 0.40088 04
1.26787 83120 -0.00230 54965 0.14779 88 0.39883 15
xl 1.26528 57796 +0.02451 01664 0.14676 44 0.39681 40
8:9 1.26005 46162 0.05078 29664 0.14575 23 0.39482 69
1.25226 64460 0.07625 79635 0.14476 16 0.39286 97
;*i 1.24202 70675 0.10069 08937 0.14379 16 0.39094 15
9:2 1.22946 51666 0.12385 04194 0.14284 16 0.38904 17
1.21473 08237 0.14552 02334 0.14191 08 0.38716 95
9':: 1.19799 38314 0.16550 09969 0.14099 87 0.38532 41
0.0.
X
s tz 1 --Jo@)
o.“ooooo
&
000
Jz L
Contents
Pege
Mathematical Properties .................... 496
12.1. Struve Function H,(z) ................. 496
12.2. Modified Struve Function L,(z) ............. 498
12.3. Anger and Weber Functions .............. 498
Numerical Methods ...................... 499
12.4. Use and Extension of the Tables ............ 499
References .......................... 500
Table 12.1. Struve Functioqs (OjzSa~) ............. 501
12.1.3
If 9v> -4,
12.1.6
5 o1(l-ty%in (zt) dt
.6
i
12.1.7 %]a 2)”
=&r(v+f) S 6
sin (2 cos 0) sinaVBdO
--I’ (l+tn)‘-3d,
-.6
Recurrence Relations
st~-v-‘H,(t)dt
=
12.1.27
r(j~)P-“-’ tan (3~~0
0 w--~P+l>
(I9Pl<h ~‘v>9k’--9)
If j,(z)= 0SH,(t)t*dt
s
12.1.28
FIGURE 12.3. Struve functions. f”+l= (2v+ l)f,< Z)--Z’+W”(Z)
++f
E,(z), 2=3, 5 + (Y+i)2’+lr(~pyY+g) (wy>-43)
Special Proper&x
Asymptotic Expansions for Large 1x1
12.1.14 H,(z) 2 0 (z>O and u> 3) 12.1.29
r(k+3)
12.1.15 H.(z)-Y.(z)=; zol *-d-R,
(n an integer>O)
r(v+3--k) (i)
H-(,+),(z)= (-l)nJn+i(z)
(larg 4<4
12.1.16 H&)=(-g (1-cos 2) where R,,,=O(IZI~-*~-~). If Y is real, z positive *
and m+ )- v 20, the remainder after m terms is
12.1.17 of the same sign and numerically less than the
first term neglected.
12.1.30
12.1.18 H,(zemr*)=e”‘n(r+*)ri H,(z) (m an integer) &(z)-yo(z)~~[~-~+q.&$~+~ . .]
12.1.19 (larg 4<*)
12.1.31
x&(z)--Yl(z)-f [1+$-y+y- . . .]
12.1.20
(larg 4<4
12.1.21 H,(z)= !!z’2)‘+1 IFS 3 3 z2 12.1.32
1; z+~, &
dr rG+*>
s[Ho(t)--YoWW -5 [In cw+r1
12.1.22
a0
st-1
L(t)&=;
Integrala
0
(See chapter 11) s0
[g-m
z4+A6- **-1
s8Bb(t)dt=~
0
12.1.33
12.1.34
2(32)” 2 k!b, 12.2.6 Lv(z)-I-p(z)
H”(Z)-Y” (z)-&r(y+f) k=O 2+1
k- 32 (-l’k+lr(k~v+l (jargzI<+a)
(larg 4<hbl<l4 a kPor(Y++--k) (i)
bo=l, h=2h, bz=6(~/2)~-&, b,=20(~/~)~-4(p/z’)
Integrals
12.1.35
12.2.7
2(W -jy!!s 4 (Iyl>lzl)
=” (4 +iJ”(z) -&++$) k-o 2+1
S 0
’ LoWt=f
[
$+12 .
“3: .
4+12 .
32ze
.
52 . 6+---1
12.2. Modified Struve Function L,(5)
Power Series
(VT
Expansion 12.2.8
SozM~~---Ldt)ldt-~ [In @d+rl
12.2.1 L”(z)=--ie-TH”(iz)
($a*”
=(4z)“+’k&r(k+4j)r(k+vt-3)
Integral Representations
?
12.2.9
S 0
’ L&)dt=L,(z)-f z
12.2.2
2($3”
Lv(z)=Gr(,+3)
S o sinh (z cos 0) sin*v e& Relation
12.2.10 L-(,++)
to Modified
(2)=1(,++)(z)
Spherical Bessel Function
(n an integer10)
wv>--3)
12.2.3
12.3. Anger and Weber Functions
I-,(s)-L,(z)= 2(z’2)”
J;;w+3> S 0
msin (tz)(l+t*)l-+
my<3, c-0)
dt
Anger’s Function
12.2.4 L”wI-L”+I=;
Recurrence Relations
L”+ (2/2)’
12.3.1 J.(z)=-I
n 0 S
* cos (vO- z sin 0) de
12.3.7
FIGURE 12.4. Modijied Struve junctions.
“+l[~lr(n-k-t)(aZ)-.+~+l
E-,,(z)=+ C --H-,(z) *
L(z), fn=C(1)5 k=O w+a
STRUVE FUNCTIONS AND RELATED FUNCTIONS 499
12.3.8 E,,(z)=--Ho(z) 12.3.10 E&)=g-H&)
12.3.9 El(z)=:-H,(z)
Numerical Methods
12.4. Use and Extension of the Tables We note that for n>6 there is a rapid loss of
significant figures. On the other hand using 12.1.3
Example 1. Compute L,(2) to 6D. From for x=4 we find H,(4)=.0007935729, H,,(4) =
Table12.1 I,(2)-L,(2)=.342152; from Table 9.11 .00015447630 and backward recurrence with 12.1.9
we have 1,,(2)=2.279585 so that L,(2)=1.937433. gives
H,(4) = .00367 1495 H,(4)= .85800 94
Example 2. Compute H,(lO) to 6D. From
H,(4) =.01510 315 H,(4)=1.24867 6
Table 12.2 for s-‘=.l. H,,(lO)-~Y,(10)=.063072;
H,(4) = .05433 519 H,(4) = 1.06972 7
from Table 9.1 we have Y,(10)=.055671. Thus,
H,(4) =.16719 87 K(4)= .13501 4
H,,(10)=.118743.
H,(4) = .42637 43
Example 3. Compute ‘Ho(t)& for 2=6 to Example 6. Compute L,(.5) for n=0(1)5 to
s 8s. From 12.2.1 we find L,(.5) =9.6307462X lo-‘,
5D. Using Tables 12.2, il.1 and 4.2, we have
L,(.5) =2.1212342X 10m5. Then, with 12.2.4 we
J)ut)dt=Jy YdWt +f In 6+fi (6) get
L,(.5)=3.82465 03X10-’ L1(.5)=.05394 2181
=-.125951+(.636620)(1.791759) L,(.5)=5.36867 34X1O-3 Lo(.5) =.32724 068
+ .816764
Example 7. Compute L,(.5) for -n=0(1)5
= 1.83148 to 6s. From Tables 12.1 and 9.8 we find L,(.5) =
.327240, L,(.5) = .053942. Then employing 12.2.4
Example 4. Compute H,(x) for x=4, -n= with backward recurrence we get
O(l)8 to 6s. From Table 12.1 we have H,,(4)=
.1350146, HI(4)=1.0697267. Using 12.1.9 we find L-,(.5) = .690562 L-,(.5) = -75.1418
L-z(.5)=-l.16177 L-,(.5) = 1056.92
H-,(4) = - .433107 H-,(4) = .689652
Lb3(.5) = 7.43824
H-,(4) = .240694 H-,(4) = - 1.21906
H-3(4) = .152624 H-,(4) = 2.82066 Example 8. Compute L,(s) for x=6 and
H-4(4) = - .439789 H-,(4) = - 8.24933 -n=0(1)6 to 8s. From Tables 12.2 and 9.8
we fhd L,,(6)=67.124454, L,(6)=60.725011.
Example 5. Compute H,(s) for x=4, n= IJsing 12.2.4 we get
O(l)10 to 7s. Starting with the values of H,(4)
and H,(4) and using 12.1.9 with forward recur- L-,(6)=61.361631 L-4(6)=16.626028
rence, we get L-*(6) =46.776680 Lm5(6)= 7.984089
L-a(6)=30.159494 L-,(6) = 3.32780
H,(4)= .13501 46 Ha(4) =.05433 54
H,(4) = 1.06972 67 H,(4) =.01510 37 We note that there is no essential loss of accuracy
H,(4) = 1.24867 51 H,(4) =.00367 SS until n= -6. However, if furt.her values were
H,(4)= .85800 95 HQ(4) =.00080 Od necessary the recurrence procedure becomes un-
H,(4)= .42637 41 H,,,(4) = .00018 %5 stable. To avoid the instability use the methods
H,(4)= .16719 87 described in Examples 5 and 6.
500 STRUVE FUNCTIONS AND RELATED FVNCTIONS
References
Texts Tables
[12.1] R. K. Cook, Some properties of Struve functions, [12.8] M. Abramowitz, Tables of integrals of Struve
J. Washington Acad. Sci. 47, 11, 365-363 (1957). functions, J. Math. Phys. 29, 49-51 (1950).
[12.2] A. Erdelyi et al., Higher transcendental functions, [12.9] C. W. Horton, On the extension of some Lommel
vol. 2, ch. 7 (McGraw-Hill Book Co., Inc., integrals to Struve functions with an application
New York, N.Y., 1953). to acoustic radiation, J. Math. Phys. 29, 31-37
[12.3] A. Gray and G. B. Mathews, A treatise on Bessel (1950).
functions, ch. 14 (The Macmillan Co., New York, [12.10] C. W. Horton, A short table of Struve functions
N.Y., 1931). and of some integrals involving Bessel and
[12.4] N. W. McLachlan, Bessel functions for engineers, 2d Struvefunctions, J. Math. Phys.29,56-58 (1950).
ed. ch. 4 (Clarendon Press, Oxford, England, 1955). [12.11] Mathematical Tables Project, Table of the
[12.5] F. Oberhettinger, On some expansions for Bessel Struve functions L.(z) and H,(z), J. Math.
integral functions, J. Research NBS 59 (1957) Phys. 25, 252-259 (1946).
RP2786.
[12.6] G. Petiau, La theorie des fonctions de Bessel,
ch. 10 (Centre National de la Recherche Scien-
tifique, Paris, France, 1955).
112.71 G. N. Watson, A treatise on the theory of Bessel
functions, ch. 10 (Cambridge Univ. Press, Cam-
bridge, England, 1958).
STRTJVE FUNCTIONS AND RELATED FUNCTIONS 501
STRUVE FUNCTIONS Table 12.1
X Ho(x) HI(X) s;Ho(W Io(+-b(4 II(X) -L1 (x) “fo(4 “j-” H$&
* 2
0.00000 00 0.00000 00 0.000000 1.000000 0.000000 0.00000 1.000000
K 0.06359 13 0.00212 07 0.003181 0.938769 0.047939 0.09690 0.959487
0: 2 0.12675 90 0.00846 57 0.012704 0.882134 0.091990 0.18791 0.919063
0.18908 29 0.01898 43 0.028505 0.829724 0.132480 0.27347 0.878819
0":: 0.25014 97 0.03359 25 0.050479 0.781198 0.169710 0.35398 0.838843
0. 5 0.30955 59 0.05217 37 0.078480 0.736243 0.203952 0.42982 0.799223
0.36691 14 0.07457 97 0.112322 0.694573 0.235457 0.50134 0.760044
2; 0.42184 24 0.10063 17 0.151781 0.655927 0.264454 0.56884 0.721389
0.47399 44 0.13012 25 0.196597 0.620063 0.291151 0.63262 0.683341
E. 0.52303 50 0.16281 75 0.246476 0.586763 0.315740 0.69294 0.645976
1. 0 0.56865 66 0.19845 73 0.301090 0.555823 0.338395 0.75005 0.609371
1.1 0.61057 87 0.23675 97 0.527058 0.80418 0.573596
1. 2 0.64855 00 0.27742 18 i* Ez: 0.500300 is ::z 0.85553 0.538719
1. 3 0.68235 03 0.32012 31 0:489655 0.475391 0:396290 0.90430 0.504803
1. 4 0.71179 25 0.36452 80 0.559399 0.452188 0.412679 0.95066 0.471907
1.5 0.73672 35 0.41028 85 0.631863 0.430561 0.427810 0.99479 0.440D86
0.75702 55 0.45704 72 0.706590 0.410388 0.441783 1.03682 0.409388
:*; 0.77261 68 0.50444 07 0.783111 0.391558 0.454694 1.07691 0;379057
1: a 0.78345 23 0.55210 21 0.860954 0.466629 1.11518 0.351533
1. 9 0.78952 36 0.59966 45 0.939643 is. :::z 0.477666 1.15174 0.324450
0.79085 88 0.64676 37 1.018701 0.342152 0.487877 1.18672 0.298634
2; 0.78752 22 0.69304 18 1.097659 0.327756 0.497329 1.22020 0.274109
0.77961 35 0.73814 96 1.176053 0.314270 0.506083 1.25230 0.250891
:*: 0.76726 65 0.78174 98 1.253434 0.301627 0.514194 1.28309 0.228992
2:4 0.75064 85 0.82351 98 1.329364 0.289765 0.521712 1.31265 0.208417
0.72995 77 0.86315 42 1.403427 0.278627 0.528685 1.34106 0.189168
22:: 0.70542 23 0.90036 74 1.475227 0.268162 0.535156 1.36840 0.171238
0.67729 77 0.93489 57 1.544392 0.258319 0.541164 1.39472 0.154618
z 0.64586 46 0.96649 98 1.610577 0.249056 0.546746 1.42008 0.139293
2:9 0.61142 64 0.99496 63 1.673465 0.240332 0.551933 1.44455 0.125242
0.57430 61 1.02010 96 1.732773 0.232107 0.556757 1.46816 0.112439
3:; 0.53484 44 1.04177 30 1.788248 0.224348 0.561246 1.49098
0.49339 57 1.05983 03 1.839675 0.217022 0.565426 1.51305 t ZEi
3: 3 0.45032 57 1.07418 63 1.886873 0.210099 0.569319 1.53440 o:oa1212
3.4 0.40600 80 1.08477 74 1.929699 0.203553 0.572948 1.55508 0.073071
3. 5 0.36082 08 1.09157 23 1.968046 0.197357 0.576333 1.57512 0.065992
3.6 0.31514 40 1.09457 16 2.001847 0.191488 0.579492 1.59456 0.059928
3. 7 0.26935 59 1.09380 77 2.031071 0.185924 0.582442 1.61343 0.054829
3.8 0.22382 98 1.08934 44 2.055726 0.180646 0.585199 1.63176 0.050642
3. 9 0.17893 12 1.08127 62 2.075858 0.175634 0.587776 1.64957 0.047311
4. 0 0.13501 46 1.06972 67 2.091545 0.170872 0.590187 1.66689 0.044781
0.09242 08 1.05484 79 2.102905 0.166343 0.592445 1.68375 0.042994
i:: 0.05147 40 1.03681 86 2.110084 0.162032 0.594560 1.70017 0.041891
+0.01247 93 1.01584 22 2.113265 0.157926 0.596542 1.71616 0.041414
::4' -0.02427 98 0.99214 51 2.112655 0.154012 0.598402 1.73176 0.041502
-0.05854 33 0.96597 44 2.108492 0.150279 0.600147 1.74697 0.042096
is -0.09007 71 0.93759 56 2.101037 0.146714 0.601787 1.76182 0.043139
417 -0.11867 42 0.90729 01 2.090574 0.143309 0.603328 1.77632 0.044571
4. a -0.14415 67 0.87535 28 2.077406 0.140053 0.604777 1.79049 0.046335
4. 9 -0.16637 66 0.84208 90 2.061852 0.136938 0.606142 1.80434 0.048376
5. 0 -0.18521 68 0.80781 19 2.044244 0.133955 0.607426 1.81788 0.050640
[(-y] [c-y] I:
(-;I6
I [
(97
I 1.
(412
I
Ho(x), HI(X), LO(Z), L1 (z), compiled from Mathematical Tables Project, Table of the Struve functions
L,(x) and H,(x), J. Math. Phys. 25,252-259, 1946(with permission).
X-’ Hoc@ - Y,(x) H,(x)- Y,(x) fdx) W) -Lo(x) I,(x)-L,(x) fdx) fA4 <x>
0.20 0.123301 0.659949 0.819924 0.133955 0.607426 0.793280
0.19 0.117449 0.657819 0.818935 0.126683 0.610467 0.794902 i* :21zi :
0.18 0.111556 0.655774 0.817981 0.119468 0.796448 0: 113505 6
0.17 0.105625 0.653818 0.817062 0.112319 i- 66:z 0.797910
0.16 0.099655 0.651952 0.816182 0.105242 0: 618598 0.799279 00.
. :Kz i
0.15 0.093647 0.650180 0.815341 0.098241 0.620955 0.800551 0.094843
0.14 0.087602 0.648504 0.814541 0.091318 0.623129 0.801721 0.088593
0.13 0.081521 0.646927 0.813785 0.084474 0.625119 0.802787
0. 12 0.075404 0.645452 0.813074 0.077706 0.626927 0.803750 i* %EB
0.11 0.069254 0.644081 0.812411 0.071010 0.628558 0.804611 0: 069761
0.10 0.063072 0.642817 0.811796 0.064379 0.630018 0.805374 0.063460 10
0. 09 0.056860 0.811232 0.057805 0.631315 0.806047 0.057147 11
0.08 0.050620 k T%5: 0.810722 0.051279 0.632457 0.806634
0.07 0.044354 0: 639696 0.810266 0.044793 0.633450 0.807140 i* E%24 :z
0. 06 0.038064 0.638888 0.809866 0.038340 0.634302 0.807572 0: 038152 17
0.05 0.031753 0.638200 0.809525 0.031912 0.635016 0.807933 0.031805 20
0. 04 0.025425 0.809244 0.025506 0.635596 0.808225 0.025451 25
0.03 0.019082 ;- 66::%! 0.809023 0.019116 0.636045 0.808450 0.019093 33
0. 02 0.012727 0: 636874 0.808865 0.012738 0.636365 0.808611 0.012731
0. 01 0.006366 0.636683 0.808770 0.006367 0.636556 0.808706 0.006366 1%
0.00 0.000000 0.636620
0.000000 0.636620 0.808738
0.808738 0.000000 CG
c-y ‘-;I1
1 1 c 1 [ 1 c 1 L- 1 [ 1 [(-;I2 1
(-!)5 92 92 c-g)1
s :[H,(1)-Y,(t)lat=~Inz+fi(x)
1
Ho(t)- Ye(t) dtzfo(x)
t
Contents
Page
Mathematical Properties . . i . . . . . . . . . . . . , . . . 504
13.1. Definitions of Kummer and Whittaker Functions . . . . . 504
13.2. Integral Representations . . . . . . . . . . . . . . . 505
13.3. Connections With Bessel Functions . . . . . . . . . . . 506
13.4. Recurrence Relations and Differential Properties . . . . . 506
13.5. Asymptotic Expansions and Limiting Forms . . . . . . . 508
13.6. Special Cases . . . . . . . . . . . . . . . . . . . . 509
13.7. Zeros and Turning Values . . . . . . . . . . . . . . . 510
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 511
13.8. Use and Extension of the Tables . . . . . . . . . . . . 511
13.9. Calculation of Zeros and Turning Points . . . . . . . . . 513
13.10. Graphing M(a, b, 2) . . . . . . . . . . . . . . . . . . 513
References . . . . . . . . . . . . . . . . . . . .. . . . . . . 514
Table 13.1. Confluent Hypergeometric Function M(a, b, s) . . . . . 516
2=.1(.1)1(1)10, a=-l(.l)l, b=.l(.l)l, 8s
Table 13.2. Zeros of M(a, b, 2) . . . . . . . . . . . . . . . . . . 535
a=--l(l)-.I, b=.l(.l)l, 7D
The tables were calculated by the author on the electronic calculator EDSACI in the
Mathematical Laboratory of Cambridge University, by kind permission of its director, Dr.
M. V. Wilkes. The table of M(a, b, z) was recomputed by Alfred E. Beam for uniformity
to eight significant figures.
504
iZONFL1TENT HYPERGEOMETRIC FUNCTIONS 505
13.1.1.5 y,=zl-bezM(l-cz, 2--b, -2) 13.1.34
13.1.25 W{ 2, 7) = -r(2-b)2-be~/r(l-a)
e: le~t~.-l(l-~)b-.-l&
13.1.26 W{ 5, 7} =e’“f(b-a’z-be* s 0
13.2.4
13.1.30
=e -A.? ’ ez,(,-A),-l(B--t)b-“-ldt
e’U(b-a, b, -2)=e(*1(1-b’e*21-bU(1--a, 2---b, -2) sA
(A=B-1)
Whittaker’s Equation L?Za>O, W2>0
13.2.5
13.1.31 !g+ [-;+;+Gp] WC0
me-art”-‘(l+l)b-.-ldt
r (a)U(a, b, 2) =
s0
Solutions:
13.2.6
Whittaker’s Functions m
13.1.32 MI,r(2)=e-tE2ffrM(4+C(-K, 1+2p, 2)
=e*s 1
e-~‘(~-~)“-l~b-“-l&
13.1.33
13.2.7
W~.r(2)=e-*z2t+rU(3+~-K, 1+2p, 2)
e-t2 Cosh@sinhbwle cothb-” (@)& * ,
(--n<arg 257r, ~=*b-a, p=#b-3)
506 CONFLUENT HYPERGEOMETRIC FUNCTIONS
13.3.7
13.2.8 r(a) U(a, b, z)
OD
=eAr
Ma, b, 4 =e+z(+bz-uz)~-*b
SA e-“(l-A)a-l(t+B)b-a-l~~
I-(b)
@=1---B) . neo A,(3z)tn(b-2a)-1nJb-1+,(J(2zb-4za))
lim
a+0
{r(i+a-b) U(a, b, z/u)} =2d-*b&,-~(2~ 13.4.2
b(b-l)M(u, b-l, z)+b(l--b-z)M(u, b, z)
13.3.4 +z(b-u)M(u, b+l, z)=O
lim( I’(l+a-b)U(a, b, -z/u)) 13.4.3
a-+-
= -7rie~*bz+~bH~~:1,(2fi) (Yz>O) (l+u-b)M(a, b, z)-aM(u+l, b, z)
13.3.5 +(b-l)M(u, b-l, z)=O
Expaneiona in Series 13.4.4
13.3.6
bM(u, b, z)-bbM(u-1, b, z)-.&(a, b+l, z)=O
M(u, b, z)=eW (b-u-$)(+)a-b++
13.4.5
. 5 (2b-2a-l),(b-%),(b--a-i+ n)
* b(a+z)M(u, b, z)+z(u-b)M(u, b+l, z)
n=O n!(b).
(-1)’ I’,-a--f+“(+) (b#O,-l,-2,. . .) -abM(u+l, b, z)=O
rt
13.6.19 + t - erf 2 Error Integral
22
&.-zn+n-I *
13.6.20 tm+t 1+n er’T(m, n, r) Toronto
r(bm+t)
Uh b, 4
- Relation Function
a b 2
-- --
13.6.21 v+t 2v+1 20 r-fe*(22)-‘K,(z) Modified Bessel
13.6.22 u+3 2v+1 -2it &,1,iMYS3) -~l(2z)-.~p(z)* EIankel
13.6.23 2v+1 2iz 3*te-i[n(“St)-Zl(2z)-.HI*‘(zjL Hankel
p+i
Numerical Methods
13.8. Use and Extension of the Tables In this way 13.4.1-13.4.7 can be used together
Calculation of M(a, b, 2) with 13.1.27 to extend Table 13.1 to the range
Kummer’s Transformation -lO<a<lO, -lOlb<lO, -lOsx<lO.
Example 1. Compute M(.3, .2, - .l) to 7s. This extension of ten units in any direction is
Using 13.1.27 and Tables 4.4 and 13.1 we have possible with the loss of about 1s. All the re-
a=.3, b=.2 so that currence relations are stable except i) if a<O, b<O
M(.3,
.2, -.l)=e-.‘M(-.l, .2, .l> and bl>lbl, x>O, or ii) b<a, b<O, lb-al>lbl,
x<O, when the oscillations may become large,
= .85784 90. especially if 1x1also is large.
Thus 13.1.27 can be used to extend Table 13.1 to Neither interpolation nor the use of recurrence
negative values of 2. Kummer’s transformation relations should be attempted in the strips
should also be used when a and b are large and b= --nf .l where the function is very large nu-
nearly equal, for x large or small. merically. In particular M(a, b, x) cannot be
Example 2. Compute M(17, 16, 1) to 7s. evaluated in the neighborhood of the points
Here a=17, b=16, and a=-m, b=-n, m<n, as near these points
M(17, 16, l)=e’M(-1, 16, -1) small changes in a, b or x can produce very large
changes in the numerical value of M(a, b, x).
=2.71828 18X1.06250 00
Example 4. At the point (- 1, - 1, x), M(a, b, x)
=2.88817 44. is undefined.
Recurrence Relations
When a=-1, M(-1, b, z)=l-zfor ati x.
Example 3. Compute M(- 1.3, 1.2, .l) to 7s.
Using 13.4.1 and Table 13.1 we have a= -.3, Hence:.mlM(-1, b,x)=l tx. ButM(b, b,x)=e”
b=.2 so that for all x, when a=b. Heocek$l M(b, b, x)=8.
M(-1.3,.2,.1)=2[.7M(-.3, .2, .l)-.3M(.7,.2, .l)] In the first case b+- 1 along the line a= - 1, and
= .35821 23. in the second case b-+-l along the line a=b.
By 13.4.5 when a= - 1.3 and b= .2, Derivatives
M(-52.5, .l, 1)=I’(.l)e*6(.05+52.5)*a6’*06 Example 11. To compute U(1, .l, 100) to 5s.
.5642 COB[(.2-4(-52.5))*“-.05~+.25s] By 13.5.2
[1+O((.05+52.5)-*6)]=-16.34+0(.2)
U(1, .l, loo)=&{ l-go+; 2go
By direct application of a recurrence relation,
M(-52.5, .l, 1) has been calculated as -16.447. -~jjjjj~+OwOb
1.9 2.9 3.9
To evaluate M(a, 5, 2) with z, a and/or b large,
13.5.17,19 or 21 should be tried.
Example 8. Compute M(-52.5, .l, 1) using =.Ol{ l-.019+.000551-.000021
13.5.21 to 3S, cos 0=&210.2. +o(lo-e) 1,
=.00981 53.
M-(-52.5, J, 1)
Example 12. To evaluate U(.l, -2, .Ol). For
=r(,l)e106.1 oode L105.1 co0 e]+s1.5641 2 small, 13.5.6-12 should be used.
52.55-i sin 2tI-*[sin (-52.5~)
+sin { 52.55(2e--sin 2e) +ir> U(.l, *2, .Ol)= r(l,l-
r(1--.2) q+“((.ol)l-‘a)
+0((52.55)-‘)I=-16.47+0(.02)
r(.8)
=,o+o((.01)~8)
A full range of asymptotic formulas to cover all
possible cases is not yet known. =1.09 to 3S, by 13.5.10.
Calculation of U(a, b, x)
To evaluate U(a, 5, 2) with a large, z small and
For -lO<z<lO, -1OSa110, -105&10 b small or large 13.5.15 or 16 should be used.
this is possible by 13.1.3, using Table 13.1 and the To evaluate U(a, 5, z) with 2, a and/or b large
recurrence relations 13.4.X-20. 13.5.18, 20 or 22 should be tried. In all these
Example 9. Compute U(l.1, .2, 1) to 5~. cases the size of the remainder term is the guide to
Using Tables 13.1, 4.12 and 6.1 and 13.1.3, we the number of significant figures obtainable.
have
Calculation of the Whittaker Functionr
U(.l, .2; l)=
Example 13. Compute M,-.,(l) and W.,,,-.,(l)
M(.1,.2,1) M(.9,1.8,1) to 5s. By formulas 13.1.32 and 13.1.33 and
~GiifSj~ r(.9)r(.2) - r(.l)r(l.8) ]* Tables 13.1, 4.4
But M(.9, 1.8, 1)=.8[M(.9, .8, 1)-iV(-.l, .8, l)] M.o,-.,(l)=e-~6M(.1, .2, 1)=1.10622,
= 1.72329, using 13.4.4. W.o,-.,(1)=e-~6U(.l, .2, 1)=.5’7469.
CONFLUENT HYPERGEOMETRIC FUNCTIONS 513
Thus the values of M,,,(x) and W,,,(Z) can M’(-3, .6,X;)
always be found if the values of M(a, 5, 2) and xi=xA [l--3M(-3, .6,X;)]
U(a, b, 2) are known. =X; [l-&Q-2,1.6, X;)/.6M(-3, .6,X;)]
13.9. Calculation of Zeros and Turning Points =.9715X1.0163=.9873 to 4s.
Example 14. Compute the smallest positive This process can be repeated to give as many
zero of M( -4, .6,~). This is outside the range of significant figures as are required.
Table 13.2. Using 13.7.2 we have, as a first
approximation
But, by 13.4.8,
Hence
X,=X,+.15M(-4, .6,X,)/M(-3,1.6,X,,),
60 -
i4
0 I
FIGURE 13.4. M(a, .5, 2).
.5 I 1.5 2 2.5 (From E. Jahke and F. Emde, Tables of functions, Dover Publlcatlons,
M Inc., New York, N.Y., 1945, with permiaslon.)
References
IO
b=l
6 Texts
z =O.l
0 1.64549 07
80 1.75647 99
0 1.86845 49
0 1.98142 05 II 0 1.21388
1.19185
1.16996
1.14822
1.12662 77
34
83
22
66
0 2.09538 12
For Olzl 1, linear interpolation in a, b or z provides 3-4s. Lagrange four-point interpolation gives 7s in a! b or 5 over
most of the table, but the Lagrange six-point formula is needed over the range 1 <s< 10. Any interpolation formula
can be reapplied to give two dimensional interpolates in a and b, a and 2: or b and z. This calculation can be checked
by being repeated in a different order.
CONFLUENT HYPERGEOMETRIC FUNCTIONS 517
CONFLUENT HYPERCEOMETRIC FUNCTION M(a, b, ix) Table 13.1
x=0.2
0"::
0:5
E
0.6
0.7
E
l:o
x=0.3
II0)
0 1.17274
1.71742 49
1.53139
1.34985
1.90800 78
94
88
56 II 0 1.11393
1.34985
1.23054
1.59674 77
1.47191 26
88
56
a\b
-1.0
-0.9
-0.8
-0.7
II-1)
-1
-1
-1
0.6
5.00000
5.45594
5.92137
6.39639
00
63
29
42
0.7 0.8 0.9 1.0
-0.6 -1 6.88112 54
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0"::
i-:
( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00
II
( 0) 1.00000 00
0 1.03645
1.07349 27
0 1.14937
08
1.11113 16
40
( 0) 1.00000 00
0:5 0 1.18822 61
CONFLUENT HYPERGEOMETRIC FUNCTIONS 519
CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x) Table 13.1
x=0.4
a\b 0.1 0.2 0.3 0.4
-1.0 O)-1.00000 00 -1 -3.33333 33 0.00000 00
-0.9 -1 I -8.32139 43 I -1 j -2.19718 27 -2 8.63057 33
-0.8 -1 -6.57495 96 -1 -1.01932 12 1-l I 1.75514 40
-0.7 -1 -4.75937 91 -2 +2.01024 24 -1 2.67677 48
-0.6 ( 0)-1.59134 63 -1 -2.87331 90 (-1 1.46463 65 -1 3.62847 08
-0.5 I 0)-1.20063 19
-0.4
-0.3
-0.2 -2)+6.90415 20
-0.1 (-1) 5.25850 66 -1) 7.66207 59
0":: I 0I 3.07676
3.64273 82
38 0 1.66597
1.84538 84
67 0 1.49182
1.62369 47
00
0 2.03014 00
i-76
0:8
0
0
2.22033
2.41605
03
02
0 2.61739 39
!:;1 0 2.82445 63
r=0.5
a\b 0.1 0.2 0.3 0.4 0.5
-1.0 ( 0 -4.00000 00 0.00000 00
-0.9 0 -3.61201 86 -2 8.38114 43
-0.8 II 0 -3.20079 89 II -1 1.71019 66
-0.7 0 -2.76573 85 -1 2.61697 96
-0.6 0 -2.30622 47 -1 3.55920 78
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
i-z
a:5
( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00
II
( 0) 1.00000 00
0 1.31281
1.15358 36
0 1.64872
87
1.47782 42
13
0 1.82563 24
( 0) 1.00000 00
1.0
-1 4.44444 44 -1 5.00000 00
$9”
-0:s
-0.7
II -1
-1
4.92975 21
5.42992 27
5.94522 72
II -1
-1
5.89284 21
5.44007 39
6.35854 17
-0.6 -1 6.47594 62 -1 6.83739 50
-0.5
-0.4
-0.3
-0.2
-0.1
0.1
::3
Fl::
CONFLUENT HYPERGEOMETRIC FUNCTIONS 521
CONFLUENT HYPERCEOMETRIC FUNCTION i)Z(a, b,x) Table 13.1
z=O.6
a\b 0.1 0.2 0.3 0.4 ( 0.5
II( 0 -1.77497
-1.00575 51
-1.27832
-1.53457
-2.00000 00
65
83
96 ( O)-1.00000 00
-0.6 ( 0)-3.05183 34
II 01 7.75149
1.14187 48
1.01404
6.63788
8.91853 04
08
45
76
85
l:o
8-98
II0 1.69207
1.82211 88
0 2.09565
45
0 1.95661 34
2.23934 57
48
II 0 1.54938
1.46364 36
57
0 1.63767 83
0
0 1.72857
22
1.82211 88
522 CONFLUENT HYPERGEOMETRIC FUNCTIONS
x=0.7
a/b 0.1 0.2 0.3 0.4 0.5
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5 Ii 0)-3.16446 06 II-1 2.05299 00
-0.4 0 -2.44543 68 -1 3.48181 61
-0.3 0 -1.67144 46 -1 4.98858 44
-0.2 -1 -8.40541 00 -1 6.57561 66
-0.1 -2 +4.92624 47 -1 8.24528 23
0.0 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00
0.1 II( 0 2.01375 27
0.2 0 3.09264 92
ki 0 4.23886
5.45463 64
06
0:5 0 6.74221 79
0.6 II 0 4.41274 94
0.7 0 5.09565 95
E 0 6.56853
5.81389 43
76
l:o 0 7.36066 31
0.6 Ii II 0
0
0
0
2.01375
2.20933
2.41306
2.62516
27
17
50
74
0
0
0
0
1.85078
2.01375
2.18318
2.35926
59
27
94
09
0 2.84585 75 0 2.54213 50
CONFLUENT HYPERGEOMETRIC FUNCTIONS 523
CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, .x) Table 13.1
z =0.8
8:;
it:
l:o
n\b 0.6
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00
0.1
00:;
8:;
II
( 0) 2.22554 09
2;
E 0 2.46668
2.98352
2.71923
3.25992 24
90
56
11
l:o
524 CONFLUENT HYPERGEOMETRIC FUNCTIONS
t-11-9.35972 27
-1 -8.20518 02
-1 -5.12058 10
(-1 -1.76920 97
-1 I +1.86021 91
t -1) 5.77931 14
0 1.45345 52
0j 1.93955 77
0 2.45960 31
0 3.01492 28
i 01 3.60688 44
0.6 I 1 I 1.16728 93 I 01 4.23689 27
0.7 1 1.39370 17
0.8 1 1.63551 72
ki:; ( 1)
1 189334
2:16782 87
94
0:8
i-b7
1"::
CONFLUENT HYPERGEOMETRIC FUNCTIONS 525
CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x) Table 13.1
II-1
-3 +7.71680
-2.72739
-5.29840
6.42974 30
3.12589 94
92
36
46
( 0) 1.00000 00 ( 0) 1.00000 00 ( 0) 1.00000 00
0.6 0.7
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0) 1.00000 00
II-1 6.28763 92
8.08383
4.60681
3.03694
1.57371 81
41
99
08 II
-1 5.27314
4.25195
8.71734 45
7.50355
6.35625 01
07
70
83
if:
0:3
8’76
II
8::
0:8
0.9
1.0
0 2.71828
3.06961 97
18
0 3.44142 89
0 3.83447 12
0 4.24952 89
0
0I
0
0
0
2.21650
2.46087
2.71828
2.98919
3.27406
01
06
18
01
39
II0
0
0
0
0
2.05491
2.26515
2.48615
2.71828
2.96190
39
76
84
18
29
526 CONFLUENT HYPERGEOMETRIC FUNCTIONS
x=2.0
II0 3.07855
7.38905 71
5.77622
4.34381
1.96790 63
17
61
05
II
-1
-1
-1
-1
-9.19616
-8.18288
-6.94107
-5.45057
98
30
82
11
( 0) 1.00000 00 ( 0) 1.00000 00
II 0 1.52511
4.35023
2.78211
2.11745
3.52448 72
19
88
92
69
CONFLUENT HYPERGEOMETRIC FUNCTIONS 527
CONFLUENT HYPERGEOMETRIC FUNCTION M(a, b, x) Table 13.1
x=3.0
a\b 0.1 0.2 0.3 0.4 0.5
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5 1 -1.78256 05
-0.4 1 -1.65079 47
-0.3 1 -1.41549 22
-0.2 1 -1.05876 41
-0.1 0)-5.60854 66
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.0000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
2:
82
0:5
0.6 2 2.05059 14
K 2 2.65765
3.36670 56
66
.i:'o 22) 4.18932
5.13805 19
80
1) 3.63241 26
528 CONFLUENT HYPERGEOMETRIC FUNCTIONS
2=4.0
a\b 0.1 0.2 0.3 0.4 0.5
-1.0 II O)-9.00000 00
-0.9 1 -1.10723 65
-0.8 1 -1.28958 24
-0.7 1 -1.43486 25
-0.6 1 -1.52885 30
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
II 12 3.20473
1.17799 42
1.44217
8.28815
5.45981 65
11
50
35
0.6 2 6.58320 17
2 8.74427 45
ii:87 3I 1.13401 20
3 1.44322 61
Zl 3 1.80888 49
x=5.0
a\b 0.1 0.2 0.3 0.4 0.5
$9" II 1 -1.56666
-2.41382 67
36
-0:s 1 -3.23511 34
-0.7 1 -3.98065 33
-0.6 1 -4.58862 62
-0.5 II 1 -4.98353 39 II 1 -2.33084 19
-0.4 1 -5.07426 08 1 -2.33646 31
-0.3 1 -4.75193 11 1 -2.15579 45
-0.2 1 -3.88754 12 1 -1.73399 46
-0.1 1 -2.32934 93 1 -1.00692 28
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
1 6.28624 01
oo*:
0:s II 2 2.62678
1.48413 16
96
8:: 2 6.01287
4.11434 26
11
0.6
t;:
0:9
Ii2 1.48413 16
2 1.98603 96
2 3.33018
2.59579 43
07
II 1 6.77444 40
1 8.98511 69
2 1.48413
1.16513 78
16
1.0 2 4.20801 74 2 1.86309 66
530 CONFLUENT HYF'ERGEOMETRIC FUNCTIONS
x=6.0
a\b 0.1 0.2 0.3 0.4 0.5
-1.0 1 -2.90000 00
-0.9 1 -6.43961 14
-0.8 2 -1.01116 95
-0.7 Ii 2 -1.37008 05
-0.6 2 -1.69209 38
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
8:: II 2 4.03428 07
1.66280 79 il 21 2.23669
9.26969 34
33
00*43 32 1.16700
7.30095 48
13 2 6.43121
4.03428 79
54
0:5 3 1.73835 48 2 9.55746 91
0.6 II 3 6.08625 44 II 3 1.35639 99
00:: 43 1.12757 14
8.38957 36 3 1.86253
2.49428 97
70
1":; 4 1.48541
1.92506 80
91 3 4.23039
3.27475 26
92
II
a\b
-1.0
-0.9
-0.8 10 -5.66666
-1.56045 67
-1.35713
-1.11025
-8.41150 64
26
62
68
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
0.1
II 1 4.32726
9.80333
2.48291 13
1.10148
6.73053 40
09
68
56
8:;
i::
0.6
2:
0:9
1.0
II2 2.34847
4.03428
1.73291
5.15728 70
3.10736 26
33
89
79
CONFLUENT HYPERGEOMETRIC FUNCTIONS 531
CONFLUENT HYPERGfEOMETRIC FUNCTION it&z, b, x) Table 13.1
x=7.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
2:
8'34
0:5
0.6
kz
0:9
1.0
532 CONFLUENT HYPERGEOMETRIC FUNCTIONS
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
82
0:3
8::
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.jl0000 00 ( 0)+1.00000 00
2;
i-98
II 3 2.98095
2.12243 80
3 4.08075
36
5.47370 48
63
l:o 3 7.22067 87
CONFLUENT HYPERGEOMETRIC FUNCTIONS 533
CONFLUENT HYPERG EOMETRIC FUNCTION M(a, b, 2) Table 13.1
z =9.0
0"::
8-i
0:5
4) 5.90279 86
4 8.44810 69
oo*sB ii 5 1.60777
1.17771 16
47
l:o 5 2.15743 14
a/b
-1.0
0.6 0.7 0.8 0.9 1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00 ( 0)+1.00000 00
0.1
00:;
x =lO.O
::t 4 3.28620
4.73642 65
75
4 6.64873 73
l:o
E II 4
5 9.13874
1.23458 32
19
Contents
Page
Mathematical Properties .................... 538
14.1. Differential Equation, Series Expansions ......... 538
14.2. Recurrence and Wronskian Relations .......... 539
14.3. Integral Representations ................ 539
14.4. Bessel Function Expansions ............... 539
14.5. Asymptotic Expansions ................ 540
14.6. Special Values and Asymptotic Behavior ......... 542
q=.5(.5)20, p= l( 1)20, 5s
The author wishes to acknowledge the assistance of David 9. Liepman in checking the
formulas and tables.
537
14. Coulomb Wave Functions
Mathematical Properties
14.1.2
w=GFds, P)+&%(T, P> (C,, Cz constants) 14.1.16 DLb)CL(d’&
PLO&L-if 2L ‘1-
14.1.5 a(~, P) =,=Z+, AE(d P’+-] PLC4 a-1 s2+q2 s-1 s
14.1.6
A:+I=l , AL.+2==&
(See Table 6.8.)
(k+L) (k-L- 1)A: =2qA;-,-A;-* (k>L+2)
14.1.20
14.1.7 CL(O)=
2Le-y Il?(L+l+itl)l
r w+q
rdd=w 91i&f2$~~’
(See chapter 6.) +2”(iq-L)(iq-L+l)+
(2L-1) (2!) ***
14.1.8 ~(7])=27r~(ez*~-l)-1
+2”(iq-L)(iq-L+1) . . . (iq+L-1)
(2L)! 1
14.1.9
14.1.21
14.1.10 (L2+v21f
cL(77)=L(2L+1) c5-167)
,Qc&- -- 2q l+p-lFL(q, P> 1
L dp GM
pL(rl) (1+q2)(4+t12) * * * (L2+q2P
14.1.11 -= +a% PI
211 w+1)[w)u2
538
COULOMB WAVE FUNCTIONS 539
14.2. Recurrence and Wronskian Relations
14.1.22 S;=$ f&(7, P) =&L(v) P+%(v, P>
Recurrence Relations
14.2.1 L ~~=(L2+112)t~~-,-(~+q)~~
GL.FL 14.2.2
(L+l) ~=[~+,1u,-[(L+1)‘+121111,,,
14.2.3
L[w+1)2+v21f~ .+,=(2L+1)[,+L(LP+1)l%
-(L+1)[L2+~21k.1
Wronskian Relations
14.2.5 FL--GL-FhGL--I=L(L2+q2)-*
14.3.2
FIGURE 14.1. &(rl, P), &(?I, P>.
FL-iGL=
v=l, p=lO
e-*q L+l
(2L+1~IcL(q)
S-yfm e-tP”(l-t)L-fl(l+t)L+i(lcEt
FL, F; GL;G;
. 14.3.3
- 24
- 22
I.1 - - 20 m
- I4
+i(l+t2)L exp [--pt+2r] arctan t]}dt
.6 -
.4- -6 14.4.1
-4
-2
-0
g2=& (7s6--309)
II-
14.5.5 eL=p-q In 2p-L -+uL
2
+Ai’ (4 [r+~+
Bi’(~) fl f2 ’ ’ ‘I } jo=l, go=o, jo*=O, go*=l-q/p
fk+l=anfn-bngr
14.4.10
gk+l=akgk+bkfk
FXrl, P) =--at(2q)-tIAW ,d+?h
G,(q, P) Bib) P
f:+l=akf:-bkgg:-fk+l/P
+d+rfi+ . . .]+Ai’!d
Bi, (2) 11 +q
&+l=ak$+bkf:-gk+dP
P2
2 WL
(27lPwGL+1rw7lP)~l
c”~(2LSl)!C,(q)
Fo-e-“?(?rp>~~l,[2(2qp)tl x=(PL-P)[-&+
1 W$l)]l,,
p;
F&e-“?(2aq)fI,,[2(2qp)*]
14.6.13 s>>o, 29-p
Go-2ecq 5 kwtlP)tl z= (2q- p) (2q) -I’3
0
dd-ii+sOn v--l)1
Co(T)) ‘v (2mj)%-*“, I
(Equality to 8S for 7>3.)
Numerical Methods
14.7. Use and Extension of the Tables Example 2. Compute GL(q, p) and GL(q, p) for
v=2, p=5, L=1(1)5.
In general the tables as presented are not si.mply Using 14.2.2 and G,,(2, 5)= .79445, G,‘= - .67049
interpolable. However, values for L>O may be from Table 14.1 we find G1(2, 5) = 1.0815. Then
obtained with the help of the recurrence relations. by forward recurrence using 14.2.3 we find:
The values of GL(q, p) may be obtained by
applying the recurrence relations in increasing
order of L. Forward recurrence may be used for L GL -CL *
References
Texta Tables
[14.1] M. Abramowitr and H. A. Antosiewicz, Coulomb [14.6] M. Abramowitz and P. Rabinowitz, Evaluation of
wave functions in the transition region. Phys. Coulomb wave functions along the transition line.
Rev. 96, 75-77 (1954). Phys. Rev. 96, 77-79 (1954). Tabulates F,, F&
[14.2] Biedenharn, Gluckstern, Hull, and Breit, Coulomb Go, Go for ~=2z=O(.5)20(2)50, 8s.
functions for large charges and small velocities. [14.7] National Bureau of Standards, Tables of Coulomb
Phys. Rev. 97, 542 (1955). wave functions, vol. I, Applied Math. Series 17
[14.3] I. Bloch et al., Coulomb functions for reactions of (US. Government Printing Office, Washington,
protons and alpha-particles with the lighter D.C., 1952). Tabulates
nuclei. Rev. Mod. Phys. 23, 147-182 (1951).
[14.4] Carl-Erik Froberg, Numerical treatment of Coulomb Mr], P>and JJ
’ (-f$’ ‘) for p=O(.2)5,
wave functions. Rev, Mod. Phys. 27, 399-411
(1955). 9=-5(1)5, L=O(1)5, 10, 11,20,21, 7D.
[14.5] I. A. Stegun and M. Abramowit,z, Generation of
Coulomb wave functions from their recurrence [14.8] Numerical Computation Bureau, Tables of Whit-
relations. Phys. Rev. 98, 1851 (1955). taker functions (Wave functions in a Coulomb
field). Report No. 9, Japan (1956).
t14.9) A. Tuhis, Tables of non-relativistic Coulomb wave
functions, Los Alamos Scientific Laboratory
La-2150, Los Alamos, N. Mex. (1958). Values
of Fo, Fb, Go,G;, P= 0(.2)40; z=O(.O5)12, 55.
546 COULOMB WAVE FUNC!CIONS
2.:
6: 5
7.0
5
E
9: 5
10.0
10. 5
11. 0
11. 5
12. 0
12.5
13.0
13. 5
14. 0
14. 5
15.0
15. 5
16. 0
16. 5
17. 0
17.5
18.0
18. 5
19. 0
19. 5
20.0
0. 5
1. 0
i:;
::i
:*z
5: 0
10.5
K
;;: ;
13:o
13. 5
14. 0
14. 5
15. 0
15. 5
16. 0
16. 5
17.0
17.5
18. 0
18. 5
19. 0
19.5
20.0
For use of this table ee Examples l-3.
COULOMB WAVE FUNC.l’IONE) 547
COULOMB WAVE FUNCTIONS OF ORDER ZERO Talde 14.1
Go(T,P)
x
Ii5
12.0
:z
13:5
14.0
:::i
15.5
2:
;;: ;
1s:o
18.5
19.0
19.5 :: k-E
20.0 16 I 117969
5 -2.0878
5 -6.3080
6 I -1.9295
10.5
:z
12:o
12.5
13.0
13.5
14.0
14.5
15.0
15.5 14 -4.66lO 11 -7.7763
16.0 15 -1.15132 12 -2.6230
16.5 15 I -6.61'?4 12 -8.8973
13 -3.0340
x 14 I -1.0399
1e:o 14 -3.5813
18.5 15 -1.2392
19.0 15 -4.3069
16 -1.5033
:::: 16 I -5.2691
- 21 3.5181
- 2 1.5740
10. 5 - 5 1.5930
11. 0 - 6 5.9782
11. 5 - 6 2.2113
12. 0 - 7 a.0697
12. 5 - 7 2.9081
13. 0
13. 5
14. 0
14. 5 - 7 5.3814
15. 0 - 7 2.0569
15.5 -10 I 5.0935
16. 0 -10 1.7129
16.5 -11 5.7147
17. 0 -ii 1.8924
17.5 -12 I 6.2217
18. 0 -12 2.0316
18.5
19. 0
19. 5
20. 0
- 1 +6.5317
- 1 -4.9515
- 1I -a.7151
- 1 I
I : g-g;
2:9346
- 5 3.8880
- 5 1.4803
10.5 - 5 I 2.3388
11. 0 - 6 9.0675
11. 5 I - 76I 5.5384
7.3981
2.0392 - 6 3.4579
12. 0
12.5
13.0
13. 5
14. 0
14. 5
15.0 - 9) 2.7940
15. 5 - 8) 2.6629
16.0
16.5
17. 0
17.5
18.0
18. 5
19. 0
19. 5
20. 0
COULOMB WAVE FUNCTIONS 549
COULOMB WAVE FUNCTIONS OF ORDER ZERO Table 14.1
Go(v,P)
6 7 8
5.5
6. 0
6. 5
7.0
.i:'o
2'0
1:::
10.5 3 1.9070
11.0 3I 4.4437
11.5 4 1.0570
12.0 4 2.5623
12. 5
13.0
13.5
14.0
14.5
15. 0
15.5
16. 0
16.5
17. 0
17.5
18. 0
18.5
19. 0
19.5
20.0
10.5 3I -1.4717
11.0 4 -1.9033
11.5 4 -4.9246
12.0 5 -1.2929
12.5 5 -3.4407
13. 0
13. 5
14. 0
14.5
15.0
15.5 8 -1.5573
16. 0 6 -4.4670
16.5 9 -1.2923
17.0 9 -3.7692
17. 5 10 i -1.1079
18.0 10 I -3.2807
18.5 10 -9.7640
19.0 11 -2.9371
19.5 11 -8.8779
20.0 12 -2.6998
550 COULOMB WAVE FUNCTIONS
q\P 11
0. 5 - l)-6.9792
::i
z
3: 0
2-z
4:5
5.0
- 2 4.3132
- 2 I 2.2096
- 2 1.0980
10. 5
11.0
11. 5
12. 0
12.5
13.0
13.5
22
15: 0
15. 5 1.2422
16. 0 4.9601
16.5 1.9580
17. 0 7.6449
17.5 2.9542
18.0 1.1303
18. 5 4.2845
19.0 1.6095
19.5 5.9943
20.0 2.2143 - 9) 1.0052
10. 5
11. 0
11.5
12. 0
12.5
13.0
13. 5
14. 0
14.5 - 5 7.9271
15.0 - 5 3.5765
15. 5
16.0
16.5
17. 0
17. 5
18.0
18. 5
19.0
19. 5
20.0
COULOMB WAVE FUNCTIONS 551
COULOMB WAWE FUNCTIONS OF ORDER ZERO Table 14.1
Go(w)
12 13 15
- 1 +8.9435
-
-
-
-
1
1
1
1
I +3.4046
-9.7085
-6.2172
+6.1593
0
- 1 I +1.1292
+5.4881
II - 01 -1.0783
-8.5560
-1.2510
- 1
0
0 I
-4.6254
-1.1612
-1.2413
10.5
11. 0
11. 5
2)
2I
3
2.5735
5.4370
1.1780
2I
2
2
1.0506
2.1519
4.5?09
1
1
I 5.0429
9.6258
2 1.8964
12. 0 3 2.6115 2 9.6(54
12. 5 3 5.9114
13.0
13.5 4 1.0421
14. 0 4 7.6408 4I 2.35'53
14.5 5 1.8544 4 5.55~78
15. 0 5 I 4.5606 5 1.3286
15. 5
16.0
16.5
4) 9.7988 4
4
I 1.1531
2.5494
I
4 5.7251
17.0 5 1.3047
17.5 5 3.0146
18.0
18.5
19. 0
19. 5
20.0
5
6)
6
6
I 7.0570
1.6726
4.0107
9.7253
7 2.3833
- 1 +6.6972 - 1 +9.7040
II - l)-4.6958
- 1I -7.2341 - 2 +5.5060
- 1 -7.2415 - 1 -9.1975
- 1 +2.8479
- 1 -1.4460
- 1I -5.0324 I - 1 -5.3907
I
II
- 1 -6.5243 (- l)-6.8002 - 1 -5.5683
- 1 -5.8597 - 1 -5.4972 - 1 -6.7414
- 1 -9.1132 - 1 -5.56,63
-5.7431
-4.9764 - 1 -4.9245 - 1 -6.2956
0I -1.6356
0 -3.0877
0 -5.9776
1I -1.1842
1 -2.4038
1 -5.0022
2 -1.0663
10.5 1I -8.8802 O)-7.7837
11. 0 2 -1.8956
11.5 2 -4.1335
12.0 2 -9.1940
12.5
13.0
13.5
14.0 4 I -9.2211 3 1-8.4644
14. 5 5 -2.3041 4 -1.9742
15. 0 5 -5.8301 4)-4.6712
I
15. 5 6 -1.4929 5 -1.1203 4 -1.1531
16. 0 6I -3.8658 5 -2.7217 4 -2.6329
16.5 7 -1.0118 5 -6.6925 4 -6.0946
17. 0 7 -2.6753 5 -1.4291
17.5 7 -7.1420 5 -3.3924
18. 0 8 -1.9243 5 -8.1473
18. 5 8I -5.2302 6 1 -1.9785
19. 0 9 -1.4335 6 -4.8557
19.5 9 -3.9609 7 1-1.2038
20.0 10 I -1.1028 7)-3.0133
552 COULOMB WAVE FUNCTIONS
10. 5
E
12: 0
12.5
13. 0
13. 5
14. 0
14. 5
15. 0
15. 5
:s
17:o
17. 5
18. 0
18. 5
19. 0
19.5
20.0
(- l)-7.4641 - 2)+1.8327
- l)-7.0977 - l)-5.3380
10.5
11. 0
11.5
12. 0
12. 5
13.0
13. 5
14.0
14.5
15. 0
15. 5
16.0
16. 5
17. 0
17.5
18. 0
18. 5
19.0
19.5
20.0
COULOMB WAVE FUNCTIONS 553
COULOMB WAVE FUNCTIONS OF ORDER ZERO Table 14.1
16
10. 5
11.0
11. 5
12. 0
12.5
13. 0
13.5
14.0
14. 5
15. 0
15. 5
16. 0
:Ei
17: 5
18.0
18.5
19. 0
:E .
554 COULOMB WAVE FUNCTIONS
II 1
(-4)5
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 556
15.1. Gauss Series, Special Elementary Cases, Special Values of
the Argument . . . . . . . . . . . . . . . . . . . . 556
15.2. Differentiation Formulas and Gauss’ Relations for Contiguous
Functions. . . , . . . . . . . . . . . . . . . . . . 557
15.3. Integral Representations and Transformation Formulas . . . 558
15.4. Special Cases of F(a, b; c; z), Polynomials and Legendre
Functions. . . . . . . . . . . . . . . . . . . . . . 561
15.5. The Hypergeometric Differential Equation . . . . . . . . 562
15.6. Riemann’s Differential Equation . . . . . . . . . . . . 564
15.7. Asymptotic Expansions . . . . . . . . . . . . . . . . 565
References . . . . . . . . . . . . . . . . . . . . . . . . . . 565
556
WPEROE~METRE mcmo~s 557
15.1.21 15.2. Differentiation Formulas and Gauss’
Relations for Contiguous Functions
Differentiation Formulas
(l+a--b#O, -1, -2, . . .)
15.2.1 ; F(u, b; c; z)=f F(uf1, b+l; c+l; z)
15.1.22
F(u, b; a-b+2;-1)=2-“7r”2(b-l)-‘r(a--b-+2)
15.2.2
[
1 1
r(3a>r(g+3a-b)-r(3+6a>r(l+~~--2;j 1 -$ F(u, b; c; z)=w F(u+n, b+n; c+n; z)
(a--b+zzo, -1, -2, . . .)
15.2.3
15.1.23 F(l, a; a+l; -1)=3a[lC1(~+3u)-~(~u)]
.$ [Za+n-l F(u, b; c; z) I= (a)&-‘F(ufn, b; c; 2)
15.2.4
15.1.25 15.2.5
F(u, b; +u+g+i; 3)=2~(a--)-‘r(l+~u+~ib)
g [Zc-atn-l (l-z)“+“-“F(a, b; c; z)]
{[r(6~>r(B+3b)l-~-[r(3+4u)r(3b)i-~:I
(~(u+b)+l#O, -1, -2, . . .) = (~-u),&-=-~(1-,z)“+~-~-“F(a-n, b; c; z)
15.1.26 15.2.6
F(u, l-u; b; *>=
2?&(b) [r(;a+$b) r (~+++~a)]-1 g [(l--z) a+b-cF(u, b; c; z)]
(b#O, -1, -2, . . .)
-“-“;;;~-““’ (l-z)=+--“F(q b; c+n; z)
15.1.27
I-- FU, 1; a+l; 3>=4lt(3+%4-e41 15.2.7
(a#-1, -2, -3,. . .)
g [ (1- z)=+“-~F(u, b; c; z)]
25.1.28
,.- e, a; a+l; 3>=~-‘a[~(3+~ua>-~(3a>l -(-l>“(Mc-b>n (l-z)=+F(u+n, b; cfn; z)
(c)n
(a#-1, -2, -3,. . .)
15.2.8
15.1.29
F (a, t-ta; g-%;-+>=(#-2a ;g;g; -$ [z”-‘(l-z)~-“+“F(u, 6;~; z)]
g [~~-‘(l--z)=+~-~F(u, b; c; z)]
(g+gu#o, -1, -2, . . .>
=(c-n),zc-“-yl-,z) a+b-c-nF(u-n, b-n; c-n; z)
15.1.31
F (a, &a+&; #.~a+#; ef*i3) Gauss’ Relations for Contiguous Functions
From 15.3.1 and 15.3.2 a number of transformation formulas for F(a, b; c; z) can bederived.
15.3.3 F(u, b; c; z)=(l--~)~-=-~F(c-u, c-b;, c; z)
15.3.6
r(c)r(c-a-b) F(a, b; a+b-c+l; 1-z)
?(c---a)r(c-b)
+(1-$-“-b
r(c)r(a+b--c) F (c-u, c-b; c-a-b+l; 1-z)
r (4 r(b)
(larg Cl--z)l<d
WSb) 2 T-
k&(b).
‘5*3*‘0 ‘(a, b;a+b; 2)=r(a)r(b) n=O (nv)2 [~(n+l)-~(u+n)-~(b+n)--In (l--z)l(l-~)~
If, and only if the numbers *(l-c), &ta-b), f (a+b-c) are such, that two of them are equal
or one of them is equal to 3, then there exists a quadratic transformation. The basic formulas
are due to Kummer [15.7] and a complete list is due to Goursat [15.3]. See also [15.2].
(1--111’--2)2
15.3.18 =(l--z)-4°F a, 2b-a; b+& -
( 4Ji--z )
l-;-G
15.3.19 F(a, a+3; c;z>=(++&‘i%>-“F 2a, 2cc-c+l; c; -
1+&-z
G-1
15.3.21 2a, 2c-2a-1; c;
26 >
Legendre functions are connected with those special cases of the hypergeometric function for which
a quadretic transformation exists (see 15.3).
15.4.7 F(a, b;2b; z)=22b-‘I’(&+b)z+-b (l-z)+cb-u-*)P:-:-t [(1-;)(l-d-q
15.4.14 F(&,~;a--b+i;d=r(a-b+i)2~b-r(i-2)-b~~~=
respectively. The general theory of differential equations of the Fuchsian type distinguishes between
the following cases.
A. None of the numbers c, c-a-b; a-b is equal to an integer. Then two linearly independent solutions
of 15.5.1 in the neighborhood of the singular points 0, 1, m are respectively
15.5.8 toa =z- “F(b, b-c+l; b-a+l; z--l)=za-c(z-l)c-a-b~(l-a, c-u; b-&l; z-l)
The second set of the above expressions is obtained by applying 15.3.3 to the first set.
Another set of representations is obtained by applying 15.3.4 to J5.5.3 through 15.5.8. This
gives 15.5.9-15.5.14.
15.5.9 ~~~,,~=(l-z)-“F a, c-b; c; -)=(l-,z)-bF(b,
zf1 c-a; c; --&)
15.5.3 to 15.5.14 constitute Kummer’s 24 solutions of the hypergeometric equation. The analytic con-
tinuation of wl ,z(O,(z) can then be obtained ‘by means of 15.3.3 to 15.3.9.
B. One of the numbers a, b, c-a, c-b is an kteger. Then one of the hypergeometric series for
instance wl ,2(0), 15.5.3, 15.5.4 terminates and the corresponding solution is of the form
15.5.15 w=zQ(l-z)@p&)
where p,(z) is a polynomial in z of degree n. This case is referred to as the degenerate case of the
hypergeometric differential equation and its solutions are listed and discussed in great detail in [15.2].
C. The number c-a-b is an integer, c nonintegral. Then 15.3.10 to 15.3.12 give the analytic continu-
ation of wr ,2Co,into the neighborhood of z=: 1. Similarly 15.3.13 and 15.3.14 give the analytic continu-
ation of wl ,2(0) into the neighborhood of z= 03 in case a-b is an integer but not c, subject of
course to the further restrictions c-a=O, fl, f2 . . . (For a detailed discussion of all possible
cases, see [l&2]).
D. The number c=l. Then 15.5.3, 15.5.4 are replaced by
15.5.16 wlto,=F(a, b; 1; z)
564 HYPERGEOMETRIC FUNCTIONS
- wmn
15.5.17 wz(O)=F(a, b; 1; 2) In 2+g1 ~@,)a z”[~(a+n)-9(u)+~(b+n)-~(~)--2~(n+l)+2~(2)1 (l4<1)
E. The number c=m+l, m=l, 2, 3, . . . . A fundamental system is
The hypergeometric differential equation 15.5.1 (a) The generalized hypergeometric function
with the (regular) singular points 0, 1, Q) is a
w=p{;,
; “,.z}
15.6.4
special case of Riemann’s differential equation
with three (regular) singular points a, b, c
15.6.1
15.6.5
0 OJ 1
w=P 1 0 a 0 21
11-c b c-u-b J
(c) The Legendre functions P!(z), &:(z)
The pairs of the exponents with respect to the
singular points a; b; c are (IL,cr’; 6, B’; y, y’ respec- 15.6.6
0 co 1
tively subject to the condition
w=P -4V 3P 0 (1-22)-1
15.6.2
r 1
cr+a’+B+B’+r+Y’=l
I%++ -3P t J
(d) The confluent hypergeometric function
The complete set of solutions of 15.6.1 is denoted
by the symbol 15.6.7
0 Q) c
15.6.3 w=P +ku -c c-k z
3-U 0 k
provided lim c+= .
HYPERGEOMETRIC FUNCTIONS 565
Transformation Formulas for Riemann’s P Function
a b c a b
z-a k z-c ’ 8-k-l
15.6.8 - -PiY B y a+k
z-b
(>( z-b I{
a’ 8’ 7’ d-j-k 8,-k--l
The P function on the right hand side is G.auss’hypergeometric function (see 15.6.5). If it is replaced
by Kummer’s 24 solutions 15.5.3 to 15.5.14 the complete set of 24 solutions for Riemann’s differential
equation 15.6.1 is obtained. The first of these solutions is for instance by 15.5.3 and 15.6.5
References
115.11 P. Appell and J. Kampe de F&et, Fonctions hyper- [15.4] E. Goursat, Propri&& generales de l’equation
geometriques et hyperspheriques (Gauthiers- d’Euler et de Gauss (Actualit& scientifiques et
Villars, Paris, France, 1926). industrielles 333, Paris, France, 1936).
[15.2] A. Erdelyi et al., Higher transcendental functions, [15.5] J. Kampe de Feriet, La fonction hypergeom&rique
vol. 1 (McGraw-Hill Book Co., Inc., New York, (Gauthiers-Villars, Paris, France, 1937).
N.Y., 1953).
[15.6] F. Klein, Vorlesungen iiber die hypergeometrische
[15.3] E. Goursat, Ann. Sci. Ecole Norm. Sup(2)19, Funktion (B. G. Teubner, Berlin, Germany,
3-142(1881). 1933).
566 HYPERGEOMETRIC FTJNCI’IONS
[15.7] E. E. Kummer, Uber die hypergeometrische Reihe, [15.11] C. Snow, The hypergeometric and Legendre func-
J. Reine Angew. Math. 15, 39-83, 127-172(1836). tions with applications to integral equations of
[15.8] T. M. MacRobert, Proc. Edinburgh Math. Sot. 42, potential theory, Applied Math. Series 19 (U.S.
84-88(1923). Government Printing Office, Washington, D.C.,
[15.9] T. M. MacRobert, Functions of a complex variable, 1952).
4th ed. (Macmillan and Co., Ltd., London, [15.12] E. T. Whittaker and G. N. Watson, A course of
England, 1954). modern analysis, 4th ed. (Cambridge Univ. Press,
[15.10] E. G. C. Poole, Introduction to the theory of linear Cambridge, England, 1952).
differential equations (Clarendon Press, Oxford,
England, 1936).
16. Jacobian Elliptic Functions and Theta
Functions
L. M. MILNE-THOMSON 1
Contents
Page
Mathematical Properties .................... 569
16.1. Introduction ..................... 569
16.2. Classifiication of the Twelve Jacobian Elliptic Functions . . 570
16.3. Relation of the Jacobian Functions to the Copdlar Trio . . 570
16.4. Calculation of the Jacobian Functions by Use of the Arith-
metic-Geometric Mean (A.G.M.) ........... 571
16.5. Special. Arguments .................. 571
16.6. Jacobian Functions when m=O or 1 ........... 571
16.7. Principal Terms ................... 572
16.8. Change of Argument ................. 572
16.9. Relations Between the Squares of the Functions ..... 573
16.10. Change of Parameter .................. 573
16.11. Reciprocal Parameter (Jacobi’s Real Transformation) ... 573
16.12. Descending Landen Transformation (Gauss’ Transforma-
tion) ....................... 573
16.13. Approximation in Terms of Circular Functions ...... 573
16.14. Ascendling Landen Transformation ........... 573
16.15. Approximation in Terms of Hyperbolic Functions .... 574
16.16. Derivatives ...................... 574
16.17. Addition Theorems .................. 574
16.18. Double Arguments .................. 574
16.19. Half Arguments ................... 574
16.20. Jacobi’s Imaginary Transformation, ........... 574
16.21. Complex Arguments ................. 575
16.22. Leading Terms of the Series in Ascending Powers of u. .. 575
16.23. Series IExpansion in Terms of the Nome p ........ 575
16.24. Integrals of the Twelve Jacobian Elliptic Functions. ... 575
16.25. Notation for the Integrals of the Squares of the Twelve
Jacobian Elliptic Functions ............. 576
16.26. Integrals in Terms of the Elliptic Integral of the Second
Kind1 ....................... 576
16.27. Theta .Functions; Expansions in Terms of the Nome p . . 576
16.28. Relations Between the Squares of the Theta Functions . . 576
16.29. Logarithmic Derivatives of the Theta Functions ..... 576
16.30. Logarithms of Theta Functions of Sum and Difference . . 577
16.31. Jacobi’s Notation for Theta Functions ......... 577
16.32. Calculation of Jacobi’s Theta Function O(ulm) by Use of
the Arithmetic-Geometric Mean ........... 577
Page
16.33. Addition of Quarter-Periods to Jacobi’s Eta and Theta
Functions . . . . . . . . . . . . . . . . . . . . . 577
16.34. Relation of -Jacobi’s Zeta Function to the Theta Functions. 578
16.35. Calculation of Jacobi’s Zeta Function Z(ulm) by Use of the
Arithmetic-Geometric Mean . . . . . . . . . . . . . 578
16.36. Neville’s Notation for Theta Functions . . . . . . . . . 578
16.37. Expression as Infinite Products. . . . . . . . ‘. . . . . 579
16.38, Expression as Infinite Series . . . . . . . . . . . . . . 579
Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 579
16.39. Use and Extension of the Tables . . . . . . . . . . . . 579
References . . . . . . . . . . . . . . . . . . . . . . . . . . 581
?Y,(rO\cYO),
&iz s,(Eydq
tYn(eO\aO),Jseccy tY,(E;\op)
~t=O~(5~)85~, e, e1=00(50)900, 9-10D
$ In GC(u)= -f(e\c~‘)
&In Sn(~)=g(to\aP)
The author wishes to acknowledge his great indebtedness to his friend, the late Professor
E. H. Neville, for invaluable assistance in reading and criticizing the manuscript. Pro-
fessor Neville generously supplied material from his own work and was responsible for many
improvements in matter and arrangement.
The author’s best thanks are also due to David S. Liepman and Ruth Zucker for the
preparation and checking of the tables and graphs.
16. Jacobian Elliptic Functions and Theta Functions
Mathematical Properties
.S .S
Similarly all the functions pq u can be expressed
.n Zi .n :: in terms of cp. This second set of definitions,
.S .C .S .C
although seemingly different, is mathematically
.n .d .n .d equivalent to the definition previously given in
terms of a lattice. For further explanation of
the pattern being repeated indefinitely on all notations, including the interpretation, of such
sides. expressions as sn (V\(Y), cn (ulm), dn (u, k), see 17.2.
569
JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS
570
16.2. Classification of the. Twelve Jacobian Elliptic Functions
According to Poles and Half-Periods
16.3.4
FIGURE 16.2. Jacobian elliptic junctions
ns u, nc u, nd u provided that when two letters are the same, e.g.,
pp u, the corresponding function is put equal to
unity.
JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS 571
16.4. Calculation of the Jacobian Functions by and then compute successively p+,, pNm2, . . .,
Use of the Arithmetic-Geometric Mean lpi, p. from the recurrence relation
(A.G.M.)
16.4.3 sin (24+,-&=~ sin q*,
For the A.G.M. scale see 17.6.
To calculate sn (ulm), cn (ulm), and dn (ulm) Then
form the A.G.M. scale starting with 16.4.4
16.4.1 ao=l, bo=Jm1, co=&, sn (ujm)=sin (potcn (ulm)=cos a
terminating at the step N when cN is negligible to
dn (ulm)=
‘OS PO
the accuracy required. Find lpN in degrees where cos (Cpl-PoPg)’
(oN=2NaNuIso0 From these all the other functions can be deter-
16.4.2
r mined.
16.5. Special Art guments
u Fm u cn u dn u
16.5.1 0 0 1 1
1 mP
16.5.2 +K m$14
(l+m:l’)“’ (l+mll’*)l’a
16.5.3 K 1 0 rn+l¶
im-‘I’ (1 + m1’*)1/2
16.5.4 $K') mll’ (l+ml’*)L/*
16.5.7 iK' co co
,szz +
az+-n m+n+n I n2zfn ,n!+n n-nz az-n az+n n-n n--n n+n n- n
- = --
n 53 ZT’t’91
n SP wL’9 t
n su OI’L’91
n 38 6’L’9T
n 311 8’L’9T
nap L’L’91
npu 9’t’91
n PS S’L’91
n ~3 p’L.91
nup s-L.91
n u3 Z’L’91.
n us I’t’9T.
--
=
*pagddns aq 03 svq (“2-n) JOS!MP aq? c~eq$smam + pus
pa!lddns aq 09 s’~q (“X--n) .ro~m3 aql ~‘t3q$smam X aJaqhi ‘sur.103asaqq sangI gsg &n~o~~o3 aq&
-n bd 30 alod ‘13JO ‘oJaz B ‘gu!od Lmup~o 11’13
s: Lx S’B Z?uyp.~oom(‘x-n)-+3 ‘(‘x-n) x& ‘8 sm.103
aql 30 auo ssq pus ma2 pd!mvd aq$ panw st uo!susdxa aqq 30 rnsa%ys.~y aq$ ‘,xz+x ‘,x! ‘;y $0
30 auo s! “x alaqM ‘&-n) 30 maMod Bmpuaoss uy papusdxa so n bd uo~~oun3Di$dma aq? uaq&
smra& ~dpupd ‘t-91
JACOBIAN ELLIPTIC :FUNCTIONS AND THETA FUNCTIONS 573
16.9. Relations Between the Squares o:f the 16.12.1 /quay> a’&*’
Functions
then
16.9.1 --dn2u+ml=-m cn2u=m sn2u-m
16.12.2 sn (ul m) _ (1+c(‘/~) sn (4~)
16.9.2 -mind2u+ml=-mmlsd2u=m d2u-m - 1+p1/2sn2 (VIP)
16.93 nd2u=‘- m ca2U, dn2u=l-m sn2u When the parameter m is so small that we may
ml
neglect m2 and higher powers, we have the
and therefore approximations
am (ujm) =gd u+i ml (sinh u cash u-u) sech u. = cn2u-sn2u. dn% cn2u-sn2u. dn2u
l-msn4u = cn2u+sn2u. dn2u
Another way of calculating the Jacobian func- 16.18.3 dn 2u
tions is to use Landen’s ascending transformation =dn2u-msn2u.cn2u=dn2u+cn2u(dn%-1)
to increase the parameter sufficiently for the above 1 - msn4u dn2u-cn2u(dn2u- 1)
formulae to become applicable. See also 16.13. 16.18.4 1 -cn 2u sn2u . dn%
l+cn 2u= cn2u
16.16. Derivatives
1 - dn 2u msn2u acn2u
16.18.5
Func- Derivative l+dn 2u= dn2u
tion
16.19. Half Arguments
sn u cn u dn u
:z% cn u --8n u an u Pole n 16.19.1
16:16:3 dn u -m sn u cn u
16.16.4 cd u -ml sd u nd u
16.16.5 sd u cd u nd u Pole d 16.19.2
16.16.6 nd u m sd u cd u
16.16.7 dc u ml sc u nc u m,+dn u-kmcn u
16.16.8 nc u SC u dc u Pole c 16.19.3 dn21yu=
16.16.9 SCu dc u nc u lfdn u
16.16.10 ns u -ds u cs u 16.20. Jacobi’s Imaginary Transformation
16.16.11 ds u -cs u ns u Pole 8
16.16.12 csu -na u da u 16.20.1 sn(iulm)=isc(ulmJ
1620.2 cn(iujm)=nc(uJ ml)
Note that the derivative is proportional to the
product of the two copolar functions. 16.20.3 dn(iulm)=dc(ul m,)
JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS 575
16.21. Complex Arguments 16.23.6
16.23. Series Expansions in Terms of the Nome 16.24.4 $ cd u du=m-‘12 In (nd u+m112sd u)
q=t?‘--*K”K and the Argument v=?ru/(2hr) 16.24.5 Jsd u du= (mmJ-1’2 arcsin (-m1’2cd u)
16.23.1 sn (ulm)=- 27r 2 pn+1/2 sin (27bfl)v 16.24.6 Jnd u du=mI-112 arccos (cd u)
rn1i2K n=O I- p2n+1
16.24.7 Jdc u du=ln (nc u+sc u)
16.23.2 cn (ul m) = - 2* 5 p.n+1/2 cos (2n.+l)v 16.24.8 Jnc u du=m;“2 In (dc u+m:bc U)
m1/2K n=O1 + p2n+l
trouble with many-valued inverse circular func- 16.27.3 &(z, a)=&(z) =1+2 $, qn2 cos 2nz
tions.
16.27.4
16.25. Notation for the Integrals of the Squares
of the Twelve Jacobian Elliptic Functions tJ4(2, *)=tY,(z)=l+2 2 (-l)“qn2 cos 2ti.2
n=1
Pqu= Upq2t dt when q#s Theta functions are important because every
16.25.1
s0 one of the Jacobian elliptic functions can be ex-
pressed as the ratio of two theta functions.
16.25.2 Ps u=s,’ (pq2t-$) dt-t See 16.36.
The notation shows these functions as depend-
Examples ing on the variable z and the nome p, 1~1<l.
In this case, here and elsewhere, the convergence
Cdu=
S 0
’ cd2t dt, Ns u=l(ns2t-$)dt-t is not dependent on the trigonometrical terms.
In their relation to the Jacobian elliptic functions,
we note that the nome p is given by
16.26. Integrals in Terms of the Elliptic Inte-
gral of the Second Kind (see 17.4) P=e--*K’IK 7
16.26.1 mSn u= -E(u) +u where K and iK’ are the quarter periods. Since
16.26.2 mCn u=E(u) -mlu Pole n q=q(m) is d et ermined when the parameter m is
given, we can also regard the theta functions as
16.26.3 Dn u=E(u) dependent upon m and then we write
16.26.4 mCd u= -E(u) +u+msn u cd u f%(z, a>=Wlm>, a=l, 2, 3, 4
16.26.5 but when no ambiguity is to be feared, we write
mm,Sd u=E(u) -mlu-msn u cd u Pole d 9,( 2) simply.
The above notations are those given in Modern
16.26.6 m,Nd u=E(u) -msn u cd u
Analysis [16.6].
16.26.7 DC u= -E(u) +u+sn u dc u There is a bewildering variety of notations, for
example the function G,(z) above is sometimes
16.26.8
denoted by 8,(z) or b(z); see the table given in
mlNc u= -E(u) +miu+sn u dc u Pole c Modern Analysis [16.6]. Further the argument
u=2Kz/~ is frequently used so that in consulting
16.26.9 m$c u= -E(u) +sn u dc u
books caution should be exercised.
16.26.10 Ns u=-E(u)+u-cn u ds u
16.28. Relations Between the Squares of the
16.26.11 Theta Functions
Ds u=-E(u)+mlu-cn u ds u Pole 8
16.28.1 ~~(z)s:(O)=s:(z)~:(O)-s:(z>st(O>
16.26.12 Cs u=-E(u)-cn u ds u
16.28.2 S:(z)S:(O)=~:(z)9:(0)--61(z)9~(0)
All the above may be expressed in terms of
Jacobi’s ieta function (see 17.4.27). 16.28.3 S:(z)r9:(0)=191(~)(93(0)-d:(z)9:(0)
8; (4
-=-tan u-j-4 n$l (-1)” i-C+ sin 2nu 16.31.1 ~(u~m)=O(u)=29&), +
fJ*(u>
16.31.2 ~,(u~m)=0~(~)=83(v)=@(u+K)
16.29.3 ‘$$=4 2 (-1)” ha sin 2nU
3 It=1
16.31.3 H(ulm)=H(u)=t$(v)
16.30.4 16.32.4
These functions are defined in terms of Jacobi’s FIQURE 16.4. Neuiue’s thetajunctions
theta functions of 16.31 by
fi,(u), G,(u), @d(u), &&)
H&SK) 1
16.36.1 s.(lL)=g$$ S,(u)= m=-
H(K) 2
JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS 579
1.0
r \. \ 16.38. Expression as Infinite Series
Let v=4(2K)
16.38.1
16.38.2 tpC(u)=
[ 1‘$2 “’ n$O qncn+l) cos (2n+l)v
l/2
FIGURE 16.5. Logarithm@
junctions
derivatives of theta 16.38.3 &(u)=
c1& { 1+2 gI qn2cos 2nvj
16.38.4
Numerical Methods
16.39. Use and Extension of the Tables
=lO-‘X7.67
Example 1. Calculate nc (1.996501.64) to 4s.
From Table 17.1,1.99650=K+.OOl. From the which is negligible.
table of principal terms From 16.12.4
Also
cc=-
0
19 ’
1+p==$? w=.19.
& is negligible to 4D. Thus
580 JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS
[(.191145)(1.01810)$.19] dn (.20].81)=.984056
= .982218+;X& (.184408)[ .384605] cn (.20].81) = .980278.
= .982218+ .000049=.982267.
Thus dn (.20] 31) = .98406.
Example 5. Use the A.G.M. scale to compute dc (.672].36) to 4D.
From 16.9.6 we have dc2(.672].36) =.36+l-snZ(~72, 36)* We now calculate sn(.672].36) by the
I.
method given in 16.4. Form the A.G.M. scale
0 58803 . 55472
a 2.
1: 0780
1533 83509
88101 00260
09789 00260
09805 1. 0048 . 00120
. 1.
3 4. 3066 -. 91879 0’ 0’ 1. i
References
Texts Tables
[16.1] A. Erdelyi et al., Higher transcendental funetions, [16.7] E. P. hdams and R. L. Hippisley, Smithsonian
vol. 2 (McGraw-Hill Book Co., Inc., New York, mathematical formulae and tables of elliptic
N.Y., 1953). functions, 3d reprint (The Smithsonian Institu-
[16.2] L. V. King, On the direct numerical calculation of tion, Washington, D.C., 1957).
elliptic functions and integrals (Cambridge [16.8] J. Hoiiel, Recueil de formules et de tables nume-
Univ. Press, Cambridge, England, 1924). riques (Gauthier-Villars, Paris, France, 1901).
[16.9] E. Jahnke and F. Emde, Tables of functions, 4th
[16.3] W. Magnus and F. Oberhettinger, Formulas and
theorems for the special functions of mathe- ed. (Dover Publications, Inc., New York, N.Y.,
1945).
matical physics (Chelsea Publishing Co., New
York, N.Y., 1949). [16.10] L. M. Milne-Thomson, Die elliptischen Funktionen
von Jacobi (Julius Springer, Berlin, Germany,
[16.4] E. H. Neville, Jacobian elliptic functions, :2d ed. 1931).
(Oxford Univ. Press, London, England, 1951). [16.11] L. M. Milne-Thomson, Jacobian elliptic function
[16.5] F. Tricomi, Elliptische Funktionen (Akademische tables (Dover Publications, Inc., New York,
Verlagsgesellschaft, Leipzig, Germany, 1948). N.Y., 1956).
116.61 E. T. Whittaker and G. N. Watson, .4 course of [16.12] G. W. and R. M. Spenceley, Smithsonian elliptic
modern analysis, chs. 20, 21, 22, 4th ed. ((Cam- function tables, Smithsonian Miscellaneous COI-
bridge Univ. Press, Cambridge, England, 1952). lection, vol. 109 (Washington, D.C., 1947).
582 JACOBIAN ELLIPTIC FUNCTIONS AND THETA FUNCTIONS
0 5 10 15 20 25
0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000
0.08715 5743 0.08732 1966 0.08782 4152 0.08867 3070 0.08988 7414 0.09149 5034
0.17364 8178 0.17397 9362 0.17497 9967 0.17667 1584 0.17909 1708 0.18229 6223
0.25881 9045 0.25931 2677 0.26080 4191 0.26332 6099 0.26693 4892 0.27171 4833
0.34202 0143 0.34267 2476 0.34464 3695 0.34797 7361 0.35274 9211 0.35907 2325
25 0.42261 8262 0.42342 4343 0.42586 0446 0.42998 1306 0.43588 2163 0.44370 5382 65
0.50000 0000 0.50095 3708 0.50383 6358 0.50871 3952 0.51570 1435 0.52497 0857 60
:; 0.57357 6436 0.57467 0526 0.57797 7994 0.58357 6134 0.59159 9683 0.60225 0597 55
40 0.64278 7610 0.64401 3768 0.64772 1085 0.65399 8067 0.66299 9145 0.67495 6130 50
45 0.70710 6781 0.70845 5688 0.71253 4820 0.71944 3681 0.72935 6053 0.74253 3161 45
0.76604 4443 0.76750 5843 0.77192 5893 0.77941 4712 0.79016 4790 0.80446 5863
0.81915 2044 0;82071 4821 0.82544 2256 0.83345 4505 0.84496 1783 0.86028 0899
0.86602 5404 0.86767 7668 0.87267 6562 0.88115 1505 0.89332 9083 0.90955 llbb
0.90630 7787 0.90803 6964 0.91326 9273 0.92214 2410 0.93489 7610 0;951as 9199
0.93969 2621 0.94148 5546 0.94691 1395 0.95611 4956 0.96935 0025 0.98700 0216
75 0.96592 5826 0.96776 8848 0.97334 6839 0.98281 0311 0.99642 3213 1.01458 4761 15
80 0.98480 7753 0;98668 6836 0.99237 4367 1.00202 5068 1.01591 0350 1.03444 0908 10
85 0.99619 4698 0.99809 5528 1.00384 9133 1.01361 2807 1.02766 2527 1.04641 6011 5
90 1.00000 0000 1.00190 8098 1.00768 3786 1.01748 5224 1.03158 9925 1.05041 7974 0
E\QI 30 36 40 45 50 55 45
0' 0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000 0.00000 0000 90"
0.09353 4894 0.09606 0073 0.09914 2353 0.10287 9331 0.10740 5819 0;11291 2907
1: 0.18636 3367 0.19139 9811 0.19754 9961 0.20501 0420 0.21405 3194 0.22506 4618 :50
0.27778 4006 0.28530 3629 0.29449 2321 0.30564 8349 0.31918 5434 0.33569 3043 75
:: 0.36710 5393 0.37706 5455 0.38924 7478 0.40405 4995 0.42204 9614 0.44403 4769 70
0.45365 1078 0.46599 3521 0.48110 6437 0.49950 2749 0.52189 9092 0.54932 5515 65
0.53676 4494 0.55141 5176 0.56937 7735 0.59127 8602 0.61799 6720 0.65080 1843
0.61581 3814 0.63268 1725 0.65339 2178 0.67868 8658 0.70961 8904 0.74770 4387 EE
0.69019 6708 0.70917 3264 0.73250 7761 0.76106 3101 0.79606 0581 0.83928 2749 50
0.75934 4980 0.78030 3503 0.80611 4729 0.83776 1607 0.87664 1114 0.92480 2089 45
0.82272 9031 0.84552 4503 0.87364 0739 0.90817 9128 0.95071 1025 1.00355 1297
:; 0.87986 2121 0.90433 1298 0.93455 6042 0.97175 1955 1.01765 9399 1.07485 2509
60 0.93030 4365 0.95626 6326 0.98837 8598 1.02796 3895 1.07692 1759 1.13807 1621
65 0.97366 6431 1.00092 3589 1.03467 8996 1.07635 2410 1.12798 8100 1.19262 9342
70 1.00961 2870 1.03795 2481 1.07308 5074 1.11651 4503 1.17041 0792 1.23801 2299
75 1.03786 5044 1.06706 1179 1.10328 blO0 1.14811 2152 1.20381 2008 1.27378 3626
1.05820 3585 1.08801 9556 1.12503 6391 1.17087 7087 1.22789 0346 1.29959 2533
s850 1.07047 0366 1.10066 1511 1.13815 8265 1.18461 4727 1.24242 6337 1.31518 2322
90 1.07456 9932 1.10488 6686 1.14254 4218 1.18920 7115 1.24728 6586 1.32039 6454
In calculating elliptic functions from theta functions, when the modular angle exceeds about 60”, use
the descending Landen transformation 16.12 to induce dependence on a smaller modular angle.
Compiled from E. P. Adams and R. L. Hippisley, Smithsonian mathematical formulae and tables of
elliptic functions, 3d reprint (The Smithsonian Institution, Washington, D.C., 1957) (with permission).
JACOBIAN ELLIPTIC FUNCTIONS AND THETA FTJNCTIONS 583
TIIETA FUNCTIONS Table 16.1
a&\4
5 lo” 15 20 25
1.00000 00000 1.00000 00000 1.0000000000 1.00000 00000 1.00000 00000
1.00001 44942 1.00005 83670 1.00013 28199 1.00023 99605 1.00038 29783
1.00005 75362 1;00023 16945 1.00052 72438 1.00095 25510 1.00152 02770
1.00012 78184 1.00051 47160 1;00117 12875 1.00211 61200 1.00337 73404
1.00022 32051 1.00089 88322 1.00204 53820 1.00369 53131 1.00589 77438
1.00034 07982 1.00137 23717 1.00312 29684 1.00564 21475 1.00900 49074
1.00047 70246 1.00192 09464 1.00437 13049 1.00789 74700 1.01260 44231
1.00062 77451 1.00252 78880 1.00575 24612 1.01039 27539 1.01658 69227
1.00078 83803 1.00317 47551 1.00722 44718 1.01305 21815 1.02083 14013
1.00095 40492 1.00384 18928 1.00874 26104 1.01579 49474 1.02520 88930
1.00111 97181 1.00450 90305 1.01026 07491 1.01853 77143 1.02958 63905
1.00128 03532 1.00515 58975 1.01173 27599 1.02119 71444 1.03383 08852
1.00143 10738 1.00576 28392 1.01311 39167 1.02369 24323 1.03781 34098
1.00156 73002 1.00631 14139 1.01436 22536 1.02594 77596 1.04141 29561
1.00168 48932 1.00678 49535 1.01543 98405 1.02789 45992 1.04452 01522
1.00178 02800 1.00716 90696 1.01631 39354 1.02947 37972 1.04704 05862
1.00185 05621 1.00745 20912 1.01695 79795 1.03063 73701 1.04889 76746
1.00189 36042 1.00762 54187 1.01735 24037 1.03134 99632 1.05003 49895
; 1.00190 80984 1.00768 37857 1.01748 52237 1.03158 99246 1.05041 79735
60 65" 70 75 80” 85
1.00000 00000 1.00000 00000 1.ooooc~ 00000 1.00000 00000 1.00000 00000 1.00000 00000
1.00313 85295 1.00406 92257 1.00534~ 44028 1.00720 88997 1.01026 06485 1.01663 88247
1.01245 94672 1.01615 50083 1.02121 95717 1.02862 79374 1.04076 43440 1.06618 38299
1.02768 16504 1.03589 51569 1.047151 56657 1.06363 90673 1.09068 07598 1.14751 59063
1.04834 57003 1.06269 75825 1.0823Ei 38086 1.11122 86903 1.15864 11101 1.25875 62174
1.07382 76019 1.09575 73598 1.12585 71388 1.17001 24008 1.24276 19421 1.39725 25218
1.10335 71989 1.13408 00433 1.17627 97795 1.23826 96285 1.34068 05139 1.55957 26706
1.13604 11010 1.17651 06705 1.23214. 31946 1.31398 80140 1.44960 33094 1.74151 57980
1.17088 93642 1.22176 77148 1.291761 91861 1.39491 71251 1.56636 90138 1.93815 19599
1.20684 51910 1.26848 10938 1.35335' 85717 1.47863 07744 1.68752 66770 2.14389 95792
1.24281 67937 1.31523 31927 1.41504 43413 1.56259 67789 1.80942 88493 2.35264 71220
1.27771 04815 1.36060 17261 1.47494 78592 1.64425 25175 1.92833 82823 2.55792 12198
1.31046 39783 1.40320 31647 1.53123 64694 1.72108 41609 2.04054 54606 2.75309 84351
1.34007 a9457 1.44173 53793 1.58218s 06891 1.79070 70015 2.14249 29245 2.93165 25995
1.36565 16965 1.47501 81348 1.62620 90720 1.85094 39670 2.23090 12139 3.08742 47870
1.38640 11169 1.50203 00916 1.66195 87940 1.89989 92030 2.30289 04563 3.21489 91220
1.40169 28947 1.52194 10514 1.68832 00831 1.93602 35909 2.35609 12550 3.30946 52989
1.41105 92570 1.53413 83232 1.70447 27784 1.95816 92561 2.38873 86793 3.36764 82512
1.41421 35624 1.53824 62687 1.70991 35651 1.96563 05108 2.39974 38370 3.38728 70037
$&lnWu)=f(t\a)
‘\a 0 5 10” 15 20 25 a/4
O0 0
11.4m3005 ll.iia29 11.;306 11.2:449 ll.iil275 10.8~811 li
1; 5.67128 5.62812 5.57427 5.49902 5.40253 80
3.73205 :%z 3.70365 3.66823 3.61876 3.55536 75
2105 2.74748 2174225 2.72658 2.70051 2.66414 2.61756 70
25 2.14451 2.14043 2.12820 2.10787 2.07952 2.04325
1.73205 1.72875 1.71888 1.70248 1.67962 1.65041 66:
;50 1.42815 1.42543 1.41729 1.40378 1.38497 1.36096 55
40 1.19175 1.18949 1.18270 1.17143 1.15577 50
45 1.00000 0.99810 0.99240 0.98296 0.96985 '0%:
. 45
50 0.83910 0.83750 0.83273 0.82481 0.81383 0.79987
0.70021 0.69489 0.68830 0.67915 0.66754
2: 0.57735 oo*% 0.57297 0.56754 0.56001 0.55047
65 0.46631 0:46542 0.46277 0.45839 0.45232 0.44464
70 0.36397 0.36328 0.36121 0.35779 0.35306 0.34708
75 0.26795 0.26592 0.26340 0.25992 0.25553 15
80 0.17633 KZ 0.17499 0.17334 0.17105 10
0.08749 0:08732 0.08683 0.08600 0.08487 KKi
98: 0.00000 0.00000 0.00000 0.00000 0.00000 o:ooooo i
c\a 30 35 40 45" 50 55
O0
5 10.6m5083 10.?7113 lO.om4914 9.;479 9.;7764 a.;657
5.28496 5.14645 4.98711 4.80696 4.60585 4.38332
:i 3.47816 3.38730 3.28290 3.16502
20 2.56090 2.49430 2.41789 2.33179 %%E
. ;z!.
25 1.99919 1.88828 1.82172 1.74793 1.66695 65
30 1.61498 :%t;3 1.52607 1.47292 1.41419 1.35001
1.33189 1:29791 1.25919 1.21591 1.16828 1.11647 565"
iz 1.11167 1.08352 0.97687 0.93462 50
45 0.93301 0.90958 xz. E595:
. 0.82139 0.78679 45
0.78307 0.76355 0.74151 0.71714 0.69066 0.66232 40
0.65359 0.61923 0.57749 0.55441
0.53902 Lx;93 0.51093 :z3:28 0.47705 0.45846 :i
0.43543 ;:M; 0.41292 0:39991 0.38595 25
0.33992 . 0.32248 0.31242 0.30168 i%::
. 20
75 0.25028 0.24424 0.23151 0.23017 0.22235 0.21419 15
0.16471 O.lb076 0.15634 0.15155 0.14645 0.14114 10
ii 0.08173 0.07977 0.07759 0.07522 0.07270
90 0.00000 0.00000 0.00000 0.00000 0.00000 %oooo~
. z
ta
\ 60 65" 70” 75 80" 85"
00
5 a.;941
4.13843
;.f;;;; ;A;: 6.4009756
3.24056
5.7:041
2.85790
4.7?263
2.37760
:i
20
2.72935
2.01530
p;;
. 2:36323
1.75208
2.15026
1.60057 YEf
.
1.60605
1.22261
25 1.57876 1.13996 65
;50
1.28047
1.06066
E%
1:00096
Xf
0193737
1.26603
tzs5 %27E
0.99169
0.83453
5";
40 0.88940 0.84142 0.79086 p;'9"4: 0:68225 :*x
45 0.75000 0.71131 0.67101 . 0.58682 0:54358 2
50 0.63242 0.60125 0.56918 0.53662 0.50411 0.47247
0.53023 0.50526 0.47987 0.45454 0.42988 0.40690
2; 0.43911 0.41932 0.39943 0.37992 0.3b140 0.34488
65 0.35605 0.34063 0.32532 0.31054 0.29684 0.28513
70 0.27885 0.26719 0.25574 0.24484 0.23497 0.22685
75 0.20584 0.19749 0.18935 0.18170 0.17490 0.16949
0.13572 0.12026 0.11601 0.11272
i50 0.06742 8%:~~ Oo.Ef 0.05988 0.05784 0.05628
90 0.00000 o:ooooo o:ooooo 0.00000 0.00000 0.00000
Contents
Page
Mathematical Properties .................... 589
17.1. Definition of Elliptic Integrals ............. 589
17.2. Canonical Forms ................... 589
17.3. Complete Elliptic Integrals of the First and Second Kinds . . 590
17.4. Incomplete Elliptic Integrals of the First and Second Kinds . 592
17.5. Landen’s Transformation ................ 597
17.6. The Process of the Arithmetic-Geometric Mean ...... 598
17.7. Elliptic Integrals of the Third Kind ........... 599
Qh>=alh>h, 15D
ml=0(.01).15
Page
Table 17.7. Jacobian Zeta Function Z(~\CY) . . . . . . . . . . . . 619
Values of K(cr)Z( ~\a)
cY=o”(20)900, 5°(100)850, (p=o”(50)900, 6D
Table 17.8. Heuman’s Lambda Function A,,(q\a) . . . . . . . . . . 622
A0 (rp\or)=F(~\go0-a)+2K(a) 2(,\90°-a), 6D
K’(a) 7r
a=0°(20)900, 5°(100)850, (p=O“(5°)900
Table 17.9. Elliptic Integral of the Third Kind lT(n; q\(y) . . . . . . 625
n=o(.l)l, cp,a=0°(150)900, 5D
The author acknowledges with thanks the assistance of Ruth Zucker in the computa-
tion of the examples, Ruth E. Capuano for Table 17.3, David S. Liepman for Table 17.4,
and Andreas Schopf for Table 17.9.
i89
=St4
Let
17.1.2 dw=u
0
y2=aG’+a1$+a2z8+a3x+a4 (la01+ Iad +O>
Elliptic Integral of the Second Kind
=b,(~-C)4+b!(x-C)3+b&-C)2+b&-c)+t’4
(lbol+
hl$0) 17.2.8 E(~\a)=E(a~m)=~z(1-t2)~~(l-mt2)~dt
1’7.1.3 I,=
s
xsy-‘dx, J,=
S [y(x-c)s]-‘dx
= y1-
0
17.1.4
(s+2)a0~~+3+3a,(2s+3)I,+2+a2(s+1)1,+,
17.2.10 = S 0
Udn2wdu,
+3~(2s+1)Is+sa4111_l=x~y (s=O, 1, 2, . . .) u
a See [17.7] 22.72. 17.2.11 =m++m
S 0
cn2 w dw
590 ELLIPTIC INTEGRALS
Elliptic Integral of the Third Kind The elliptic integral of the third kind depends
17.2.14
on three variables namely (i) the parameter,
n(n;P\4=Jo~ Cl- n sin2 6)-l [l -sin2 (Ysin2 B]-1’2dB (ii) the amplitude, (iii) the characteristic n.
When real, the characteristic may be any number
If 2=sn (ulm), in the interval (- 03, a). The properties of the
integral depend upon the location of the charac-
17.2.15 teristic in this interval, see 17.7.
n(n;ulm)=
S’ (l-nt2)-1[(l-t2)(l-mt2)]-1/2dt
0
17.3. Complete Elliptic
and Second
Integrals
Kinds
of the First
17.2.16 =
S
o” (l- n sn2 (wlm))-ldw
The Amplitude p
Referred to the canonical forms of 17.2, the ellip-
tic integrals are said to be complete when the
amplitude is +n and so z=1. These complete in-
17.2.17 cp=am u=arcsin (sn u) =arcsin z tegrals are designated as follows
can be calculated from Tables 17.5 and 4.14. 17.3.1
The Parameter m
[K(m)]=K=~1[(l-t2)(l-mt2)]-1~2dt
Dependence on the parameter m is denoted by a
vertical stroke preceding the parameter, e.g., =S 0
*I2
(l-m sin2 O)-1/2dB
Fblm>.
Together with the parameter we define the 17.3.2 K=F($rlm)=F(%r\cr)
complementary parameter ml by
17.3.3
17.2.18 m+ml= 1
E[K(m)]=E=~01(l-t2)_‘Il(l_mt2)112dt
When the parameter is real, it can always be
arranged, see 17.4, that 0 <rn < 1. =S 0
*I2
(I- m sin2 e)‘i2d8
The Modular Angle cc
17.3.4 E=E[K(m)]=E(m)=E(&\F\ol)
Dependence on the modular angle (Y, defined in
terms of the parameter by 17.2.1, is denoted by a We also define
backward stroke \ preceding the modular angle,
thus E((P\(Y). The complementary modular angle 17.3.5
is r/2--a or 90°--a! according to the unit and rt !
thus ml=sin2 (90°-a)=cos2 LY. K’=K(m,)=K(l-mm)=
S 0
(l-ml sin2 8)-1/2d8
17.3.10 E=*aF(-*, 3; 1; m)
Infinite Series
17.3.11 17.3.24 am u=v+g 2’ sin 2sv where v=ru/(2K)
.9=1 SC1+a’“>
K(m)=& [l+c>’ rn+(Ey m2 Limiting Values
17.3.19 17.3.34
K(m)-[ao+arml+ . . . +a4m~l+[bo+bIm,+ ...
17.3.20 +b4mtl ln Wml>+4m>
le(m)l12XlO-*
log,, i log,, i= (?rlog,, e)a= 1.86152 28349 to 10D
ao= 1.38629 436112 bo= .5
al= .09666 344259 b,= .12498 593597
17.3.21 a~= .03590 092383 b2= .06880 248576
a3= .03742 563713 b3= .03328 355346
q=exp [--lrK’/K]=E+8 (Gy+y (zy a(= .01451 196212 b4= .00441 787012
8 The approximations 17.3.33-17.3.36 are from C. Has-
+992 g 4+ . . . (Id<l) tings, Jr., Approximations for Digital Computers, Prince-
0 ton Univ. Press, Princeton, N. J. (with permission).
ELLIPTIC INTEGRALS
592
17.3.35
~(m)=[l+a~m~+azm~l+[blm,+bzm~l In (lh>
f&d
Jr(m)J<4X10S6
al= .46301 51 b1= .24527 27
a,=.10778 12 b2=.04124 96
17.3.36
E(m)=[l+aIm,+ . . . +a4mf]+[blmI+ . ..
+b4m:] In (l/m,) +6(m)
Ie(m)1<2X10-g
al= .44325 141463 bl= .24998 368310
a3= .06260 601220 bz= .09200 180037
a3= .04757 383546 ba= .04069 697526
al=.01736 506451 bd= .00526 449639
17.4.6
E(u+2mK+2niK’) =E(u) +2mE+2ni(K’-E’)
17.4.7 E(K-u) =E-E(u)+msn u cd u
Imaginary Amplitude
If tan @=sinh cp
17.4.8 F(icp\cY) =iF(B\&r-a)
17.4.9
E(+\a) = -iE(e\+-cz) +iF(e\+-a)
+i tan e(i -cos2 ar sin2 e>i
Jacobi’s Imaginary Transformation
17.4.10
E(iulm) =&+dn(ulmI)sc(ulml) -E(ulmdl
Complex Amplitude
FIWRE 17.2. Complete elliptic ini%grd of the
second kind. 17.4.11 F(q+i$lm) =F(Xlm)+iF(~ImI)
ELLIPTIC INTEGRAL& 593
where cot2 X is the positive root of the equation Parameter Greater Than Unity
x2-[cot2 cp+m sinhv csc2p--mJx-ml cot:“(a=O 17.4.15 F(cp(m)=m-+F(Olm-l), sin O=m* sin cp
and m tan2 PI= tan2p cot2X- 1.
17.4.16 E(ulm) =mtE(um~lm-l) - (m- 1)~
17.4.12
by which a parameter greater than unity can be
E((P+i#\cY) =E(X\a) -iE(p\90°-a) replaced by a parameter less than unity.
+iF(p\90°-a)+qp
Negative Parameter
where 17.4.17
bl=sin2 a sin X cos X sin2 ~(l-sin2 (Ysin2 A)*
I.0-
.6 -
406
.6 -
30”
.4-
2o”
.2 -
IO’
0 I, I I I I I I I I I I I I I I <---,a 00,00
0o IO’ 20° 3o” 4o” so” 60° 7o” 60° 9’f 100 200 ,o* 4OQ 300 600 70* SO’ SO’
FIGURE 17.3. Incomplete elliptic integral qf the FIGURE 17.4. Incomplete elliptic integral of the
first kind. .first kind.
F(v\d, p constant F(y~\4, a constant
594 ELLIPTIC INTEGRALS
2
I.2 - 60'
2 7.5.
1.0 - .
.6 -
I
.6 -
I
.4 -
I
0' IO' 20' 30' 40° 50' 60' 70' 60' 90' '
goe.ELQp-Q.
F((P\~ a! constant.
FIGURE 17A. p-90’ 7~
1.6 -
LO -
.6 - 500
E((o\4
FIGURE 17.8. 90’ 7-q’ (Yconstant.
17.4.21
F((p\90°) =ln (set pftan .p) =ln tan
FIGURE 17.6. Incomplete elliptic integral of the 17.4.22 F(icp\SOO) =i arctan (sinh (p>
second kind.
E(v\a), 9constant 17.4.23 Eb\O) = cp
Special Casee 17.4.24 E@o\O) =ilp
17.4.34 Z(uIl)=tanh u
Addition Theorem
17.4.35
Z(u+v)=Z(u)+Z(v)--msn u sn 2, sn(u+E)
Jacobi’s Imaginary Transformation
17.4.36
iZ(iu~m)=Z(zllm,)+~,-dn(ulml)sc(zllm,) .2 -
sin (I = b/a (a
cd-1
(I >
? b”
b a2 ~a2(P-x2) l=b
I &la~=bqa2Lx2) cd v
a>b
m = Plaa 17.4.47
&-I " !! sin2 (p=-
x2-ae t=a dc v
(I)a a2 39-F
ns-*(Ixb2
-a-aa
> sin +J=: t=a ns v aJzrn( 7) [(P--a2);lLP)]*~”
b t=b nc v ba z ta+a2
cota=- ab 17.4.50
cos ‘p=i (aa+b2)*” s b ta [(P+a2)(EP)]l/*
m=as/(as+bl)
sin* cp=-d+bn t=(d+P)*f*dsv dt
I a2+xa (a2+@‘qm& [(p+aa)(&2-~)]l/’
xa(aa+ P)
sina p= p(a2+xs)
t=(aa+P)Ila
ab sdv 1
aa(aa+W1’z s o’(f2+a*)
dt
[(c+aZ)@2-@)]1/2
CO8 $2 t=bcnv
b
ELLIPTIC INTEGRALS ~597
Some Important Special Cases
Z!Z =
fF(9\4 008 9 a &(9\4 co9 (p
_- _-
17.4.53 17.4.57
- dt X2-1 - dt 2-l-h
45O
s + (1Sl”P x2+1 s z (ta-i)* x-1+&
17.4.34 17.4.58
= dt l-29 = dt &+1-x
45O
s 0 u+w 1+x1 s 1 (ta-i)+ &1+x
L s-
2 “1
17.4.55
= dt
* (P-l)+
1
X
45O
17.4.59
’
s I (1-W
dt &-1+x
&+1-x
17.4.56 17.4.60
1-43-x
1-2 111 1 dt X 45O dt
s 0 (l---t’)4 s d-(1--P)+ 1+&x
-
sina--x-83
= dt 17.4.72
A 9-B2-83
s4 (x-8) -AZ
F(9\4 CO8 9=(x-p)+p
17.4.63
x h dt (81-/%)(x-A)
s JF cos8
9=(h-83)(81-4
17.4.64
* dt x--81
A F(v\4 sin’ 17.4.74
s 81 JF
9=-
z-82 u dt
A --
s z J(-P) F(cD\W’-4)
17.4.63
A F(v\a) COB%X-81
(p=- -
x-83 17.5. Landen’s Transformation
17.4.66
Descending Landen Transformation 6
F(v\ (90” -a”))
Let CY*,CY,,+~be two modular angles such thpt
17.4.67
17.5.1 (1 +siIl (Y&I) (1 +cos CYJ=2 (%‘+I $a”)
on-63
F(v\ (90” - ~‘1) cod c-81-2
and let Q,,, Q,,+~ be two corresponding amplithdes
17.968 such that
= dt b--Ba)@--8s)
X- F(9\ (90” -a”))
s 8s d-P sin’ 9=(8*-/9*)(x-883j 17.5.2 tan (Q,,+~- an>=cos (Y, tan pn (vn+l>cp.)
17.4.69 5 The emphasis here is on the modular angle sine
x h dl is an argument of the Tables. All formulae
-- Z-81
s- = CP
cosla ‘-&-& Landen’s transformation may also be expressed in
of the modulus k=mi=sin a and its complement k’
=cos (1.
598 ELLIPTIC INTEGRALS
Thus the step from n to n+ 1 decreases the modular With ~,=cr we have
angle but increases the amplitude. By iterating
17.5.13 F(cp\ru) =2(l+sin ff)-lF&\~~~)
the process we can descend from a given modular
angle to one whose magnitude is negligible, when
17.4.19 becomes applicable. 17.5.14 F(cp\ol) =2n 2: (1 +sin a,)-lF((p,\(y,)
With LY~=(Ywe have
17.5.3 17.5.15 F(Cp\cY) = 8i,(1 +cos CYJF((On\(Y,)
F(cp\cr)=(l+cos 4-1F(P1\LYl)
17.5.16 F(p\a)=[ csc or8Qsin (r8]$In tan ($?r+*@)
=Hl+sin +Y(P~\~
17.5.17 *=lim (pn
17.5.4 F(p\~4=2-“~~~(l+sinol,)F(q,\a,) n+m
Neighhorhood of a Right Angle (see also 17.4.13)
17.5.5 I%\4 = Cpjl (1 +sin a,> When both (p and (Y are near to a right angle,
interpolation in the table F(cp\a) is difficult.
Either Landen’s transformation can then be used
17.5.6
with advantage to increase the modular angle and
decrease the amplitude or vice-versa.
17.5.7 K=F(b\a)=$r 8tI (l+sin a,)
17.6. The Process of the Arithmetic-Geometric
Mean
17.5.8 F&\a) =27r-‘Ka
Starting with a given number triple (a,, b,,, c,)
17.5.9 we proceed to determine number triples
(al, h, cd, (a~, b2, cd, . . . , (aN, bN, CN) according to
I&\,) =F((p\a) [ 1 -i sin2 (Y(1 +f sin cyl the following scheme of arithmetic and geometric
1 means
-tF sin a1 sin K+ . . .)]+sinct![a (sinar,)1/2sin~,
1
)I
co
+Psinculsina2sinaQ+. ..
cl=3 (ao--bo)
c2=3 (al-h)
Ascending Landen Transformation
whence &(8)
---Cot fl+4 5 Q2S(1-2q28 cos 2fi+q*“)-l sin 2j3
$1 (P) a=1
17.6.6 K’(a)=&
In the above we can also use Neville’s theta
functions 16.36.
17.6.7
17.7.6 n(n\~)=K(ar)+~lK(cu)Z(e\cr)
=z1 [c;~+2c:*+2%;*+
K’(a)-,??(a)
. . . $2%&2,21
K’ (4 Case (ii) Hyperbolic Case n> 1
To calculate F(cp\cr), E(cp\cy) start from 17.5.2 The case n>l can be reduced to the case
which corresponds to the descending Landen O<N<sin* (Yby writing
transformation and determine cpl, cpz, . . , q,V
successively from the relation 17.7.7 N=n-’ sin2 cr, pl=[(n-l)(I-nn-1sin2 cw)]’
17.6.8 tan (cpn+l-cpJ=(bn/~n> tan G, CPO=(O 17.7.8
Then to the prescribed accuracy H(n; (~\a, = -n(N; (~\a, +F(v\d
17.6.9 F(v\4 =v%/(2%nr) *
+&In1 [(A(v) -I-& tan cp>
(A(v)--P, tan (PI-Y
17.6.10
where A(q) is the delta amplitude, 17.2.4.
a\4 = E(cp\4 - u-m7 F(cp\4
* =cl sin (Pl+c2 sin qn+ . . . +cN’sin @ 17.7.9 &\a> =KW -WV\4
17.7. Elliptic Integrals of the Third Kind Case (iii) Circular Case sin* ff<n< I
17.7.12
17.7.3
17.7.13
p=
C2 .a@*sinh 2sfi I[ 1+2 2 q”” cash 1
2s/3
-1
a=1 a-1
Ww\4=~1 I-3 ln [&(v+@/~&-i31
17.7.14 n(n\a)=K((y)+3as2[1-~o(~\a)]
Special Cases
17.7.18 n=O
no% (0\4 =Fb\4
17.7.19 n=O, a=0
NO; cp\O) =cp
17.7.20 cY=o
n.l.y=45’
The case n<O can be reduced to the case -f d In (l+n+ sin cp)(l-n) sin cp)-‘1 n#l
sin2 a<N< 1 by writing
17.7.22 n= fsin (Y
17.7.15 (1Tsin (~){2II(fsin ar; cp\cx)--F(cp\cr)}
N= (sin2 a-n) (1 -TX)-’ =arctan [(lTsin CY)tan +LY/A(c,o)]
Numerical Methods
17.8. Use and Extension of the Tables
(10+8X)2-4(3+X)(9+10X)=0; i.e., if X=-i or f
Example 1. Reduce to canonical form y-l&c,
s and then
where
y2=-3x4+3423-1192+1722-90 Q,+; Q2=; (z--1)2, Q,-f Q2=; +2)2
By inspection or by solving an equation of the
fourth degree we find that Solving for Q1and Qz we get
S-l s s
y dx= f T-‘dt= f [(3t2+2)
=
The computation could also be made using
n a, b, c, common logarithms with the aid of 17.3.20. The
I -_
point of this procedure is that it enables us to
0 1.00000 00000 .lllll 11111 .99380 79900 calculate p1 without the loss of significant figures
1 .55555 55555 .33333 33333 .44444 44444
2 .44444 44444 .43033 14829 .11111 11111 which would result from direct interpolation in
3 .43738 79636 .43733 10380 .00705 64808 Table 17.1. By this means In (l/al) can be found
.43735 95008 .43735 94999 .00002 84628
d .43735 95003 .43735 95003 0 without loss of accuracy.
- - Example 6. Find m to 10D when K’/K=.25
1
and when K’/K=3.5.
Thus K(80/81)=; ?ru,-l=3.59154 510(Il. From 17.3.15 with K’/K=.25 we can write the
iteration formula
Example 4. Find E(80/81).
First Method m(“+“= 1- 16em4” exp [ --~rL(m(“‘)/K’(m’“‘)].
Use 17.3.30 which gives, with m=80/81 Then by iteration using Tables 17.1 and 17.4
=f [1.854075-.535623]=.659226
A rough estimate now sEows that Qlies between
where
40’ and 41’. We therefore form the following 2 2 2
table of F(q\56.789089’) by direct interpolation sin (o~=--I sin p2=-7 sin2 a=--’
2 4 4
in the foregoing table Thus
F
4o:oo .74003 Second Method, Numerical Integration
40.5O .75040
41.0° .76082 If we wish to use numerical integration we must
observe that the integrand has a singularity at
whence by linear inverse interpolation t=2 where it behaves like [8(t--2)1-t.
We remove the singularity at t=2, by writing
$0=40.5Of.5O
.75342- .75040 =40 644go
.76082- .75040 ’ 1
and so sin cp=.65137=sn (.75342].7).
This method of bivariate interpolation is given
where
merely as an illustration. Other more dfiect
methods such as that of the arithmetic-geomletric f(t)=[(t2-2)(t2-4)]-f-[8(t-2)]-*.
mean described in 17.6 and illustrated for the
Jacobian functions in chapter 16 are less laborious. If we definef(2)=0,
Example 8. Evaluate
3 [(2Pfl) (P-2)]-“W.
S‘f (W
2
where
l+
S4f
2
(t)dt=l-.340773=.659227.
S 0
20
(24-12t+2t2-ta)-1'2 dt. n co9alI I sin Q*
We have ; :02943
33333 333
725 .99956 904
.94280 663
3. 00021 673 .99999 998
There is only one real zero and we therefore sin (2~~-90°)=sin 30°, Qpl=60°
use 17.4.74 with P(t)=t?-22t2+12t-24, fl=2 so sin (2Q2-Q1)=Sin. al sin (01, lp2=57.367805'
that P’(2)=16, P”(2)=8, X=2 and therefore
Sin (2Qa-&=Sin a2 Sin e, Q~= 57.348426'
m=sh2 ,=I, a=30° Sin (2Qp-Q~)=Si.tl cU3 Sin Qa, ~,=57.348425'=@.
4
From 17.5.16
Therefore the given integral’is
2
F(go0\300) =A . 1 .942:O 904 1.999:6 663
,:,=; [F(Q,\~Q~)--F(Q~\~Q~)~
s-s0
where
1 =1.37288 050 In tan 73.674213’
CO8 Ql=--1 ~,=70.52877 93’
3
=1.37288 050(1.22789 30)
1 F(90°\300) = 1.68575 to 5D.
CO9 Q2=-, ~2=6O'
2 Example 13. Find the value of F(89.5’\89.5’).
and the integral=3[1.510344-1.212597]=.148874. First Method
Example 12. Use Landen’s transformation to
evaluate This is a case where interpolation in Table 17.5
is not possible. We use 17.4.13 which gives
F(89.5°\89.50) = F(90’\89.5’) - F(#\89.5O)
First Method, Descending Transformation where
We use 17.5.1 to give cot $=sin (.5O) cot (.5O)=cos (.5O)
$=45.00109 084O
2
1 +sin cxl= .=1.071797 and F($\89.5’) = .881390 from Table 17.5.
1+cos 30
cos ,,=[(l-sin CYJ(1 +sin CXJ]~/~= .997419 F(90’\89.5’) =K(sin2 89.5’) =K(.99992 38476)
=6.12777 88
1 +sin (Ye= 2 = 1.001292 ; co9 CY2=.999999
1 +cos al Thus F(89.5q89.5’) z5.246389.
Second Method
1 +sin cys= 1 +cos (Y2= 1 .oooooo
2
Landen’s ascending transformation, 17.5.11,
Thus from 17.5.7, gives
ELlUPTIC INTEGRALS 605
CO8 Cq=(l-sin 89.5°)/(l+sin 89.5O) This is case (i) of integrals of the third hind,
sin q=[(l-cos CYJ(l+cos q)]*=.99999 99997 O<n<sin2 (Y, 17.7.3
CO8 as=0
n=$ (0=45~,ar=30~,
sin rYz=l.
17.5.12 ‘then gives c=arcsin (n/sin2 0r)*=30°,
sin (2~--89.5°)=sin 89.5Osin 89.5’ /3= $rF(30°\300)/K(300) = .49332 60
=.99992 38476 V=&rIq45°\300)/K(300)=.74951 51,
2+q-89.5°=89.29290490, q,=89.39645 24L5' 61= (16/45)*
q= .01797 24.
1 t
In (tan 89.69822 801°)=5.24640.
.99996 19231 > Using the q-series, 16.27, for the d functions we get
Example 14. Evaluate
II (A; 45’\30’) = (16/45)+ { - .02995 89
S 1
* [(9-P)(16+ta)3]-fdt to5D.
Table
+(1.86096 21)(.74951 51)) =.813845.
17.9 gives .81385 with 4 point Lagrangian
From 17.4.51 the given integral interpolation.
II (~\30°)=K(300)+;~~ [1-~~(e\30~)]
=S 0
*/4
(l++ sin2 @-‘(l-t sin2 0)-k&
to 5D.
where e=arcsin [(~--~)/cos~cx]~~~=~~~. Thus using Here the characteristic is negative and we there-
Table 17.8
fore use 17.7.15 with n=-i, sin2 cx=i
l-I ($\3OO) =2.80099.
References
Texts [17.6] F. Tricomi, Elliptische Funktionen (Akademische
Verlagsgesellschaft, Leipzig, Germany, 1948).
[17.1] A. Cayley, An elementary treatise on elliptic
functions (Dover Publications, Inc., New York, [17.7] E. T. Whittaker and G. N. Watson, A course of
N.Y., 1956). modern analysis, chs. 20, 21, 22, 4th ed. (Cam-
[17.2] A. Erdelyi et al., Higher transcendental functions, bridge Univ. Press, Cambridge, England, 1952).
vol. 2, ch. 23 (McGraw-Hill Book Co., Inc.,
New York, N.Y., 1953). Tables
[17.3] L. V. King, On the direct numerical calculation of
[17.8] P. F. Byrd and M. D. Friedman, Handbook of
elliptic functions and integrals (Cambridge
elliptic integrals for engineers and physicists
Univ. Press, Cambridge, England, 1924).
(Springer-Verlag, Berlin, Germany, 1954).
[17.4] E. H. Neville, Jacobian elliptic functions, 2d ed.
(Oxford Univ. Press, London, England, 1951). [17.9] C. Heuman, Tables of complete elliptic integrals,
[17.5] F. Oberhettinger and W. Magnus, Anwendung der J. Math. Phys. 20, 127-206 (1941).
elliptischen Funktionen in Physik und Technik [17.10] J. Hoiiel, Becueil de formules et de tables numb-
(Springer-Verlag, Berlin, Germany, 1949). riques (Gauthier-Villars, Paris, France, 1901).
ELLIPTIC INTEGRALS 607
[17.11] E. Jahnke and F. Emde, Tables of functions, 4th [17.14] L. M. Milne-Thomson, The Zeta function of
ed. (Dover Publications, Inc., New York, N.Y., Jacobi, Proc. Roy. Sot. Edinburgh 52 (1931).
1945). [17.15] L. M. Milne-Thomson, Die elliptischen Funktionen
[17.12] L. M. Milne-Thomson, Jacobian elliptic function von Jacobi (Julius Springer, Berlin, Germany,
tables (Dover Publications, Inc., New York, 1931).
N.Y., 1956). [17.16] K. Pearson, Tables of the complete and incomplete
(17.131 L. M. Milne-Thomson, Ten-figure table of the elliptic integrals (Cambridge Univ. Press, Cam-
complete elliptic integrals K, K’, E, E’ and a bridge, England, 1934).
1 [17.17] G. W. and R. M. Spenceley, Smithsonian elliptic
table of 8&, sm, Proc. London IMath. function tables, Smithsonian Miscellaneous Col-
sot. 2, 33 (1931). lection, vol. 109 (Washington, D.C., 1947).
608 ELLIPTIC INTEGRALS
T&k 17.1 COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS
AND THE NOME p WITH ARGUMENT THE PARAMETER m
0.10 1.61244 13487 20219 2.57809 21133 48173 0.00658 46515 53858 0.90
0.11 1.61688 90905 05203 2.53333 45460 02200 0.00728 28484 49518 0. a9
0.12 1.62139 31379 80658 2.49263 53232 39716 0.00798 89058 49815 0.88
0.13 1.62595 48290 38433 2.45533 80283 21380 0.00870 30002 35762 0. a7
0.14 1.63057 55488 81754 2.42093 29603 44303 0.00942 53141 02678 0.86
0.15 1.63525 67322 64580 2.38901 64863 25580 0.01015 60362 37153 0. a5
0.16 1.63999 98658 64511 2.35926 35547 45007 0.01089 53620 10173 0.84
0.17 1.64480 64907 98881 2.33140 85677 50251 0.01164 34936 87540 0. a3
0.18 1.64967 82052 94514 2.30523 17368 77189 0.01240 06407 58856 0.82
0.19 1.65461 66675 22527 2.28054 91384 22770 0.01316 70202 86392 0.81
0.20 1.65962 35986 10528 2.25720 53268 20854 0.01394 28572 75318 0.80
0.21 1.66470 07858 45692 2.23506 77552 60349 0.01472 83850 66891 0.79
0.22 1.66985 00860 83368 2.21402 24978 46332 0.01552 38457 56320 0. 78
0.23 1.67507 34293 77219 2.19397 09253 19189 0.01632 94906 37206 0.77
0.24 1.68037 28228 48361 2.17482 70902 46414 0.01714 55806 74605 0.76
0.25 1.68575 03548 12596 2.15651 56414 99643 0.01797 23870 08967 0.15
0.26 1.69120 81991 86631 2.13897 01837 52114 0.01881 01914 93399 0.74
0.27 1.69674 86201 96168 2.12213 18631 57396 0.01965 92872 66940 0. 73
0.28 1.70237 39774 10990 2.10594 83200 52758 0.02051 99793 66788 0.72
0.29 1.70808 67311 34606 2.09037 27465 52360 0.02139 25853 82708 0. 71
0.30 1.71388 94481 78791 2.07536 31352 92469 0.02227 74361 57154 0.70
0. 31 1.71978 48080 56405 2.06088 16467 30131 0.02317 48765 35013 0. 69
0.32 1.72577 56096 29320 2.04689 40772 10577 0.02408 52661 67250 0.68
0.33 1.73186 47782 52098 2.03336 94091 52233 0.02500 89803 73177 0. 67
0.34 1.73805 53734 56358 2.02027 94286 03592 0.02594 64110 66576 0.66
0.35 1.74435 05972 25613 2.00759 83984 24376 0.02689 79677 51443 0. 65
0.36 1.75075 38029 15753 1.99530 27776 64729 0.02786 40785 93729 0.64
0.31 1.75726 85048 82456 1.98337 09795 27821 0.02884 51915 76181
0. 38 1.76389 83888 a3731 1.97178 31617 25656 0.02984 17757 44138 kz
0.39 1.77064 73233.33534 1.96052 10441 65830 0.03085 43225 51033 0: 61
0.40 1.77751 93714 91253 1.94956 77498 06026 0.03188 33473 13363 0. 60
0. 41 1.78451 88046 81873 1.93890 76652 34220 0.03292 93907 86003 0.59
0.42 1.79165 01166 52966 1.92852 63181 14418 0.03399 30208 70043 0.58
0.43 1.79891 80391 87685 1.91841 02691 09912 0.03507 48344 66773 0. 57
0.44 1.80632 75591 07699 1.90854 70162 81211 0.03617 54594 93133 0.56
0.45 1.81388 39368 16983 1.89892 49102 71554 0.03729 55570 75822 0. 55
0.46 1.82159 27265 56821 1.88953 30788 53096 0.03843 58239 43468 0.54
0.47 1.82945 97985 64730 1.88036 13596 22178 0.03959 69950 38753 0.53
0.48 1.83749 13633 55796 1.87140 02398 11034 0.04077 98463 75263 0.52
0.49 1.84569 39983 74724 1.86264 08023 32739 0.04198 51981 67183 0. 51
0. 50 1.85407 46773 01372 1.85407 46773 01372 0.04321 39182 63772 0.50
ml K’(m) K(nd 91(m) m
Table 17.2 COMPLETE ELLIPTIC INTEGRALS OF THE FIRST AND SECOND KINDS
AND THE NOME q WITH ARGUMENT THE MODULAR ANGLE a
E’(a)=E(90%)
ql(“)=q(9V-a)
Kc4 K’ (4 Q(Q)
1.57079 63267 94897 0.00000 00000 00000
1.57091 59581 27243 5.43490 9m8296 25564 0.00001 90395 55387
1.57127 49523 72225 4.74271 72652 78886 0.00007 61698 24680
1.57187 36105 14009 4.33865 39759 99725 0.00017 14256 42257
1.57271 24349 95227 4.05275 81695 49437 0.00030 48651 48814
1.57379 21309 24768 3.83174 19997 84146 0.00047 65699 16867
1.57511 36077 77251 3.65185 59694 78752 0.00068 66451 27305
1.57667 79815 92838 3.50042 24991 71838 0.00093 52197 97816
1.57848 65776 88648 3.36986 80266 68445 0.00122 24470 64294
1.58054 09338 95721 3.25530 29421 43555 0.00154 85045 16579
1.58284 28043 38351 3.15338 52518 87839 0.00191 35945 90170
1.58539 41637 75538 3.06172 86120 38789 0.00231 79450 15821
1.58819 72125 27520 2.97856 89511 81384 0.00276 18093 29252
1.59125 43820 13687 2190256 49406 70027 0.00324 54674 43525
1.59456 83409 31825 2.83267 25829 18100 OiOk76 $2262 86978
1.59814 20021 12540 2.76806 31453 68768 0.00433 34205 09983
1.60197 85300 86952 2.70806 76145 90486 0.00493 84132 64213
1.60608 13494 10364 2.65213 80046 30204 0.00558 45970 58517
1.61045 41537 89663 2.59981 97300 61099 0.00627 23946 95994
1.61510 09160 67722 2.55073 14496 27254 0.00700 22602 97383
1.62002 58991 24204 2.50455 00790 01634 0.00777 46804 16442
1.62523 36677 58843 2.46099 94583 04126 0.00859 01752 53626
1.63072 91016 30788 2.41984 16537 39137 0.00944 92999 75082
1.63651 74093 35819 2.38087 01906 04429 0.01035 26461 44729
1.64260 41437 12491 2.34390 47244 46913 0.01130 08432 78049
1.64899 52184 78530 2.30878 67981 67196 0.01229 45605 27181
1.65569 69263 10344 2.27537 64296 11676 0.01333 45085 07947
1.66271 59584 91370 2.24354 93416 98626 0.01442 14412 80638
1.67005 94262 69580 2.21319 46949 79374 0.01555 61584 97708
1.67773 48840 80745 2.18421 32169 49248 0.01673 95077 33023
1.68575 03548 12596 2.15651 56474 99643 0.01797 23870 08967
1.69411 43573 05914 2.13002 14383 99325 0.01925 57475 39635
1.70283 59363 12341 2.10465 76584 91159 0.02059 05967 10437
1.71192 46951 55678 2.08035 80666 91578 0.02197 80013 16901
1.72139 08313 74249 2.05706 23227 97365 0.02341 90910 88188
1.73124 51756 57058 2.03471 53121 85791 0.02491 50625 23981
1.74149 92344 26774 2.01326 65652 05468 0.02646 71830 76961
1.75216 52364 68845 1.99266 97557 34209 0.02807 67957 17219
1.76325 61840 59342 1.97288 22662 74650 0.02974 53239 19583
1.77478 59091 05608 1.95386 48092 51663 0.03147 42771 20286
1.78676 91348 85021 1.93558 10960 04722 0.03326 52566 95577
1.79922 15440 49811 1.91799 75464 36423 0.03511 99625 22096
1.81215 98536 62126 1.90108 30334 63664 0;03704 02001 87133
1.82560 18981 35889 1.88480 86573 80404 0.03902 78889 26607
1.83956 67210 93652 1.86914 75460 26462 0.04108 50703 79885
1.85407 46773 01372 1.85407 46773 01372 0.04321 39182 63772
K'(a) KM 9lH
cI
C-i’9
4164 -w E’(a)
1.00000 00000 00000 1.15707963267 94897 1.00000 00000 00000
0.40330 93063 38378 l.!j7067 67091 27960 1.00075 15777 01834
0.35316 56482 96037 l.fi7031 79198 97448 1.00258 40855 27552
0.32040 03371 34866 l.!i6972 01504 23979 1.00525 85872 09152
0.29548 83855 58691 l.!i6888 37196 07763 1.00864 79569 07096
0.27517 98048 73563 1.15678090739 77622 1.01266 35062 34396
0.25794 01957 66337 l.!i6649 67877 60132 1.01723 69183 41019
0.24291 29743 06665 l.!i6494 75629 69419 1.02231 25881 67584
0.22956 71598 81194 1.56316 22295 18261 1.02784 36197 40833
0.21754 89496 99726 l.!j6114 17453 51334 1.03378 94623 90754
0.20660 97552 00965 l.!i5688 71966 01596 1.04011 43957 06010
0.19656 76611 43642 l.!i5639 97977 70947 1.04678 64993 44049
0.18728 51836 10217 1.55368 08919 36509 1.05377 69204 07046
0.17865 56628 04653 1.55073 19509 84013 1.06105 93337 53857
0.17059 45383 49477 l.!i4755 45758 69993 1.06860 95329 78401
0.16303 35348 21581 l.!i4415 04969 14673 1.07640 51130 76403
0.15591 66592 65792 1.54052 15741 27631 1.08442 52193 72543
0.14919 73690 67429 1.53666 97975 68556 1.09265 03455 37715
0.14283 65198 36280 1.53259 72877 45636 1.10106 21687 57941
0.13680 08474 28619 1.52830 62960 54359 1.10964 34135 42761
0.13106 18244 99858 1.52379 92052 59774 1.11837 77379 69864
0.12559 47852 09819 1.51907 85300 25531 1.12724 96377 57702
0.12037 82455 07894 1.51414 69174 93342 1.13624 43646 84239
0.11539 33684 49987 1.50900 71479 16775 1.14534 78566 80849
0.11062 35386 78854 1.50366 21353 53715 1.15454 66775 24465
0.10605 40201 85996 1.49811 49284 22116 1.16382 79644 93139
0.10167 16783 93444 1.4.9236 87111 24151 1.17317 93826 83722
0.09746 47524 70352 1.4!8642 68037 44253 1.18258 90849 45384
0.09342 26672 88483 l.d.8029 26638 27039 1.19204 56765 79886
0.08953 58769 52553 1.4.7396 98872 41625 1.20153 81841 13662
0.08579 57337 02195 1.4.6746 22093 39427 1.21105 60275 68459
0.08219 43773 66408 1.4.6077 35062 13127 1.22058 89957 54247
0.07872 46415 92073 1.4.5390 77960 65210 1.23012 72241 85949
0.07537 99738 58803 1.4,4686 92406 95183 1.23966 11752 88672
0.07215 43668 98737 1.4,3966 21471 15459 1.24918 16206 07472
0.06904 22996 09032 1.43229 09693 06756 1.25867 96247 79997
0.06603 86859 10861 1.4.2476 03101 24890 1.26814 65310 65206
0.06313 88302 96461 1.4,1707 49233 71952 1.27757 39482 50391
0.06033 83890 33716 1.4,0923 97160 46096 1.28695 37387 83001
0.05763 33361 79494 1.40125 97507 85523 1.29627 80079 94134
0.05501 99336 98829 1.39314 02485 23812 1.30553 90942 97794
0.05249 47051 04844 1.2~848865913 75413 li31472 95602 64623
0.05005 44121 29953 1.3'7650 43257 72082 1.32384 21844 81263
0.04769 60340 17056 1.26799 91658 73159 1.33286 99541 17179
0.04541 67490 83529 1.35937 69972 75008 1.34180 60581 29911
0.04321 39182 63772 1.35064 38810 47676 1.35064 38810 47676
d") E'(a) E(4
C-95)3
[I 1
612 ELLIPTIC INTEGRALS
K’
c 1 (-32
K’
For K >3.0, K < 0.3, see Example 6.
Table 17.4
AUXILIARY FUNCTIONS FOR COMPUTATION OF THE NOME q AND THE PARAMETER m
y(m)2$
9 Q (4 L(m) ml Q(4 ’ L(4
0. 00 0.06250 00000 00000 0.00000 00000 0.08 0.06513 95233 36060 0.02111 58281
0. 01 0.06281 45660 38302 0.00251 65276 0. 09 0.06549 04937 14101 0.02392 34345
0.02 0.06313 33261 60188 0.00506 66040 0.10 0.06584 b5155 38584 0.02677 14110
0.03 0.06345 63756 34180 0.00765 09870 0.11 0.06620 77131 77434 0.02966 07472
0.04 0.06378 38128 42217 0.01027 04595 0.12 0.06657 42154 15123 0.03259 24678
0;06411 57394 13714 OiO1292 58301 0.13 0.06694 61556 59704 0.03556 76342
0.06445 22603 66828 0.01561 79344 0.14 0.06732 36721 61983 0.03858 73466
0.06479 34842 57396 0.01834 76360 0.15 0.06770 69082 47689 0.04165 27452
See Examples
[5716 1
3, 5 and 6.
c(-;I6 3
ELLIPTIC INTEGRALS 613
ELLIPTIC INTEGBlAL OF THE FIRST KIND F(v\a) Table 17.5
0.08726 730 0.17453 962 0.26182 180 0.34911 842 0.43643 361 0.52377 095
0.08727 387 0.17459 198 0.26199 739 0.34953 092 0.43722 998 0.52512 754
0.08728 623 0.17469 061 0.26232 912 0.35031 330 0.43874 792 0.52772 849
0.08730 289 0.17482 397 0.26277 965 0.35138 244 0.44083 848 0.53134 425
0.08732 185 0.17497 630 0.26329 709 0.35261 989 0.44328 233 0.53562 273
0.08734 084 0.17512 935 0.26382 007 0.35388 123 0.44580 113 0.54009 391
0.08735 756 0.17526 454 0.26428 466 0.35501 092 0.44808 179 0.54419 926
0.08736 998 0.17536 525 0.26463 238 0.35586 223 0.44981 645 0.54735 991
0.08737
_ . 659 . 0.17541 895 0.26481 840 0.35631 976 0.45075 457 0.54908 352
The table can also be usedinversely to find cp=amu where u=F(lp\a) and so the Jacobian elliptic
functions, for example sn u=sin (D, cn u=cos IP, dn u= (l-sin2 a sin2 P) 112. See Examples 7-11.
Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals, Cambridge
Univ. Press,Cambridge,England, 1934(with permission). Known errorshave beencorrected.
614 ELLIPTIC INTEGRALS
5 13589544 22344604 1.31101 537 1.39859 928 1.48619 317 1.57379 213
::
II*
14740244
1: 17069811
:: 23727471
26548460
1.32732 612
1.36083 467
1.41751 762
1.45663 012
1.50780 533
1.55273 384
1.59814 200
1.64899 522
1. 20625 660 :: 30915104 1.41338 702 1.51870 347 1.62477 858 1.73124 518
i: 1. 25446 980 36971948 1.48788 472 1.60847 673 1.73081 713 1.85407 468
31490567 :: 44840433 1.58817 233 1.73347 444 1.88296 142 2.03471 531
2: :: 38443225 54409676 1.71762 935 1.90483 674 2.10348 169 2.30878 680
75 45316 359 :* 64683711 1.87145 396 2.13389 514 2.43657 614 2.76806 315
85 :: 49977412 1: 72372395 2.00498 776 2.38364 709 2.94868 876 3.83174 200
616 ELLIPTIC INTEGRALS
[ 1
(-i)4 [ (-l)3] [‘-p”] [‘f’“] [ (-;)“I
[c-y1
0.08726 562 0.17452 624 0.26177 698 0.34901 329 0.43623 105 0.52342 670
0.08725 905 0.17447 391 0.26160 165 0.34860 188 0.43543 791 0.52207 785
0.08724 671 0.17437 550 0.26127 157 0.34782 632 0.43394 028 0.51952 597
0.08723 006 0.17424 275 0.26082 567 0.34677 648 0.43190 776 0.51605 197
0.08721 113 0.17409 157 0.26031 693 0.34557 562 0.42957 525 0.51204 932
0.08719 220 0.17394 015 0.25980 639 0.34436 714 0.42721 938 0.50798 838
0.08717 554 0.17380 680 0.25935 592 0.34329 797 0.42512 769 0.50436 656
0.08716 317 0.17370 770 0.25902 064 0.34250 043 0.42356 271 0.50164 622
0.08715 659 0.17365 493 0.25884 192 0.34207 467 0.42272 556 0.50018 720
See Example 14.
Compiled from K. Pearson, Tables of the complete and incomplete elliptic integrals, Cambridge Univ.
Press, Cambridge, England, 1934 (with permission). Known errors have been corrected.
ELLIPTIC INTEGRALS 617
ELLIPTIC INTEGRAL OF THE SECOND KIND E(p\ol) Table 17.6
J:(v\a)=Jl(l-sin2 n sin2 S) ‘do
a(P
\ 35” 40” 45” 50” 55” 60”
0 0.61086 524 0.69813 170 0.78539 al6 0.87266 463 0.95993 109 1.04719 755
2” 0.61082 230 0.69806 905 0.78531 125 0.87254 a83 0.95978 la4 1.04701 051
4 0.61069 365 0.69788 136 0.78505 085 0.87220 la3 0.95933 459 1.04644 996
6 0.61047 983 0.69756 935 0.78461 792 0.87162 487 0.95859 083 1.04551 764
a 0.61018 171 0.69713 427 0.78401 409 0.87081 998 0.95755 301 1.04421 646
0.60980 055 0.69657 784 0.78324 162 0.86979 001 0.95622 460 1.04255 047
:2” 0.60933 793
0.60879 577
0.69590 226
0.69511 023
0.78230 343
3.78120 308
0.86853 863
0.86707 031
0.95461 005
0.95271 478
1.04052 491
1.03814 615
:“6
la
0.60817 636
0.60748 229
0.69420 492
0.69318 999
D.77994473
3.77853323
0.86539 034
0.86350 481
0.95054 522
0.94810 a78
1.03542 177
1.03236 049
20 0.60671 652 0.69206954 il.77697 402 0.86142 062 0.94541 386 1.02897 221
22 0.60588 229 0.69084 al4 10.77527316 0.85914 545 0.94246 984 1.02526 804
0.60498 319 0.68953 083 0.77343 735 0.85668 781 0.93928 709 1.02126 023
:z 0.60402 308 0.68812 308 0.77147 387 0.85405695 0.93587 699 1.01696 224
28 0.60300 616 0.68663 077 0.76939 059 0.85126 295 0.93225 la6 1.01238 a73
0.60193 687 0.68506 023 0.76719 599 0.84831 663 0.92842 504 1.00755 556
0.60081 994 0.68341 al7 Cl.76489908 0.84522 958 0.92441 083 1.00247 977
0.59966 035 Oi68171170 Cl.76250947 0.84201 414 0.92022 452 0.99717 966
0.59846 332 0.67994 830 Cm.76003726 0.83868 340 0.91588 234 0.99167 469
0.59723 431 0.67813 578 k75749 309 0.83525 115 0.91140 150 0.98598 560
0.59597 a97 0.67628 229 0.75488 a09 0.83173 189 0.90680 017 0.98013 430
0.59470312 0.67439 630 0.75223 383 0.82814 080 0.90209 742 0.97414 397
0.59341278 0.67248 651 0.74954 234 0.82449 369 0.89731 325 0.96803 a99
0.59211 406 0.67056 191 0.74682 605 0.82080 700 0.89246 a58 0.96184 497
0.59081 324 0.66863 167 0.74409 773 0.81709 775 0.88758 513 0.95558 a73
0.58951 664 0.66670 515 0.74137 047 0.81338 346 0.88268 551 0.94929 830
0.58823 065 0.66479 la3 0.73865 766 0.80968 217 0.87779 305 0.94300 285
0.58696 171 0.66290 130 0.,73597286 0.80601 230 0.87293 la4 0.93673 272
0.58571 622 0.66104 317 O-73332979 0.80239 262 0.86812 660 0.93051 931
0.58450 056 0.65922 707 0.,73074229 0.79884 217 0.86340261 0.92439 505
0.58332 103 0.65746 255 0.72822 416 0.79538 015 0.85878 561 0.91839 329
0.58218 382 0.65575 905 0.72578 915 0.79202 582 0.85430 169 0.91254 a21
0.58109 497 0.65412 585 0.72345 085 0.78879 a39 0.84997 709 0.90689 460
0.58006 032 0.65257 197 0.72122 260 0.78571 685 0.84583all 0.90146 778
0.57908 549 0.65110 612 0.71911 737 0.78279 987 0.84191 082 0.89630 323
0.57817 584 0.64973 667 0.71714 767 0.78006 562 0.83822 090 0.89143 642
0.57733 641 0.64847 154 0.71532 545 0.77753 157 0.83479 335 0.88690 237
0.57657 la9 0.64731 al2 0.71366196 0.77521 434 0.83165223 0.88273 530
0.57588 663 0.64628 328 0.71216 766 0.77312 952 0.82882 031 0.87896 al0
0.57528 450 0.64537 322 0.71085210 0.77129 143 0.82631 a79 0.87563 la5
“8:
a4
0.57476 897
0.57434 302 0.64459
0.64394 347
879 0.70972
0.70879 381
019 0.76840
0.76971 298
544 0.82416
0.82238 694
177 0.87275
0.87036 520
381
0.57400 912 0.64344 316 0.70805 745 0.76737 830 0.82097 770 0.86847 970
it 0.57376 921
0.57362 470 0.64307
0.64286 973
075 0.70721
0.70753 289
050 0.76663
0.76619 912
339 0.81996
0.81935 631
604 0.86712
0.86629 990
068
90 0.57357 644 0.64278 761 0.70710 678 0.76604 444 0.81915 204 0.86602 540
[ (-;)l] [ Qw] [ (-i)3] [ (-l)4] [ (-i)5] [(-(y]
5 0.61059 734 0.69774 083 0.78485 586 0.87194 199 0.95899 964 1.04603 012
0.60849 557 0.69467 152 0.78059 337 0.86625 642 0.95166 385 1.03682 664
:: 0.60451 051 0.68883 790 0.77247 109 0.85539 342 0.93760 971 1.01914 662
35 0.59906 618 0.68083 664 0.76128 304 0.84036 234 0.91'807la6 0.99445 152
0.59276 408 0.67152 549 0.74818 650 0.82265 424 0.89489 714 0.96495 146
5”: 0.58633 563
0.58057 051
0.66196 758
0.65333 a44
0.73464 525
0.72232 215
0.80419 500
0.78723 a20
0.87052 066
0.84788 276
0.93361 692
0.90415 063
;:a5 0.57621 910
0.57387 732
0.64678 548
0.64324 351
0.71289 304
0.70776 799
0.77414 195
0.76697 232
0.83019 625
0.82042 232
0.88079 972
0.86773 361
618 ELLIPTIC INTEGRALS
1.13303 553 1.22001 a78 1.30698 342 1.39393 358 1.48087 384 1.56780 907
1.12176 337 1.20649 962 1.29106 728 1.37550 358 1.45984 990 1.54415 050
1.10005 236 1.18039 569 1.26026 405 1.33976 099 1.41900 286 1.49811 493
1.06958 479 1.14359 al3 1.21665 a53 1.28896 903 1.36076 208 1.43229 097
1.03292 bb0 1.09900 a29 1.16345 846 1.22661 050 1.28885 906 1.35064 388
0.99358 365 1.05063 981 1.10513 448 1.15755 065 1.20849 656 1.25867 963
0.95606 011 1.00378 508 1.04769 389 1.08838 943 1.12673 373 1.16382 796
0.92579 978 0.96518 626 0.99915 744 1.02823 305 1.05342 b32 1.07640 511
0.90857 a73 0.94269 al3 0.96992 212 0.99022 779 1.00394 027 1.01266 351
ELLIPTIC INTEGRALS 619
JACOBIAN ZETA FUNCTION +-\a) Table 17.7
K(,)z(~\,)=K(,)E(,\o)-E(~)F(~\\n)
K(gO”)2(~\a)=K~~90’)Z(~~~l)=K(9~) tanh IL=- for all ?L
5” 10” 15” 20” 25” 30”
0.000000 0.000000 0.000000 0.000000 0.000000
0.000083 0.000164 :- K% 0.000308 0.000367 0.000414
0.000332 0.000655 0:000957 0.001231 0.001467 0.001658
0.000748 0.001474 0.002155 0.002770 0.003302 0.003734
0.001331 0.002621 0.003832 0.004928 0.005875 0.006644
0.002080 0.004098 0.007706 0.009188 0.010393
0.002997 0.005905 t "OK;;: 0.011107 0.013246 0.014987
0.004082 0.008043 0:011765 0.015136 0.018055 0.020433
0.005337 0.019796 0.023621 0.026740
0.006761 it. EE! is. E:E 0.025094 0.029951 0.033919
0.008357 0.016470 0.024105 0.031035 0.037055 0.041981
0.010125 0.019958 0.029216 0.037627 0.044942 0.050941
0.012067 0.02379-I 0.034834 0.044878 0.053626 0.060814
0.014186 0.027972 0.040968 0.052799 0.063119 0.071617
0.016483 0.032508 0.047624 0.061401 0.073438 0.083373
0.018962 0.037403 0.054811 0.070696 0.084599 0.096103
0.021625 0.042664 0.062540 0.080700 0.096624 0.109834
0.024476 0.048298 0.070823 0.091430 0.109534
0.027520 0.054315 0.079674 0.102905 0.123356 is :GE;
0.030761 0.060725 0.089108 0.115148 0.138120 0:157347
0.034205 0.067540 0.099145 0.128185 0.153860
0.037860 0.074774 0.109807 0.142046 0.170614 00.:z;
0.041734 0.082444 0.121118 0.156765 0.188428 0:215197
0.045835 0.090569 0.133109 0.172383 0.207353 0.237025
0.050177 0.099172 0.145813 0.188947 0.227450 0.260240
0.054771 0.108280 0.159273 0.206513 0.248789 0.284929
0.059634 0.117925 0.173536 0.225145 0.271452 0.311193
0.064786 0.128146 0.188661 0.244921 0.295538 0.339150
0.138989 0.204716 0.265933
: . EZ 0.150510 0.221785 0.288294 "0.
. ::zl: i*. z%
0.082227 0.162776 0.239971 0.434726
0.088818 0.175872 t :::'b:i t 34707sK 0.471170
0.095876 0.189901 i* z;; 0:364981 0:442321 0.510371
0.103468 0.204994 0:302637 0.478462 0.552710
0.111676 0.221320 0.326895 i*. :z;: 0.517644 0.598675
0.120612 0.239097 0.353322 0.461145 0.560402 0.648900
0.130420 0.258615 0.382351 0.499384 0.607444 0.704225
0.141301 0.280272 0.414575 0.541857 0.659739 0.765737
0.153537 0.304631 0.450832 0.589673 0.718657 0.835238
0.167542 0.332519 0.492356 0.644462 0.786214 0.914934
0.183967 0.365230 0.541075 0.708771 0.865556
0.203902 0.404937 0.600229 0.786884 0.961976 :- E56E
0.229402 0.455734 0.675918 0.886859 11268462
0.265091 0.781873 1.026844 :* E%! 1.472953
0.325753 2. E: 0.962000 1.264856 1:552420 1.820811
m m 00 m al m
m on 0 m L-a
AO(P\4 =- F(p’go”-~)+~K(~)Z(p\90”-~)=H
K’(ol) {K(rr)E(v\90”-a)-[K(a)-E(~)]qco\SOo-+
a%+
\0 5” 10” 15” 20” 25” 30”
0.087156 0.173648 0.258819 0.342020 0.422618 0.500000
F! 0.087129 0.173595 0.258740 0.341916 0.422490 0.499848
4 0.087050 0.173437 0.258504 0.341604 0.422104 0.499391
0.086917 0.173173 0.258111 0.341084 0.421462 0.498633
86 0.086732 0.172804 0.257562 0.340359 0.420566 0.497574
c 1 [(-j)l1
(-;I1 [
c-y
I [(-p1 [
(-$)l
I [c-y1
0.572487 0.641567 13.705765 0.764592 0.817600 0.864388
0.563926 0.632010 13.695307 0.753346 0.805703 0.852010
0.547600 0.613936 11.675748 0.732623 0.784220 0.830282
0.524935 0.589127 0.649283 0.705094 0.756337 0.802903
0.497760 0.559735 11.618381 0.673501 0.724985 0.772830
0.468167 0.528076 0.585512 0.640369 0.692612 0.742291
0.438541 0.496661 11.553214 0.608153 0.661480 0.713246
0.411857 0.468546 0.524506 0.579721 0.634200
0.392679 0.448417 0.504034 0.559529 0.614903
624 ELLIPTIC INTEGRALS
[ C-f)5
1 c 1 [
c-:)4 (-72
1 [C-f)7
1
$3~ Examples 15-20.
626 ELLIPTIC INTEGRALS
[ Q5)5
1 [ C-$5
1 [(-f)l1
18. Weierstrass Elliptic and Related
Functions
THOMAS H. SOUTHARD ’
Contents
Page
Mathematical Properties ............ + ....... 629
18.1. Definitions, Symbolism, Restrictions and Conventions . . 629
18.2. Homogeneity Relations, Reduction Formulas and Processes. 631
18.3. Special Values and Relations ............. 633
18.4. Addition and Multiplication Formulas ......... 635
18.5. Series Expansions. .................. 635
18.6. Derivatives and Differential Equations ......... 640
18.7. Integrals, ...................... 641
18.8. Conformal Mapping ................. 642
18.9. Relations with Complete Elliptic Integrals K and K’ and
Their Parameter m and with Jacobi’s Elliptic Functions , 649
18.10. Relations with Theta Functions ............ 650
18.11. Expressing any Elliptic Function in Terms of Q.3 and @’ , . 651
18.12. Case A=0 ...................... 651
18.13. Equianharmonic Case (g2=0, g3= 1) ........... 652
18.14. Lemniscatic Case (g,=l, g3=O). ............ 658
18.15. Pseudodemniscatic Case (g,= - 1, g3=O) ....... 662
Table 18.1. Table for Obtaining Periods for Invariants g2 and g3 Page
G2=g2g3-2fi). . . . . . . . . . . . . . . . . . . . . . . . 673
The author gratefully acknowledges the assistance and encouragement of the personnel
of Numerical Analysis Research, UCLA (especially Dr. C. B. Tompkins for generating the
author’s interest in the project, and Mrs. H. 0. Rosay for programming and computing,
hand calculation and formula checking) and the personnel of the Computation Laboratory
(especially R. Capuano, E. Godefroy, D. Liepman, B. Walter and R. Zucker for the prepara-
tion and checking of the tables and maps).
629
18. Weierstrass Elliptic and Related Functions
Mathematical Properties
A>0 Y A<0
2w’ 202
J
R2 R3
co'
R,= R4
+FPP
r r WX
0 w 2w
w1=w
w=wfw’ w&w’-w
w=w’
w REAL wa REAL
w’ PURE IMAG. w; PURE IMAG.
1W’ 12 w, since g3 Z 0 Iw:I 2 02, since ga2 0
Fundamental Rectangles
Study of all four functions (,f$ , @ ‘t 1, u) can be reduced to consideration of their values in a Funda-
mental Rectangle including the origin (see 18.2 on homogeneity relations, reduction formulas and
processes).
A>0 A<0
Fundamental Rectangle is i FPP, which has ver- Fundamental Rectangle has vertices 0, w2, w&$
tices 0, w, waand w’
UP
FUNDAMENTAL
RECTANGLE
(+ FPP)
-x
FIGURE 18.2
There is a point on the right boundary of Fundamental Rectangle where @ =O. Denote it by 4.
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 631
18.2. Homogeneity Relations, Reduction The Case ga<O
!W’
Reduction from l/r FPFa; T;u<ngdamental Rectangle in
00 A<0
(w*=w+w’)
18.2.38
18.2.39
18.2.40
18.2.41
18.2.42
18.2.44
(i=1,2,3) (i=1,2,3)
@‘, @‘9 and { are infinite, u is zero at 2.=2w1, i=1,2,3 and at 2oi(A<O).
00 A<0
Half-Periods
18.3.2 @‘(wJ=O(i=1,2,3)
18.3.10 $l$O if
Quarter PerioQ
A>0 A<0
ew18 er291~
18.3.21 a(w/2)=21/4~/8(2H1+3e1)l/8 ~(~2~2)=21,4~/8(2~2+3,2)1/8
~&fI~
18.3.25 u(w’~2)=21/4H~“(2H3-3e3)1/8
At z=2w,/3(i=l, 2, 3) or 20;/3,@“~=12@@‘~;
equivalently :
18.3.38 E+z+G=Ql2/4
18.3.39 WW-~P3+~~=0
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 635
18.3.40 LT;H;fl= --A/l 6
18.3.41 16H:-l:!gzH:+A=O(i=l, 2, 3)
18.4.1
18.4.2
18.4.3
18.4.5
18.4.6 p ‘(22)=
18.4.9 4 09 ‘3(2>
~(32)=3Kz)i-~,(,)~,,,(~)- gj”“&.)
and
18.5.3
18.5.4
18.5.5
18.5.6
18.5.9 c*=cy3
18.5.10 cs=3c2c3/11
18.5.12 c,=2&3/33
18.5.13 cs=5c~(11&+36c~)/7293
18.5.14 c,=c,(29c;+llc:)/2717
18.5.16 c,,=14c,c~(389c;+369c;)/3187041
18.5.17 c~2=(114950c~+1080000c;cc:+166617c~)/891678645
18.5.18 c,,=10c~c3(297c;+530c~)/l1685817
2c2(528770c!:+7164675c;c;+2989602cj)
18.5.19 c14=
(306735) (215441)
4c3(62921815cs,+179865450c;c~+14051367c’,)
18.5.20 c15= (179685) (38920531)
cl~=cj(58957855c~+1086511320c~~+875341836c~)
18.5.21
(5909761)(5132565)
c,,,czc~(30171955c:+126138075c~c~+28151739c:)
18.5.22
(920205)(6678671)
18.5.23
c1o=2c~c3(3365544215c~+429852433.45c~cf+8527743477c~)
18.5.24
(91100295)(113537407)
J NOTES: 3. cl0 is given incorrectly in [18.12] (factor 13 is
1. c,-clp were computed and checked independently missing in denominator of third term of bracket) ;
by D. H. Lehmer; these were double-checked by this value was computed independently.
substituting ga=20 CZ, g3=28 cz in values given in 4. No factors of any of the above integers with more
[18.10]. than ten digits are known to the author. This is not
2. clrcls were derived from values in [18.10] by necessarily true of smaller integers, which have, in
the same substitution. These were checked (numer- many instances, been arranged for convenient use
ically) for particular values of gz, ga. with a desk calculator.
Value * of Coefficients a,, n
= = = = = =
1 I,
-2.3. 6.69
3
.107893773
I 10 I 1,
-2.3. 6.23 -2.3. 6.69
7
.257.13049 .107896773
6 II 8 10
-2:&.~ -2.3. 6.7 -2.3. 6.7
li 10
2:3% &:f#.~ -2.3. b -2$’ 6.7
6 ,nc*r\.nn
ii .0:03 .*-n,?on Nulyuy In .m,l
.“il.l,O
t
J42ql
II, 1 10 t
2.3.6 2:3:6.9103 -2!&.7 -2f3fb.691 -2!3:6.11.31 -2.3. 6.7.2.3
4
.31 13.37.41 33609 313.190337 263.43489b3 .17b2630144977
IO
2f3f23 2.3.6.31 : zf35.17 -&&.,j3 -2!3:b.b03 -2:3:3$ -2!3:6.7.11.29 -2!3$.7.613
a
.109 3911 166217 ,316939669 a33.1129.9bbl .176052%081
-2.3’ 2f3i3 : !:3:b.b3 i3s.37 -2.3:5X -2f3h.151 -213’ -2f3k17.63 -2.3yb.71117 ;a!&;
2
167 3037 ,853 2337260103 .2967.41X39 R7.19b6blObQ 35866647631801
+m
‘Value3 Of a.. .I in unfactored form for 4m+‘3n+113b are given in k3.251, p. 7; of (ar, n)3-m in factored form in [13.15], Vol. 4, p. 89 for 4m+6n+l<25. Additional values were computed and cheeked on
de& calculators; primality of large factors was established with the aid of SWAC (National Bureau of Standards Western Automatic Computer).
638 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
18.5.35 All=8a2a3A~/11
7
+zoo (33~~+lS0~~~~+10~~)Uz6
18.5.36 A13=% (az”+S&)
+?$ (lla;+loa~) u2’
18.5.37 AIS= -96&a3/175
18.5.4Q z=ufAguSfA,u’+A,us+. . .
18.5.41 where u= {-’
+ s; . . (195a~+456&;+42a:)U3~
18.5.42 &=---6&I
18.5.48 82 (13496;-41166;)
+ o(u4y
1f
A17=9163
(1154316;--2256861,)
18.5.26 where a2=g2/8 18.5.49 A,,=&
~~~;u[1--3e2u4~-4e,u'+~2L8+~u~lo
18.5.58
(P w=[-
+702+-5&
13
un 4@3c:~3~14
33
+u2 -5cy--14c3u2
1 [ lOc2u-56c3u3+
+21c3ef+5ci)u3+(18cae:+30c~ei -~)u’+24(~~2~3e:+~~e,
+3h&)U4+ 22c~~+92c2c3ei+105~
( +42c$e,+E dc3) a?'+70 (g de:
-y Us+ ‘z w~e~+~4e,+84&
> ( +gc$e:+T c$eset+E c24+& 4) al3
7c3u”
26
5&L’
u,4+61&3u’6
66 I[ + 5s~~
3
~““(2)=4~3(z)-g*~(z)-g~=4(~3-~c*~-7c3)
18.6.4 QY’(z)=6~2(z)-~g~=6@2-10cz
llczc& (loc;-84c%) ul* @“‘(Z) = 12
+2-F-- 18.6.5 @Q’
--3-+ 33
1 ...
18.6.6
1363&a ,,+&316+-55~~) utg +
+ 429 429 @(4)(2)=12( QQ”, @‘@‘>
~(‘“‘(2)=5.9!~‘[~‘-33~z~~-2ca~ +2c1,/3]
18.6.13
~~“0’(z)=ll![~“-6c~~4-6c~@8+7~~a
+ (342cgQ +84+ lOc,8)/33]
18.6.14
+x681 c%3el+~9150 wi+~ 5 4 > &- --.,
~‘“‘(z)=6~11!~‘[~*-4c~~8-3cg~2
18.5.69
where a= (z- WJ + (774 L? +57cd331
18.6.15
Reversed Series for Small 1~1
p’yZ)=13![&07-7c~p-7ca@4+35C$p/3
18.5.70 +210c~a~2/11+(84c+35c~))/13-1363&~/4291
18.6.16
~‘~~~(z)=7.13!~‘[~~-5q~4-4c*~~
+5~~2+60c~c~~/11+(12+55Q)/13]
1%2% u 11 +3fW+W+ ula+ 18.6.17
+- . . . ,
55 6006
~(14)(z)=15![~8-8c&P-8c~$iP+52~~4/3
18.5.71 where -ya= gal48 +32&m @ “Ill + (444&---3284) @ *I39
-488&s@ /33+s(55+2316eo,)/429]
18.5.72 Ya=gdl20 18.6.18
For reversion of Maclaurin series, see 3.6.25 and ~‘1s~((z)=8~15!~‘[~7-6c&‘6-5ca~4+26e’,~’/3
[18.18]. f123~q @a/11+(111t+82c$@ /39-61&/33]
WEIERSTRASS ELILIPTIC AND RELATED FUNCTIONS 641
Partial Derivatives with Respect to Invariants y”= [a @ (2) +a]y (LamtYs equation&see 118.81,
2.26
18.6.19 For other (more specialized) equations (of
orders l-3) involving p(z), see [18.8], nos.
A~=~‘(3g2l-~g~z)+6g2~‘-9$~~~-$~ 1.49, 2.28, 2.72-3, 2.439-440, 3.9-12.
For the use of @ (z) in solving differential equa-
18.6.20 tions of the form y’“+A(z,y)=O, where A(a,y) is
a polynomial in y of degree 2m, with coefficients
A~=~‘(-;g&+$)-9g.lP’+$+;gzg~ which are analytic functions of z, see [18.7],
p. 312ff.
18.6.21 18.7. Integrals
hfk$nite
d”--33.($2P +&)
Adg,- 18.7.1 @P(z)dz=; @ W+~ $22
s
+; 2 (%I@ +;,,>-; $2P
4:-3a2 18.7.5
$a =-- 27
18.6.27
(y+a)“(y+b)”
w’=i
s #3
-Jqj
- dt
w=@ (a) maps the Fundamental Rectangle w=@(z) maps th e Fundamental Rectangle
onto the half-plane vI0; if Iw’l=&7~=0), the onto the half-plane ~50; if jw:)=wz(g3=O), the
isoscelestriangle Owwzis mapped onto ~20, vIO. isosceles triangle OQW’ is mapped onto ~10,
VlO.
w=@‘(z) maps the Fundamental Rectangle w=@ ‘(2) maps the Fundamental Rectangle
onto the w-plane less quadrant III; if lw’l=u, the onto most of the w-plane less quadrant III; if
triangle Owwzis mapped onto v 20, a 2~. jwil =w2, the triangle Owzw’is mapped onto v 20,
v2u.
(a = period ratio)
w={(z) maps the Fundamental Rectangle onto w={(z) maps the Fundamental Rectangle onto
the half-plane ~20. If a 11.9 (approx.), ~50; the half-plane u>O. The image is mostly in
otherwise the image extends into quadrant I. quadrant IV for small a, entirely so for (approx.)
For very large a, the image has a large area in 1.3 la I3.8. For very large a, the image has a
quadrant I. large area in quadrant I.
w=a(z) maps the Fundamental Rectangle onto w=a(z) maps the Fundamental Rectangle onto
quadrant I if a < 1.9 (approx.), onto quadrants quadrant I if a<3.8 (approx.), onto quadrants I
I and II if t.9 <a<3.8 (approx.). For large and II if 3.8<a<7.6 (approx.). For large
a, w&d41 =z; consequently the image winds a, arg[r (az+$)]-$ consequently the image
around the origin for large a. winds around the origin for large a.
Other maps are described in [18.23] arts. 13.7 Other maps are described in [18.23] arts, 13.8
(square on circle), 13.11 (ring on plane with 2 (equilateral triangle on half-plane) and 13.9
slits in line) and in [18.24], p. 35 (double half (isoscelestriangle on half-plane).
equilateral triangle on half-plane).
Obtaining @ ’ from @ ‘2
I I I
-X X
0 .4 w=1
FIGURE 18.4
In region A In region A
LZ?(@‘) 20 if ~2.4 andxS.5; 9(@‘) 20 elsewhere (1) If ~~11.05, use criterion for region A for
A>O.
(2) If 1 $z<1.05: a(@‘) 20 if ~2.4 and
x1.4, -1~/4<arg (@‘)<3*/4 if .4< ~1.5 and
.4<z<.5. Y(@ ‘) 1 0 elsewhere
WEXERSTRASS ELLLPTIC AND RELATED FUNCTIONS 643
In region B In region B
The sign (indeed, perhaps one or more significant Use the criterion for region B for A>O.
digits) of @ ’ is obtainable from the first term,
-21.9, of the Laurent series for @‘.
A>0 w=l
Map: @(z)=u+iv
w'=1*4i
-4 * -3 -2 -I 0 I 2 3
1.5
4
--
__
--_
--
-
----
-
-/I[[
_._ - -
- -
”
-
- -
-
r
&_
-
-- - - -
- _ - -
1.0 - - - i -
- - -
/.3 ’
,‘.3i - L
Ltd-~-i
-- c--
- -t -
.5- M --
0’=2.Oi
l--y’ I
0 1.0 I
FIGURE 18.5
644 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
A<0 wa=l
Map: @ (z)=u+iv
wj=1.5i
-2 -
-5-
lE,l5 .07
4=2.oi
lC215 .03
-6- Iv
FIGURE 18.6
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 645
00 w=l
Map: c(z)=u+iv
FIGURE 18.7
646 WEIERSTRASS ELLIPTIC AND RELATED F’TJNCTIONB
Map:
Near zero:
Y
strq--y
-1
-__-
-
___
###L
0 .5 1.0 x.
If, IS.04
IQ .0002
w&i
"
.5 I.0 I.5 -_2&L2.5 3.0 3.5 ”
0 ’ .9 H .7- 6 .5 i--or---
_ Y
-.
I I (I -. ._
-.5 ,, \\ -.
(1’ .75 _ _- 1 _-
4si!F \ ‘\ ‘-, .li
1 ; I _ _ -_
‘.
.75i, \ ’ .5 - 1
-1.0 _ _ _-
I’ \
“!7i ,I. ’ _ _-
xi2i I __ - _
‘I I “& ,!5i j \
I.31 \ -. --
1.41
-4.5 - ,.’ , kik __
_- & ’ .3 \ 0 .5 -i.o x
\
,I ,: ’ 1
: ,
-2.0 _--* I
/’ I
,’ :’
-2.5 ------ ., .2
#’
,/
0
I’
-3.0 b
;=1.5i __---
*-
/’
-3.5 -
0;=2.oi
-3.5 t-----G
FIGURE 18.8
WEIERSTRASS ELLWTIC AND RELATED FUNCTIONS 647
00 w=l
1.0
lI--~l
---_-- --
kY ---_--__--.
_
-
- _
I.0
--
.5--------- --_-_
-_-
-------- --.
.5
w’=2
.
0
-
- ---_-- - .5 I.0 x
,\J,’ . ,1.2i jy
1.4 - - -- -- --
--- --
-- - _
- --
1.0 - - -- -
-- -__
-i-d---/.81
-- - -- _
-- - _
- - - -
0.5- -- -
-- -- -
-#I
. - a !4i
--
.2i
/ill!!l~
--
--
--
- -
0 - .5 -- co x
2.Oi ’ 1.8i
r, I\I ’
‘6i
2.0
IY
------ --~ -_--.
-._--.- --.- -- -
A<0 t&=1
It-1
Map: u(z) =u+iv
.5i Y
1.liiiiit
#’’ \ .ai
*5 - --- -
- -- _
---- -_ _
-- - -
--- _
0- -- -- -
.5 7.0 x
1.0 1.2
0 .5 1.0
U
0 .2 .4 .6 .8 I.0
BB~
V - -
1.0
- -- -- -
-- ---
-- -- -
- - --
.5 -- ---
- - - -
- -
_-- - - -
-- -
0 - .5- - I.0 x
U
0 *2 .4 .6 .u 1.0
FIGURE 18.10
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 649
18.9. Relations with Complete Ellipltic Integrals K and K' and Their Parameter m
A>0 A<0
-mW(m)
18.9.1 e1=(2 3w2
(2m-l)IC(m) 2(1--2m)Kym)
18.9.2 e2= e2=
3wa 34
m-1)-6iJm-m2
18.9.3 e3=(2 3w2 - JWd
a
gl,4w-m+l~K4(m) gl=4(16mz-16m+l)K4(m)
18.9.4
30’ 34
g3=4(m-2)(2m-l)(m+l)~(m) g,,8(2m-1)(32mz-32m-l)IP(m)
. .
18 9 5
27~’ 274
A~16ma(m-1)a~2(m)
18.9.6 A =-256(m-m2)K12(m)
WI2 Wp
1 3e2
18.9.9 m= (e2-e3Y(el-e3)
m=Z-4H,
18.9.10 D<m I*, f&e g3201
18.9.11 ~(z>=e3+(el-e3>/sn”(z*lm) @Yz)=e2+& :+z[z:/ti
-
18.9.12
@‘(z)=-2(el-e3)3/2 ~cn(z*~m)dn(z*~m)/sn3(z*~n) -4IWsn( z’ Im)dn( z’ Im)
P(z)=
[l-cn(z’(m)]2
where where
z*=(el-e3)b .z~=22Hp
18.9.14 q’=l(w’)=SW’;W
A>0
18.10.1 r=w‘/w
18.10.2 p,eir+,e-rK‘lK
18.10.3
q is real and since g320(]o’]20), O<q_<eBr P & q-;e-$2@av andsinceg~2O(Iw~I2wl),
P-
18.10.4 (v=7rz/2w) (v=rz/2wJ
j=l,2,3
18.10.7
18.10.15 g2=;(@':(0)+@(0)+~:(0)]
18.10.16 g3=4eIe2e3
18.10.17 Ai=-&2(0) (-A)i=&8;2(0)e-(r/4
2
7&?:"(o)
18.10.18 m4=-12wg;(0) &;“(o)
Ir2=r(~2~=-12y&(o)
18.10.19 +f(~')="~';~
WEIERSTRASS ELLIPTIC AND RELATED FUNCYl’IONSr 651
Series
18.10.20 81(O) =o
9z&(O)>O; 84(0)=s3o
If f(z) is any elliptic function and P(Z) has same periods, write
18.11.2 3 where A is
a constant. If any a, or b, is congruent to the origin, the corresponding factor is omitted from the
product. Factors corresponding to multiple poles (zeros) are repeated according to the multiphcity.
18.12.3 18.12.12
18.12.35 u(d) =o
18.12.36 u(q) =o
18.12.37 g (w/Z) = 5c
18.12.38
18.12.39
18.12.40 e**/%Jz
u(w/Z)=
7r
18.12.41 @(d/2)=-c
18.12.42 @‘(d/2)=0
FIGURE 18.11
18.12.43 {(d/2) = +im
18.12.44 u&‘/2) =o 6 This value was computed and checked by multiple
precision on a desk calculator and is believed correct to
18.12.45 @ 6.d~) = --c 30s.
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 653
18.13.16
W=Wl 0
18.13.17
w 0
18.13.18
W"6.Q 0
18.13.19
Y' 0
- - -- -
values ’ along (0, wJ
= = =
.P’
_- _- --
18.13.N
- p[ GG-GF + 4/cos 407
20*/9
i/cos-- 2/‘m 4o”
18.13.21
WI3 l/(21”- 1) - p(21”+ 1)/(2’!‘- 1)
18.13.22
- P[ 2/cos + Qcos so01
402/9
D- %:os 80”
18.13.23
WI2 a+&
18.13.24
ad3 1
18.13.25
8wl9
18.13.26
2012 -21/ae2 3i
18.13.27
32014 l a(ez - HZ) i(3”‘) &qi
18.13.28
20 0 i
Conformal Maps
-60’ .2 .4 .6 .6 1.0 1.2 I.4
Equianharmonic Case
Map: f(z)=u+iw
4 i I\ +-----‘\, ‘..
WEIERSTRASS ELLIPTIC AND RELATED FUNCTION& 655
.2 .4 .6 .6 1.0 1.2
0
s-(z)
1 -.4
Near zero: {(z)=,+q
x
-.6
1.5, I 5 .007
-3 (.$*I 5 I x10-5
t”
v
1.0 1.6
.8
FIGURE 18.12
656 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
:\~~3.2s) = l/10192 3.
9.8116
5714 28571
16954 42857
47409 733iZx %88X lo-6
l/(13.19.28”) = l/5422144 1.8442 88901 21693 55885 78983X10-’
3/(5.13’.19.28’)=234375/(7709611X10*) 3.0400 36650 35758 61350 20301X lo-10
4/(5.13*.19.31.28’)=78125/(16729 85587X 108) 4.6697 95161 83961 00384 33643X10-1’
(7.43)/(13”~19*.31.37.283 6.8662 18676 79393 36788 98X10-‘@
(6.431)/(5.13”.19’.31.37.43.28’) 9.7990 31742 57961 41839 66X10-”
(3.7.313)/(5’.13’.19’.31.37.43.28*) 1.3685 06574 79360 13026 87X10-n
(4.1201)/(5*.134.1~.31.37.43.280) 1.8800 72610 01329 79236 40X10-x’
(2’.3.41.1823)/(5.13’.19*.31’.37.43.61.28i”) 2.5497 66946 68202 63683X10+
(3.79.733)/(5.13’.196.31’.37.43.61.67.28”) 3.4222 48599 51463 05316X lo-80
3.1153.13963.29059
4.5541 38864 99184 3O391X1O-J”
53.136.194.31s.37s.43.61.67.73281’
2’.3*.7.11.2647111
6.0171 15776 98241 99591 X 10-s’
5’.13’.19’.31s.37s.61.67.73.79.28’6
First 5 approximate values determined from exact values of cxk; subsequent values determined by using exact ratios
cak/c3k+ using at least double precision arithmetic with a desk calculator. All approximate c’s were checked with the
use of the recursion relation; cx -CH are believed correct to at least 215; C~O-C~O are believed correct to 205.
Ca
c3k< 13k-1 . 2@-1t k=2, 3, 4, . . .
18.13.39
Series near SO
18.13.43 where u=(z-z,J 18.13.49
18.13.44
Series near w2 (Lo’--i)==x[-
2-ir+A x2+$ a++o(xq]
~c(z)~<3xlo-~
18.13.64 w= (t-722)2/e2
*=(-1)9.99999 98 a4= (- 10)6.12486 14
Values at Half-periods
18.14.8
w-0, a=3 0 y=*/4w e*b(21/4)
18.14.9
w-a a-0 0 7+4 erl4(~&4*/4
18.14.10
co’=01 s--t 0 ‘)’ = - *i/40 &/8(21/4)
Valuesalong(O,w)
18.14.11
WI4 &$&+ 2’14) (1 + 21’4)
18.14.12
,r1”‘(21/16)
42 aI2 --Q l+Q
8w 2&i af
18.14.13
edla(31/.9)
2013 ,m -G55*
& (2+ &)l’l’
18.14.14
3014 $(&-2$(1+2f)
a=1+fi
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
659
Values along (0, zO)
P r
18.14.15
-
%I4 -&++Jz,)
18.14.16
42 -i/2
18.14.17
@ (2.zo/3) ‘la
22013 2 dsec 30”- 1
3 I
18.14.18
32014 -- ; (a-&q
18.14.28 a(iz)=iu(z)
Conformal Maps
Lemniscatie Case
Map: f(z)=u+iu
P(z)
Near zero: @(z)=-$+c,
md=$+g+~2, lzl<l
I---zol<&
~(~)~-(~-~o)2+(z-zo)6+6,
4 80
660 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
--(z-z32
Near z,,: g’(z)= +Q
Near zero: @ ‘( z)=$+Q
-(2-2cJ+3(2-20)6+c,
P’(z)= 2 40
l(z)
1
Near zero: c(z)=--+Q -.2
-.4
(z-zo)3
Near 20: T(z)=T~+T+Q, -.6
12-r I
FIGURE 18.14
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 1661
Coefficients for L.aurent Series for !?,@‘,andI
Cc,.=0 for m odd)
Z-Z
l/20 .05
l/(3.202) = l/1200 .8333 , .~.
. . . x10-3
2/(3.13.203) = l/156000
5/(3.13.17.20’) = l/21216000 .641025
.47134 23831
641025 07bi8’ ;8::X 10-T
2/(3z.13.17.205) = 1/(31824X 105) .31422 82554 04725 99296X10-Q
10/(33.13*.17.206)=1/(4964544X 105) .20142 83688 49183 32882% lo-11
4/@132.17.29.2d’) =‘ij(7998432X 167) .12502 45048 02941 37651X10+3
2453/(3’.11.132.172.29.2o8)=9582O3125/(1262OO2599X 10’6) .75927 19109 76468 59917X 10-l”
2.5.7.61/(33.13a.172.29.37.2O~)=833984375/(18:194643943X 10’7) .45338 43533 93461 06092X 10-I”
c&+9 z k=l, 2, . . .
Series near w
18.14.34 18.14.45 5=(2--w)
18.14.46
18.14.35 v=(z-co)%
SO@”
___- 819@‘l’
18.14.36 8 +O(@‘13)
+ 3
f)=y 1-y+6yz-@ 1 172Y4
[ 5 5 75 Other Series Involving {
Reversed Series for Large [rj
52@ 1064g
-ij+ 195 +wi7>
1 , 18.14.47 z=c-’
18.14.37 !I=(@-ed 18.14.48 v=t-4112
662 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
18.14.52 x= (z-w)
18.14.58 SC”@‘(x) =~a,x4n+a(x)
18.14.53
Ie(x):)1<4x10-7
x=w-p+~-~+~-‘+g~~~~~~3
891
ao= -2.00000 00 a,=(-7)6.58947 52
+ O(d5) al=(-l)1.00000 02 a,=(-9)5.59262 49
18.14.54 w= -2( c-7) a2=(-3)4.99995 38 as=(-11)5.54177 69
Series Involving (r u3= (--5)6.41145 59
18.14.55
25 3%+’ -3.23~~~ 3~107~‘~+3~~7~23~37z~~
~
a=z--2.5!-22.91+ 23.13! + 24.17! 25.21!
18.14.59 x{(x) =&7.x4’+E(x)
+
(e(x)[<3X10-8
18.14.56
f&=(-1)9.99999 99 a4= - (-9)2.57492 62
aI=-(-2)1.66666 74 a,=-(-11)5.67008 00
a2=-(-4)1.19036 70 a,=(-13)9.70015 80
a,=-(-7)5.86451 63
Values at Half-Periods
18.15.7 w2 0 92 = d2w2
18.15.8 CO’ = 03 -i/2 8 f(B2+$2’)
18.15.9 02’ 0 0 112’
= --ill,
Numerical Methods
18.16. Use and Extension of the Tables
@‘-Use Laurent’s series directly “near” 0 (if Izi< 1, four terms give at least 8S, five terms at least
11s). Elsewhere, either proceed as for @ and {, or get @‘2=4@3-1 and extract appropriate square
root (g@‘ZO>.
(b) Given @ (@‘,l,a) corresponding to a point in the Fundamental Triangle, compute z more
accurately than can be done with the maps. Only a few significant figures are obtainable from the use
of any of the given (truncated) reversed series, except in a small neighborhood of the center of the series.
For greater accuracy, use inverse interpolation procedures.
Example 3. Given period ratio a, find parameters m (of elliptic integrals and Jacobi’s functions
of chapter 16) and p (of 6 functions).
m-In both the cases A>0 and A<O, the period ratio is equal to K’(m)/K(m) (see 18.9). Knowing
K’/K, if I< K’/Ks 3, use Table 17.3 to find m; if K’/K>3, use the method of Example 6 in chapter 17.
An alternative method is to use Table 18.3 to obtain the necessary entries, thence use
q----In both the cases A>0 and A<O, the period ratio determines the exponent for p[p=e+ if A>O,
p=ie+j2 if A<O]. Hence enter Table 4.16 [e+, zc=O(.Ol)l] and multiply the results as appropriate
[e.g., e-4.72r= (e-r)4(e-.72r)]a
Determination of Values at Half-Periods, Invariants and Related Quantities from Given Periods (Table 18.3)
A>0 A<0
Given w and u’, form ~‘/iw and enter Table 18.3. Given w2 and w2’, form w2’/zIw2and enter Table
Multiply the results obtained by the appropriate 18.3. Multiply the results obtained by the appro-
power of w (see footnotes of Table 18.3) to obtain priate power of w2 (see footnotes of Table 18.3)
value desired. to obtain value desired.
Example 4. Example 4.
Given w=lO, w’=lli, find ef, gt, and A. Given w2=10, w2’=lli, find ef, gc and A.
Here w’/iw= 1.1, so that direct reading of Table Here w3w2= 1.I, so that direct reading of Table
18.3 gives 18.3 gives
el(l)=1.6843 041 cl(l) = -.2166 2576+3.0842 589i
e2(l)=-.2166 258 (=-el--e3) e2(1)=.4332 5152=--2LS(eJ
e3(l) = - 1.4676 783 e3(l) =&Cl>
92(l) = 10.0757 7364 g2(1) = -37.4874 912
ga(1) =2.1420 1000. g3(1) = 16.5668 099.
A<0
Given g2>0 and g3>0 such that A=d-27gi’>O Given g2 and g3>0 such that A=d--27d<O! (if
(if g3=0, Jw’J= w ; see lemniscatic case), compute g3=0, Jw2’l= WZ;see pseudo-lemniscatic case), com-
&=g2gr2”. From Table 18.1, determine wg;/e pute j2=g2gi21a. From Table 18.1, determine
and w’g3lt8, thence w and w’. w2g3’ I6 and w2’g31/6,thence w2 and w’.
Example 6. Example 6.
Given g2=10, g3=2, find w and w’. With g2= Given 2= -10, g3=2, find Q and w;l. With
g2g3-2/a=6.2996 05249, from Table 18.1 wg3’16= g2z-g2g3-2L - 1O/1.5874 0105=-6.2996 053, from
1.1267 806 and w’g3‘16=1.2324 295i whence w= Table 18.1 w2g31/6= 1.5741 349 and wkg31’6=1.7J24
1.003847 and w’= 1.09797Oi. 396i whence w2=1.40239 48 and wj=1.52561 02i.
Example 7. Example 7.
Given g2=8, g3=4, find w and w’. With Given g2=7 s=6, find w2 and 0;.
J2=g2g3-2/a=3.1748 02104, from Table 18.1 wg31’B= With &=g2g3-2 9 =7/3.3019 2725=2.119974, from
1.2718 310 and w’g,“6=1.8702 425i Table 18.1 w2g31’6=1.3423 442 and w2’gJ’6=3.1441
whence w=1.009453 and w’=1.484413i. 141i whenoew2=.99579 976 and wi=2.33241 83ti.
FIGTJEE 18.16
666 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
A>0 A<0
Method 2 (for @ or @ ’ only) Method 2 (for @ or @ ’ only)
Compute et(i= 1,2,3) (if only 92, g3 are given Compute e2 and Hz (if only g2,g3 are given, use
use Table 18.1 to get the periods, then get ei in Table 18.1 to get the periods, then get e( in
Table 18.3; if periods are also given, use Table 18.3 Table 18.3; if periods are also given use Table 18.3
directly). In any case, obtain m(=[ez-es]/ directly). In any case, obtain m(=$--3e2/4H2)
[el- e3]), thence Jacobi’s functions sn(z*jm), thence Jacobi’s functions sn(z’)m), cn(z’lm),
cn(z*lm), dn(z*)m), from 16.4 and 16.21 and dn(z’jm), from 16.4 and 16.21 and @ or @ ’ from
@ or @,’ from 18.9.11-18.9.12. 18.9.11-18.9.12.
Method 3 (accuracy limited by Table 4.16 of Method 3 (accuracy limited as in the case A>O).
eenz and by the method of getting periods). Obtain periods, their ratio a, thence q2=e+J2
Obtain periods, their ratio a, then q=e-” from from Table 4.16. Then proceed as in the case
Table 4.16. Hence get St(O), i=2,3,4 from A>O, using corresponding formulas.
truncated series 18.10.21-23. Compute appro-
priate d functions for z=x and for z=iy, whence
get P(x), @‘(4 and/or JW), Q (in), @ ‘(iy>
and/or [(iy), then use an addition formula (if
either x or y is “small”, it is probably easier to
use Laurent’s series).
Example 8. Given 2=.07+.li, g2=10, g3=2, Example 8. Given z=.1+.03i, g2=-10, g3=2,
find @. find@.
Using Laurent’s series directly with Using Laurent’s series directly with
cz=.5 C2=-.5
c,=.O7142 85714 ~a=.07142 85714
cq= .08333 33333 ch= .80333 33333
c5= .00974 02597 z.-2=76.59287 938-50.50079 960i
2-2= -22.97193 820-63.06022 25i c&= - .00455 ooo- .00300 oooi
+ c2z2= - .00255 000+ .00700 OOi c3z4= $ .OOOOO 334 + .OOOOO 780i
+c3z4=- .OOOOl214- .OOOOl02i c4z6= - .OOOOO 002 + .OOOOO 01 li
fc4z6= f .OOOOO 024- .OOOOO Oli
@ (z)=76.58833 270-50.50379 169i.
@ (z) = -22.97450 010-63.05323 28i.
Example 9. Given z=15+73i, g2=8, g3=4, Example 9. Given 2=1.75-t-3.6( g2=7, g3=6,
find @. From Example 7, w=1.009453, c,+= find @. From Example 7, o2=.99579 98, &=
1.484413i. From Table 18.3, e,=1.61803 37, 2.33241 83i. Using 18.2.18 with M= 1, N= 1,
e3= -.99999 96, whence m=.14589 79. From @(1.75+3.6i)=@(-.24159 96-l.O64836i)=
18.2.18 with M=7 and N=24, 9(.867658+ @(.24159 96+1.0648 36i). WithA<O fromTable
1.748176i)=@(l5+73i). Since z lies in Rz, by 18.3,el=-.81674362+.50120 90i,e2=1.63348 724,
18.2.31 @(15+73i)=F(.867658+1.220653). e,=-.81674 362--50120 9Oi whence m=.01014
From 16.4 with z*=1.40390+1.97505i, sn(z*lm) 3566, Hi=1.58144 50, so that z’=2zH!=.76415
=2.46550+1.96527i. Using 18.9.11, ,f$(15+733) 29+3.367959i. From 16.4, cn(z’lm)=4.00543 66
= - .57743t .067797i. -12.32465 69i. Applying 18.9.11, @ (1.75+3.6i)
= - .960894- .383068i.
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 667
A>0 A<0
Example 10. Given w=lO, w’=2Oi, find Example 10. Given wZ=5, wi=7i find
{(9+19i) by use of theta functions, 18.10 and @‘(3+2i) by use of theta functions, 18.10 and
addition formulas. addition formulas.
For the period ratio a=w’/wi=2 with the aid of With the use of Table 4.16 and 18.10.2, q=ie-.‘r
Table 4.16, p=e-2”=.00186 74427. =.11090 12784i.
Using the truncated approximations 18.10.21- The theta functions are computed for argument
18.10.23 we compute the theta functions for zero using 18.10.21-18.10.23 and the theta
argument zero. Using 16.27.1-16.27.4 we com- functions for arguments vl and vz corresponding
pute the theta functions for arguments v where to z=zlfzz using 16.27.1-16.27.4. Using 18.l.O.5-
Z=X and z=iy. Then, with 18.10.5-18.10.7 18.10.6 together with 18 10.10, we find @ (3) =
together with 18.10.9 and 18.10.18 we obtain .10576 946, @ (2i) = -.24497 773, @’ (3) =
l(9) =.09889 5484, s(19i) = -.00120 0 155i, - .07474140, Q ‘(Zi) = - .25576 007i. The addi-
@ (9) = .01706 9647, p’(9) = - .00125 3460, tion formula 18.4.1 yields @(3+2i)=.O1763 210
@ (19i) = -.00861 2615, @‘(19i) = -.00003 :757i. -.07769 187i, and 18.4.2 yields @‘(3+21)=
Using the addition formula 18.4.3, we obtain -.00069 182 t.04771 305i.
[(9 + 19i) = .07439 49- .00046 8%.
Use of Table 18.2 in Computing @, @‘, r for Special Period Ratios
If the problem is reduced to computing @ , @ ‘, { in the Fundamental Rectangle for the case when the
real half-period is unity and pure imaginary half-period is ia, for certain values of a Table 18.2 may be
used. Consider @ as an example. If Iz( is “small”, then use Laurent’s series directly for @ (z) [invar-
iants for use in the series are given in Table 118.31.
If 2 is “large” and y “small” use Table 18.2 to obtain x2@ (2) and S?@‘(X), thence @ (x) and g’(x) ;
use Laurent’s series to obtain @ (iy) and @‘(iy) ; finally, use addition formula 18.4.1.
For z “small” and y “large”, reverse the procedure. For both 2 and y “large,” use Table 18.!2 to
obtain JB (x), P’(X), @ (iy) and @‘(iy), thence use addition formula 18.4.1.
Similar procedures apply to @ ’ or [. For @ ‘, one can also first obtain $J5, then compute @ I2
=4&?‘--g,@ -g3 and extract the appropriate square root (see 18.8 re choice of sign for @‘).
A>0 A<0
Example 11. Compute @(.8+i) when a==1.2. Example 11. Compute @(.9+.li) for a=1.05.
Using Table 18.2 or Laurent’s series 18.5.1-4 with Using Table 18.2 or Laurent’s series 18.5.1-4 #with
g2=9.15782 851 and g2=-42.41653 54 and
g,=3.23761 717 from Table 18.3, g3=9.92766 62 from Table 18.3,
@(.8)=1.92442 11, @ (.9) = .34080 33,
@‘(A) = -2.76522 05, @‘(.9) = -2.164801,
@ (i) = - 1.40258 06 and @ (.li) = -99.97876,
p’(i)=-1.19575 5%. Using the addition for- @‘(.li) = -2000.4255i. With the addition for-
mula 18.4.1 mula 18.4.1
@(.8+i)=-.3 81433-.149361i. @(.9+.1i)=.231859-.215149i.
Example 12. Compute ((.02+3i) for (2~4. Example 12. Compute @‘(.4+.9i) for o=2.
Using Table 18.2 or Laurent’s series 18.5.1-5 with Using Table 18.2 or Laurent’s series 18.5J-4,
g2=8.11742 426 with
g3=4.45087 587 g,=4.54009 85,
from Table 18.3, g,=8.38537 94
[(.02) =49.99999 89, from Table 18.3,
@ (.02) =2500.00016, @ (.4) =6.29407 07,
@‘(.02) = -249999.98376, @‘(.4) = -30.99041,
p(3i) = .89635 173i, @ (.9i) = - 1.225548,
@(3i)=-32326 511, @,‘(.9i) = -3.19127 03i.
@ ‘(3i) = - .00249 829i. Using the addition formulas 18.4.1-2,
Applying the addition formula 18.4.3, 9’(.4+.9i)=1.10519 76-.56489 OOi.
{(.02+3i)=.O16465+.89635i.
668 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
(or periods from which g2 and g3 can be computed-in any case, periods must be known, at least
approximately)
First reduce the problem (if necessary) to computation for a point z in the Fundamental Rectangle
(see 18.2). After final reduction let z denote the point obtained.
A>0 A<0
If $?a>~/2 or, If $?z>w,@ or
$z>w:/4, use duplication formula as in case
Nz>w’/2, use duplication formula A>O. Otherwise, use Maclaurin series for u
directly.
A>0 A<0
Example 15. Given {=lO-15i, g2=8, 9:$=4, Example 15. Given a=.4+.li, g2=7, g$=6,
find Z. Using the reversed series 18.5.40 with find z. Using the reversed series 18.5.70 with
y,=.14583, y3=.05
A5= -.I3333 333,
u= +.40000 000+.10000 oooi
A,= - .02857 14286,
$=+.OOOll 783+.00032 696i
u= .03076 923076+.04615 38461%
appropriate figure will give the value of P(Z) approximation to (zol by Graeffe’s process, we
[c(z) or u(z)] to 2-35. If @’ is wanted instead, may use the fact that z,,=wi-iyo(A>O), zo=w2
get @, use 18.6.3 to obtain @ I2 and select sign (s) +ivo(A<O) to obtain an approximation to zoJ.
of @ ’ appropriately. (See Conformal Mapping It is noted that ye/w is a monotonic decreasing
(18.8) for choice of sign of square root of @ “>. function of (period ratio) a2 1 for A>0 and
Since z$@ (zo) =O, the Laurent’s series gives yo/wz is a monotonic increasing function of a for
A<0 and
O=1+C&&L3+C~U4+ . . .
[0 5 yo/wz<z arccosh fl]
where u= .zt. We may solve this equation [by
Graeffe’s (root-squaring) process or otherwise] Further data is available from Table 18.2 or from
for its absolutely smallest root [having found an Conformal Maps defined by .Cp(2).
References
Texts and Articles [18.14] E. S. Selmer, A simple trisection formula for the
elliptic function of Weierstrass in the equian-
[18.1] P. Appell and E. Lacour, Principes de la theorie des harmonic case, Norske Vid. Selsk. Forh. Trond-
fonctions elliptiques et applications (Gauthier- heim 19, 29, 116-119 (1947).
Villars, Paris, France, 1897).
[18.2] A. Erdelyi et al., Higher transcendental functions, [18.15] J. Tannery and J. Molk, Elements de la theorie des
vol. 2, ch. 13 (McGraw-Hill Book Co., Inc., fonctions elliptiques, 4 ~01s. (Gauthier-Villars,
New York, N.Y., 1953). Paris, France, 1893-1902).
[18.3] E. Gmeser, Einfiihrung in die Theorie der ellip- [18.16] F. Tricomi, Elliptische Funkt.ionen (Akademische
tischen Funktionen und deren Anwendungen Verlagsgesellschaft, Leipzig, Germany, 1948).
(R. Oldenbourg, Munich, Germany, 1950). [18.17] F. Tricomi, Funzioni ellittiche, 2d ed. (Bologna,
[18.4j G. H. Halphen, Trait6 des fonctions elliptiques et Italy, 1951).
de leurs applications, 1 (Gauthier-Villars, Paris, [l&18] C. E. Van Orstrand, Reversion of power series,
France, 1886).
Phil. Mag. (6) 19, 366-376 (Jan.-June 1910).
[18.5] H. Hancock, Lectures on the theory of elliptic
functions, vol. 1 (John Wiley & Sons, Inc., New [l&19] E. T. Whittaker and G. N. Watson, A course of
York, N.Y., 1910, reprinted, Dover Publica- modern analysis, ch. 20, 4th ed. (Cambridge
t,ions, Inc., New York, N.Y., 1958). Univ. Press, Cambridge, England, 1952).
[18.6] A. Hurwite and R. Courant, Vorlesungen iiber
allgemeine Funktionentheorie und elliptische
Funktionen, 3d ed. (Springer, Berlin, Germany, Guides, Collections of Formulas, etc.
1929).
[18.7] E. L. Ince, Ordinary differential equations (Dover [18.20] P. F. Byrd and M. D. Friedman, Handbook of
Publications, Inc., New York, N.Y., 1944). elliptic integrals for engineers and physicists,
j18.81 E. Ksmke, Differentialgleichungen, LBsungs- Appendix, sec. 1030 (Springer-Verlag, Berlin,
methoden und Liisungen, vol. 1, 2d ed. (Akade- Germany, 1954).
mische Verlagsgesellschaft, Leipzig, Germany [18.21] -4. Fletcher, Guide to tables of elliptic functionst
1943). Math. Tables Aids Comp. 3, 247-249 (1948-49).
(18.91 D. H. Lehmer, The lemniscate constant, Math. [18.22] S. Fliigge, Handbuch der Physik, vol. 1, pp. 126
Tables Aids Comp. 3, 550-551 (1948-49). 146 (Springer-Verlag, Berlin, Germany, 1956).
[18.10] S. C. Mitra, On the expansion of the Weierstrassian
and Jacobian elliptic functions in powers of the j18.231 H. Kober, Dictionary of conformal representations
argument, Bu!l. Calcutta Math. Sot. 17, 159-172 (Dover Publications, Inc., New York, N.Y.,
1952).
(1926).
[lS.ll] F. Oberhettinger and W. Magnus, Anwendung der [18.24] L. M. Milne-Thomson, Jacobian elliptic function
elliptischen Funktionen in Physik und Technik tables (Dover Publications, Inc., New York,
(Springer, Berlin, Germany, 1949). N.Y., 1950).
(18.12) G. Prasad, An introduction to the theory of ellipt,ic [18.25] K. Weierstrass and H. A. Schwarz, Formeln und
functions and higher transcendentals (Univ. of Lehrsiitze zum Gebrauche der elliptischen Func-
Calcutta, India, 1928). tionen. Nach Vorlesungen und Aufzeichnungen
(18.13) U. Richard, Osservazioni sulla bisesione delle des Herrn K. Weierstrass bearbertet und
funzioni ellittiche di Weierstrass, Boll. Un. Mat. herausgegeben von H. A. Schwarz, 2d ed.
Ital. 3, 4, 395-397 (1949). (Springer,Berlin, Germany, 1893).
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 671
Tables [18.28] T. H. Southard, Approximation and table of the
[18.26] Chih-Bing Ling, Evaluation at half-periods of Weierstrass @J function in the equianhatmonic
Weierstrass’ elliptic function with rectangular case for real argument, Math. Tables Aids IComp.
primitive period-parallelogram, hlath. Comp. 149 11, 58, 99-100 (Apr. 1957). f(u)=@(u)-$
69, 67-70 (19GO). Values of ei (i-1, 2, 3) to 15D
for various period ratios in case A>O. to 7D . with modified central differences,
[18.27] E. Jahnke and F. Emde, Tables of functions, 4th 1~=0(.1).8(.05)1.55.
ed., pp. loo-106 (Dover Publications, Inc., New
York, N.Y., 1945). Equianharmonic case, real [18.29] D. A. Strayhorne, A study of an elliptic function
(Thesis, Chicago, Ill., 1946). Air Documents
argument,.P(u),P (4, T(u),u (u),u=O (go) Division T-2, AMC, Wright Field, Micro-
film No. Rc-734F15000. @,(a; 37, -42), 4D,
$ mostly 4D.
z= .04i(.O4i) 1.36i.
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 673
Table 18.1
TABLE FOR OBTAININC: PERIODS FOR INvARIANTS g, AND g,
@2-=&G 3
Non-Negative Discriminant Non-Positive Discriminant
[‘;;I-11
0.00 1.85407 47 1.85407 47 - 1.28254 90 3.04337 67 a- 0
[ 1
(,3)3
i:
1s=0.20412 4145 $‘=0.40824 829
674 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
[(-jP1
0.82239 820 0.82246 703
['-;'"I ['-j'"] ['-;I"] ['-;'"] [c-y]
z/i=y\u 4.0
1.1 _1.39585
____^ 80
^^
, 1.5Z55Y tlu
::1 3 1.67719 97
1.' 1 If the real half-period ~1, see 18.2 Homogeneity Relations. Interpolation with 1.85056 87
1.5 2.04521 26
respect to ‘6 will, in -general, be diflicult because of the non-uniform subintervals
I. 6 involved. Aitken’s interpolation may be used in this case. As few as 3s may 2.26025 62
$z be obtained. For the computation of @, Ip’ or I at z-.r+,!/, an addition formula
;: i may be used (18.4 and E&plea 11-12).
2.0
678 WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
[ 1
(-p4 [‘-;‘“] [C-i'"3
::; 0.44345
0.32282
14
70
i-98
2:o
0.18790
+0.03858
-0.12508
92
90
40
680 WJGIERSTRASS ELLIPTIC AND RELATED FUNCTIONS
,- 1 [95 I
[(-j"] [c-y] [c-y]
Forn=l: ,j2=d, q3=U, q=&‘/Z,
^
I?:,=-02/2,~=*‘4, n’jr=-r/4.
.
For a=-: 92-r4/12, 9s-(i/216, c, =r2/6, ‘;I = -+/12, v&/12, v’/;= m.
(w= 1.85407 4677 is the real half-period in the Lemniscatic case 18.14.)
-For 4<n<
. -, .to obtain V’ use Legendre’s relation q’ = W’ -+/2.
‘ro obtain the corresponding values of tabulated quantities when the real half-period o+l,
multiPIY 92 by u-4, !lR by w-6, ei by o--2 and 7 by w--l.
WEIERSTRASS ELtLIPTIC AND RELATED FUNCTIONS 681
For a=-: g2=?r4/12, c7s=i+/216, G?e,= -7+2/12, Se, =o, 7,=+/12, $/;=-.
(m= 1.8540’7 4677 is the real half-period in the Lemniscatic case 18.14.)
For 4<a< -, to obtain 11;use Legendre’s relation ni = tt&*i.
To obtain the corresponding values of tabu!ated quantities when the real half-period wz+1,
multiply c12by OF*, 83 by G6, ei by $? and 7 by 02’.
WEIERSTRASS ELLIPTIC AND RELATED FUNCTIONS 683
INVARIANTS AND VALUES AT HALF-PERIODS Table 18.3
(Non-Positive Discriminant-Unit Real Half-Period)
u(cq/i R+‘) Su(w’)
1.00 1.18295 13 1.182951 0.474949 0.474949
1.02 1.17091 79 1.219157 0.475654 0.483826
1.04 1.15940 62 1.255842 0.476433 0.492792
1.06 1.14841 45 1.292964 0.417275 0.501851
1. 08 1.13793 68 1.330480 0.478169 0.511006
1.10 1.12796 39 1.368342 0.479107 0.520259
1.12 1.11848 38 1.406502 0.480078 0.529611
1.14 1.10948 26 1.444910 0.481074 0.539064
1.16 1.10094 49 1.483513 0.482085 0.548616
1.18 1.09285 44 1.522257 0.483104 0.558268
1.20 1.08519 40 1.561089 0.484122 0.568019
1.22 1.07794 61 1.599952 0.485132 0.577866
1.24 1.07109 31 1.638790 0.486126 0.587809
1.26 1.06461 72 1.677548 0.487098 0.597843
1.28 1.05850 11 1.716167 0.488041 0.607968
1.30 1.05272 75 1.754591 0.488949 0.618179
1.32 1.04727 97 1.792765 0.489817 0.620474
1.34 1.04214 12 1.830630 0.490639 0.638850
1.36 1.03729 63 1.868133 0.491410 0.649302
1.38 1.03272 96 1.905218 0.492126 0.659828
1.40 1.02842 64 1.941832 0.492783 0.670422
1.42 1.02437 26 1.977922 0.493376 0.681082
1.44 1.02055 48 2.013437 0.493902 0.691804
1.46 1.01696 00 2.048327 0.494357 0.702582
1.48 1.01357 57 2.082544 0.494739 0.713414
1.50 1.01039 05 2.116040 0.495045 0.724295
1.52 1.00739 28 2.148771 0.495272 0.735221
1.54 1.00457 23 2.180693 0.495418 0.746189
1.56 1.00191 88 2.211766 0.495480 0.757192
1.58 0.99942 27 2.241950 0.495458 0.76a229
1.60 0.99707 51 2.271208 0.495348 0.779295
1.65 0.99179 98 2.340071 0.494687 0.807059
1.70 0.98727 79 2.402437 0.493456 0.834917
1.15 0.98340 36 2.457895 0.491645 0.862812
1.80 0.98008 56 2.506120 0.489246 0.890687
1.85 0.97724 49 2.546866 0.486255 0.918490
1.90 0.97481 36 2.579972 0.482673 0.946170
1.95 0.97273 30 2.605345 0.478503 0.973680
2.00 0.97095 31 2.622973 1.000975
2.05 0.96943 05 2.632902 2 E," 1.028011
2.10 0.96812 82 2.635245 0.462516 1.054750
2.15 0.96701 46 2.630169 0.456054 1.081151
2.20 0.96606 23 2.617892 0.449041 1.107179
2.25 0.96524 80 2.598678 0.441488 1.132799
2.30 0.96455 19 2.572828 0.433405 1.157978
2.4 0.96344 79 2.502604 0.415693 1.206881
2.5 0.96264 13 2.410244 0.395997 1.253647
2.6 0.96205 18 2.299090 0.374417 1.298044
0.96162 12 2.172666 0.351055 1.339858
;:7: 0.96130 65 2.034544 0.326022 1.378884
0.96107 67 1.888235 0.299435 1.414929
0.96090 89 1.737097 0.271420 1.447812
0.96078 62 1.584242 0.242114 1.477367
0.96069 67 1.432486 0.211664 1.503441
0.96063 12 1.284291 0.180224 1.525899
0.96058 34 1.141740 0.147962 1.544621
0.96054 86 1.006520 0.115052 1.559512
0.96052 31 0.879924 0.081678 1.570495
0.96050 44 0.762869 0.048028 1.57751a
0.96049 08 0.655914 +0.014297 1.580552
0.96048 09 0.559298 -0.019318 1.579595
0.96047 37 0.472982 -0.052618 1.574671
0.96045 40 0.000000 0.000000 0.000000
[(93
1 [. 1
-(-;I2
[(95
1
&;+$ ,>:{=,p f+$ +, /,,=a’ (1) = -2Be,, f-f ;+f =f (v2+ni).
( > ( )
g~~i: .(i)d+, +;)=b(i), ~(W,)=pXIRp’~/4/21i4~.
=: .(1)=2&4/r, ++o, +‘)=o.
(w=1.85407 4677 is the real half-period in the Lemniscatic case 18.14.)
To obtain the corresponding values of tabulated quantities when the real half-period
wzfl, multiply c by 02.
19. F’arabolic Cylinder Functions
J. C. P. MILLER l
Contents
Page
Mathematical Properties . . . . . . . . . . . . . . . . . . . . 6i6
19.1. The Pare.bolic Cylinder Functions, Introductory . . . . . 686
The Equation G- ($z”+a)y-0
References . . . . . . . . . . . . . . . . . . . . . . . . . . 700
The author acknowledges permission from H.M. Stationery Office to draw freely from
[19.11] the material in the introduction, and the tabular values of W(a, 5) for a= -51(1)5,
+2=0(.1)5. Other tables of W(a, z) and the tables of U(a, 2) and V(a, z) were prepared on
EDSAC 2 at the University Mathematical Laboratory, Cambridge, England, using a program
prepared by Miss Joan Walsh for solution of general second order linear homogeneous
differential equations with quadratic polynomial coefiicients. The auxiliary tables ‘were
prepared at the Computation Laboratory of the National Bureau of Standards.
Introductory
These are solutions of the differential equation
=e -L”(1-t(a++) $+(a++) (uf8) $+ . . .}
d*
19.1.1 a+y (ax2+~x+c)Y=o
=e-fz2,F~(+a+j-; $; +x2)
with two real and distinct standard forms
19.2.2
d2y
&*-- ($x”+a)y=o
19.1.2
=e~x2M(-&a++, 4, --$X2)
These have been chosen to have the asymptlltic r($-+u) cos n(++$u)
&2&-f
yI=2 sin n($++u).Y,
behavior exhibited in 19.8. The first is Whit-
taker’s function [19.8, 19.91 in a more symmetrical =U(u, 2) +U(u, -c-x)
notation.
19.4.5
19.3.1
- r(f-+uj sin?r($++a)
y2=2 cos a($++z)-Yz
u(a,2)=D~,~~(2)=COSa(~+~a)~Y~ J;;2w:
-sina(~+&z) .Y, =U(u, 2) -U(u, y>
19.4.6
19.3.2
$zrU(--a, iiz)=
V(a, 5)=& {sin7r($++u).Y1
2 r(g+u) (e-i”‘@-t’U(u, *z) +ei”‘+f’U(u, rix) }
scos n(~+$u)*:F~j
in which 19.4.7
csc n(;++a)
19.3.4 Yz=& “;*:2”’ yz=fi 2~a-t~(5+3u) yz
19.5. Integral Representations
19.3.5 A full treatment is given in [19.11] section 4.
fi Representations are given here for U(u, z)
U(“F0)=2#a+tr(p+$u) others may be derived by useof the relations
19.4.
4%
u’(“‘o)=-2*=-,r(a+~u)
r (B-4 e+
19.3.6
2@+t sin?r(z-$3)
19.5.1 U(u, Z)=F
S (I ers-lsaSa-td,y I
V’(u, O)=
r($-+u) where (Yand p are the contours shown in Figares
19.1 and 19.2.
In terms of the more familiar D,(X) of Whit-
taker, When a++ is a positive integer these integrals
become indeterminate; in this case
19.3.7 U(u, 2)=D-a-*(2) m
19.3.8 19.5.3 U(u, z)=& e -pa
S 0
e-zs-+8zsa-+dsi
t PLAN4
19.4. Wronskian and Other Relations S PLANE
--IJ+~sls-a-~ds 19.5.9
19.5.4 U(a, z)=~-& efr’ (e
s U(a, 4
=iIyf-+a>
19.5.5 =Jz;;;
,ca-ud
S
efz2 (3 ezs+@‘S-a%?S 2wtr Sml) 3ze-fzaf(l+t)-f”-f(r-t)~-fdt
19.5.10
19.5.6 =iIy*-$a) e-“(tz”+v)-‘“-‘(tz”-v)f”-‘dv
21W~ S ‘11
where E, e3 and e4 are shown in Figures 19.3 and The contour cl is such that (+z2+v) goes from
19.4. coevfr to mei* while v= is2 is not encircled ;
s PLANE ($z2-v)+-t has its principal value except pos-
sibly in the immediate neighborhood of the
branch-point when encirclement is being avoided.
Likewise ql is such that (a~“-v) goes from me*,
to me -** while encirclement of v= -2~” is simi-
larly avoided. The contours (11) and (sl) may be
obtained from cl and qI by use of the substitution
v=$z”t.
The expressions 19.5.7 and 19.5.8 become inde-
terminate when a=~, 8, J+, . . .; for these values
FIGURE 19.3 FIGURE 19.4 19.5.11
-t*<arg s<h On ea I*<arg a<;s
19.5.7
On r, --:r<arg a<-**
rl
Ub, Z)=r(&a) m-f22 me-~~~o-f(Z2+2S)-ta-tds S 0
19.5.8 19.5.12
=W--;;‘S,, ~2e~(~22+v)ta-f(~z2-v)-*a--1dv
U(a, ~)=~(~l+~~)
S
e-tr2 ,me-‘s~a-~(Z2+2s)-to-tdg
Barnee-Type Integrals
where the contour separates the zerosof r(s) from those of l’(a+$-2s). Similarly
V(a, s) =
19.10.2 U(a, z)=~% exp
2
ear sin { +s++7ra+e+v,}
(27r)tJl?(+-a)
19.10.3 U(a, -z)=~% exp {e+v(a, -2))
where
where
19.10.12 vlm-& In Y-$+$&. ..
19.10.4
d3 -- &
v(a, x)-- *In X-kg1 (-1>“d3JX3* “f-y3 yg+ *. . (x2+4a-+- m)
(a>O, x2+4a-++ a)
In each case the coefficients d3r are given by
and d3a is given by 19.10.13.
19.10.13
(ii) a negative, x2+4a large and positive. Write
=S
+ ‘YG%=txY
0
+ Ial arcsin 2.L.
24 al
where v,, vr are given by 19.9.5 and p=1/--a.
Alternatively, with p=Jla(, and again -a>>x2,
19.10.10
U(a , x) =2w
(27r)“4
ear co9 ++*?ra+e+v, +w -144x2
1 1 (4p)B + ’ * *)
PARABOLIC CYLINDER FUNCTIONS 691
and 19.15.16
V(-3, x)=8. 5(3x):(5~~+g9.~~-5~~--~)
19.14.5
-
19.15.17 U(-+, x)=Ja/~($x)K+
V(-n-3, x)=e-fz2
19.15.18 U(-3, ~)=&$(32)~2K~
-__- sin ina
~ WO)
aW(3n+l) > 19.15.19 U(-8, x)=&$~($z)~(~K~-K;)
Here V(-a, 2) is closely related to Dawson’s 19.15.20 VC+, x>= (3x> (It+I-d
integral
Set2at
z
19.15.21 V&, x) = (+r)“(2r*+2I+)
0
19.15.22 V(;, x) = ($x)“(514+51-+-I;-I-;)
These relations give a second solution of 19.1.2
whenever 2a is an odd integer, and a second solu-
tion is unobtainable from U(a, 2) by reflection in The Equation $$+(i +-a) y=O
the y-axis.
19.16. Power Series in x
19.15. Explicit Formula in Terms of Bessel
Functions When 2a Is an Integer Even and odd solutions are given by 19.2.1 to
19.2.4 with -ia written for a and xeii* for x;
Write the series involves complex quantities in which the
imaginary part of the sum vanishes identically.
19.15.1 I-,--I,=(2/?r) sinna. K, Alternatively,
19.15.2 I-,fI,=cos 7m * 9, 19.16.1
where the argument of all modified Bessel func- yl=l+a $+(a’ -&) $+(a3-fa) $
tions is ax2. Then
At x=0, 19.20.1
T=-(~a,y
19.20.2
T= +(9w
Complex Solutiom3
cl2Sz
$2 Gx)“, -$s($x)3++ (6x)’ Y=J4a-x2 8=4~6,(~//2&)
(2P)4 (2p)5 + ***
Ydx=+xY+a arcsin 2
0 2&
The upper sign gives the first function, and the (see Table 19.3 for S4=+r-G3), then
lower sign the second function.
(ii) a negative 19.23.6 W(a, x)=exp{ --8+v(a, x)}
When -a>>x2, with p=JTi, then 19.23.7 W(a, --2)=exp{e+v(a, -2)}
19.22.4 where
W(a, x)+iW(a, -x)
19.23.8
=JzW(a, 0) exp {v,+i(pz+r+vJ }
v(u, x)-- 6 da do
+ In Y+p+y~+p+ ...
where W(a, 0) is given by 19.17.4, and
(x2--4a+- a)
19.22.5 and d3, is again given by 19.23.12.
-- (6x)” 2k44
__- 9(+x)2+v (&z)B+ (iii) a negative, x2-4a>>O
&- (zp)“+ (2p)4 (2p)6 ..’
Write
,*,t(M3 Bx+8&Y +wBd3+w’- . .
2P (2P) 3 (2P)5 19.23.9
(a+- ~0)
x=J22+4/a/ e=4[albl(z/2y’la[)=+~ Xdx
Further expansions of a similar type will be found
in [19.3].
19.23. Darwin’s Expansions =+xX-a In -x+x
2m
(i) a positive, x2--4a>>O
Write = ~x~x2+41ul - a arcsinh --?--
Niq
19.23.1 (see Table 19.3 for 0,) then
19.23.12 19.24.8
8x4+8 J$xe+152x2 i
d3=-; ($ax) -&+(4p)i- (4p)6 + . * ’
d6=%xz+ 2a 19.24.9
$2' ~x4+16++x6+yx2
lfo”-(4p)4 (4p)B - . i. .
h=$ &j ~~-~~az’+~a~x~+~ a3x3+19a4x
( > 19.24.10
d12=- 153 x4+186ax2+80a2 +
$x6+168x2
(4p)6 - . . .
8
19.24.11
See [19.11] for d15, . . ., d24, and [19.5] for an al-
ternative form. $x4--16
+- (4p)4 +
19.24. Modulus and Phase
When a is positive, the function W(a, x) is Again, when a<O, x2-4a>O, with X=dx2m,
oscillatory when x< - 2&i and when x>:!JZ; then
when a is negative, the function is oscillatory for 19.24.12 F---\i%?“r x=$r+e+vi
all x. In such cases it is sometimes convenient
to write where 8, v, and vr are given by 19.23.4 and 19.23.9.
Another form also when a>O, x2--4a+m is
19.24.1
19.24.13
k-W(a, x)+ikhW(a,-x)=E(a, x)=Fe”” (XX)
19.24.2 F-
k-t dWa,
dx 4 +ikt my(~~mx)=,f(a, x)=-Gc!W 19.24.14
(XX) G-
Then, when x2>>lal,
19.24.3 while J, and x are connected by
F- 19.24.15
J/-X-
19.24.4
4a2-3 4a3-19a+ 19.25. Connections With Other Functionp
X-4x2---a In x+#~++T+,,+ 8x4 ...
Connection With Confluent Hypergeometric and Be/ssel
Functions
19.24.5
19.25.1
G- zz +y-14a;;663a- ...
J( > W(a, &x)=2-f $ H(-$7 +a, +x2)
(J 3
19.24.6
4a2+5+4a3+29a
*- $x-j-a In x+$#~~-$r+- + ... where
8x2 8x4
where +2 is defined by 19.17.10. 19.25.2
H(m, n, x) =e-fZIFl(m+ 1 --in; 2m+2; 2i;c)~
When a<O, lal>>x”
19.24.7 19.25.3 =e- fZM(m+l-iin, 2m+2, 2ix) 1
F~&W(a, 0) eor
19.25.4
where V, is given by 19.22.5 with p=J=o. AIso
W(0, ~x)=2-~~?rz{J-t(4x2)fJ~(i~2) I (xqx
696 PARABOLIC CYLINDER FUNCTIONS
19.26.6
as an approximation to a zero of the derivative,
we obtain for the corresponding zero c or c’, with
-a =p2 t,he expressions
c=BJii[ (a>>01
19.26-J ,,;+!&$+52a5-240a3$315a+ ...
768Op with the aid of Table 19.3. For a zero of W(a, x)
we replace a, by b,. When a is negative we solve,
19.26.2 c’ = P
- 2/33+3P+52P5+280/33-285/3+ again with the aid of Table 19.3,
P+-- 48p5 7680~~ ’ ’’
These expansions, however, are of little value 19.26.7
in the neighborhood of the turning point x=2-a.
zr),=+{[Jl2+1+arcsinh [} =@$$@
Here first approximations may be obtained by
use of the formulas of 19.7. If a, (negative) is a
zero of Ai( the corresponding zero c of U(a, 2) c=wl45 (--a>>01
19.26.17 =
-sx g “‘f, $1
wa, c’)=U{ -%(a,
da: -x) 1 z-c’ =-l/W’(a, --c’) I
As before, Bessel functions of other orders ma Jt be
19.27. Bessel Functions of Order f 2, f 2 as obtained by use of the recurrence relation 10.21.23,
Parabolic Cylinder Functions which here becomes
Most applications of these functions refer to 19.27.8 ~221y+l(~x2)+2~I~(~x2)-~x21v-~(~x2) =fj
cases where parabolic cylinder functions would be
more appropriate. We have 19.27.9 ~xz~,+,(~x2)-2~~v(~x2)-~x2~v~,(~x2)~=0
Numerical Methods
19.28. Use and Extension of the Tables reverse direction, from arbitrary starting va ues
(often 1 and 0) for two values of a some hat
For U(u, x), V(u, x) and W(u, x), interpolation
beyond the last value desired. This is because4 the
x-wise may be carried out to 5-figure accuracy
almost everywhere by using 5-point or 6-point recurrence relation is a second order homogen ous
Lagrangian interpolation. For la] 5 1, com.pa- linear difference equation, and has two i de-
rable accuracy u-wise may be obtained with 5- or pendent solutions. Loss of accuracy by cant lla-
6-point interpolation. tion occurs when the solution desired is diminis i: ing
For [aI> 1, U(u, x) and V(u, x) may be obtained as a varies, while the companion solution is in-
by use of recurrence relations from two values, creasing. By reversing the direction of prog ess
possibly obtained by interpolation, with Ial :I 1; in a, the roles of the two solutions are in er-
such a procedure is not available for W(u,. &a), changed, and the contribution of the desired s lu-
lal> 1. tion now increases, while the unwanted solu ion
In cases where straightforward use of the a-wise diminishes to the point of negligibility. By st rt-
recurrence relation results in loss of accuracy by ing sufkiently beyond the last value of a for w ich
cancellation of leading digits, it may be worth the function is desired, we can ensure that the
while to remark that greater accuracy is usuSally unwanted solution is negligible but, because : the
attainable by use of the recurrence relation in the starting values were arbitrary, we have an un-
698 PARABOLIC CYLINDER FUNCTIONS
known multiple of the solution desired. The com- selations 19.6.1-2. If less accuracy is needed they
putation is then carried back until a value of a :an be found by use of mean central differences of
with (al < 1 is reached, whan the precise multiple U(u, s), V(a, x) and also of W(u, x) with the formula
that we have of the desired solution may be deter-
du
mined and hence removed throughout. Compare hu’=h &=& -~pccs3u+&63u- . . .
also 9.12, Example 1.
Example 1. Evaluate U(a, 5) for u=5, 6, 7, using h= .l; this usually gives a 3- or 4-figure
. . ., using 19.6.4. value of duldx.
If greater accuracy is needed for dW(u, x)/dx it
(a++) U(a+ 1,~) +zU(a, 2) - U(a- 1,z) =0
may be obtained by evaluating d2W/d2 with the
help of the differential equation satisfied by W
a Forward Backward Final Values and integrating this second derivative numerically.
Recurrence Recurrence
This requires one accurate value of dW/dx to
start off the integration; we describe two methods
3 (-6) 5. 2847’ 12) 1. 59035 (-6) 5.2847**
4 (-7) 9. 172* 11) 2. 76028 (-7) 9. 1724 for obtaining this, both making use of the differ-
5 i-7j 1. 5527 10) 4. 67131 (-7) 1. 55227 ence between two fairly widely separated values
9) 7. 72041 (-8) 2.5655
; [I;; 2. 5609
4. 1885 9) 1. 24785 ( -9) 4. 1466 of W, for example, separated by 5 or 10 tabular
8 (-10) 6. 2220 8) 1. 97488 (- 10) 6.5625 intervals.
9 (-lO)+l. 2676 7) 3. 06369 (-10) 1.01806
10 (-11)-O. 1221 6) 4.66352 (-11) 1. 5497 (i) Write j,, j& j:’ for W(u, xO+rh) and its
11 (-ll)+l. 2654 0) 697082 (-12) 2. 3164
12 (-12)-5.6079 102444 (-13) 3.404 first two derivatives, then j: may be found from
13 (-12)+3. 2555 14789 (-14) 4.91
14 2111 (-15) 7.01
15 292 (-16) 9.7 hj;-& (jn-j-.)-gm$ (n-r)(j:‘-ji’?)
16 42
17 5
18 1+ h2
-- 2n
19 I +-& s2+* 64- . . .} (f;--jQ
c=1.4-.012730+,000002=1.38727 y'(c)=2.0637(1+.000203)=2.0641
which is correct to 5 decimals, while 19.26.11 gives compared with the correct value 2.06416.
References
Texts [19.9] E. T. Whittaker and G. N. Watson, A course of
modern analysis, 4th ed. (Cambridge Univ.
[19.1] H. Buchholz, Die konfluente hypergeometrische
Press, Cambridge, England, 1952).
Funktion (Springer-Verlag, Berlin, Germany,
1953). Tables
[19.2] C. G. Darwin, On Weber’s function, Quart. J.
Mech. Appl. Math. 2, 311-320 (1949). [19.10] British Association for the Advancement of Science,
[19.3] A. Erdelyi et al., Higher transcendental functions, Mathematical Tables, vol. I, Circular and hyper-
vol. 2 (McGraw-Hill Book Co., Inc., New York, bolic functions, exponential, sine and cosine
N.Y., 1953). integrals, factorial (gamma) and derived func-
[19.4] J. C. P. Miller, On the choice of standard solutions tions, integrals of probability integral, 1st ed.
to Weber’s equation, Proc. Cambridge Philos. (British Association, London, England, 1931;
Sot. 48, 428-435 (1952). Cambridge Univ. Press, Cambridge, England,
[19.5] F. W. J. Olver, Uniform asymptotic expansions for 2d ed., 1946, 3d ed., 1951).
Weber parabolic cylinder funcjions of large [19.11] National Physical Laboratory, Tables of Weber
order, J. Research NBS 63B, 2,131-169 (1959), parabolic cylinder functions. Computed by
RP63B2-14. Scientific Computing Service Ltd. Mathemati-
[19.6] G. N. Watson, A theory of asymptotic series, Philos. cal Introduction by J. C. P. Miller. (Her
Trans. Roy. Sot. London, A 211,279-313 (1911). Majesty’s Stationery Office, London, England,
[19.7] H. F. Weber, Ueber die Integration der partiellen 1955).
Differential-gleichung: &@a?+ @r&1/2+ k*u=O, [19.12] National Physical Laboratory Mathematical
Math. Ann. 1, l-36 (1869). Tables, vol. 4, Tables of Weber parabolic cyl-
(19.81 E. T. Whittaker, On the functions associated with inder functions and other functions for large
the parabolic cylinder in harmonic analysis, Proc. arguments, by L. Fox (Her Majesty’s Stationery
London Math. Sot. 35, 417-427 (1903). Office, London, England, 1960).
702 PARABOLIC CYLINDER FUNCTIONS
Table 19.1
0)-3.0982
0 I -3.0617
0 -2.9073
0 -2.6435
0 -2.2824
:*: i 0I 4.3739
4.8038
314 0 5.1246
5.0 ( 0) 1.8800 (-1) 9.2276 (-1) 4.4586 (-1) 2.1235 (-2) 9.9802 -2) 4.6331 (-2) 2.1262 (-3)9.6523
For interpolation, see19.28.
PARABOLIC CYLINDER FUNCTIONS : 703
Tab14 19.1
V(p5.0, z) V(-4.5, .r) V(-4.0, .r) TT(-3.5,J) V(-3.0, cc) V(-2.5, z) l7-2.0, z)
(-2)-5.8311 0.0000 0.0000 -1
-1
-1
-1
-1
1.8125 (-1) 1.5812 - .5385
(-1) 1.1580 r; - .9387
-1 - .2553
-1 - .4995
-1 ii - .6835
1.0 6.7728 i-2 -7.b762
:*;
1:8
1.9
-l)-1.0927
-2 1 -6.9034
-2 -2.7540
-2 +1.4424
-2) 5.6116
-1 3.4421
-1 1 3.7545
-1) 4.0712
-1) 3.8093
-1) 4.1462
t-78
4:9 -1) 5.5449
5.0 (-1) 1.8370 (-1) 3.3533 (-1) 6.2047 ( 0) 1.1734 ( 0) 2.2757 ( 0) 4.5254 ( 0) 9.2067 ( 1) 1.9107
704 PARABOLIC CYLINDER FUNCTIONS
Table 19.1
2.0
2.1
-1
-1
-1
-1
I
4.9754
4.5701
4.1741
3.7910 II II II II
-1 4.2938
2.8559
3.1876
3.5391
3.9086 -1 2.9165
3.2511
3.6054
3.9779
2.6029 -1 2.3693
2.6647
3.3204
3.6788
2.9820 -1 ;*;;;;
1,' 2.4313
2:7312
2.1538
214 (-lj3.7407 -1 3.4238
2.5
f:4
3.0
3.1
3.2
z*z
.
II
-1 1.8488
-1 1.6124
-1 1.3985
-1 1.2064
1.0351
3.5
3.6
3.7
3.8
3.9 II II
1; 4.4657
-2 2:5638
;.:;g
5.3190
-2 4.6771
r; 2:7052
2.2315
33';;;; II -2 3.4324
1.9411
2.3589
4.1098
2.8525
t*10
4:2
4.3
4.4
4.5
ii
-2 3.6903
1.6688
2.0512
2.5079
3.0502
(-2j1.1618 -3 I 9.9881
Ii-2 2.4280
1.9923
1.0676
1.3211
1.6265 Ii
-2
-3 1.7268
9.1898
1.1397
2.1094
1.4064
4.6
4.7
t:9"
5.0 (-3)4.3375
II
-3 9.3333
4.6914
5.9310
7.4594
(-3)3.6919
-3 8.0067
-3 6.3856
-3 5.0667
-3 3.9996
(-3)3.1412
II
-3 6.8657
3.4085
4.3266
8.5831
5.4641
(-3)2.6716 (-3)2.2714 (-3)1.9305 (-3)1.6401
PARABOLIC CYLINDER FUNCTIONS RO5
Table 119.1
I I0.0000
-1 3.1502
0.7972
2.3760
1.5905
iI
-1 3.3194
5.0435
4.1939
(-1) 6.0492
(-1) 6.4993
I 0) 1.3191 f Oj1.3881
2.3
2.4
2.5 ( 0) 1.5902 t 0) 1.7975
0) 1.9338
22::
22:: ( 0) 2.4881
3.0 t 0) 2.1558
3.1
;::
3.5
3.6
9.9377
1.1805
1.4113
Table 19.1
U( -0.2,x) U(O.3,x)
Ii -10 1.0421
9.8431
1.1000
1.1581
1.2163
82
2;
0:9
1.0 t-1)6.8072
:::
(-lj4.9087
1.5
II -1 4.0657
3.6765
4.4769
3.3102
(-1)2.9673 (-1)2.5142
(-lj1.3136 II
-1 1.8627
1.0695
1.2363
1.4232
1.6315
2.5
ii
II
il -1
-2 1.0248
6.4422
5.4703
7.5534
8.8173 -2 9.2134
5.7406
6.7502
7.9031
4.8608
2.9 (-2)6.9114
II
3.0
-2 4.0978
1.9799
2.3907
2.8739
3.4393
;-21
3:3
3.4 -2 4.4006
2.5730
5.2146
3.0912
3.6967
t-213.1669 t-2)2.7772 t-2)2.4340 f-2)1.8659
22
32
II ii
3:9
II
4.0
-3 5.8057
2.3371
2.9546
4.6568
3.7179
-3)7.7613
-3 3.1779
4.0011
5.0135
6.2526 -3 4.3184
3.4390
5.3973
6.7143
2.7259
II
t:;
4.5 (-3J3.9954 t-3)2.9336 t-3)2.5122 t-3)2.1504
-3 2.4912
-3' 3.1626
1.5233
1.9528
i-87
4:9
4.6
5.0
I
-3 2:9173
2.2914
-3 1.7909
I'I:';.;G;l"
(-3)1.3929 (-3)1.1825 (-3)1.0035 (-4)8.5136 (-4)7.2201 (-4)6.1210 (-4)5.1875
PARABOLIC CYLINDER FUNCTIONS 707
Table 14.1
Ij II
-1)5.7994
-1 6.2358
-1 I 6.6661
-1 4.5280
5.0724
5.6069
6.1307 -1 7.0905
(-1 6.6436 -1 7.5093 -1 7.5184
8.1782
7.1901
6.8621
7.8474
t-1)6.1992 t-1)7.1460 -1)7.92:38
Ii -1 8.3353
9.15138
8.7460
(-lj8.4321 (-lj9.0756 (-1)9.5771
(-1)8.9640 0 I 1.0591
0 1.1013
0 1.1490
II -10 1.0019
1.1100
1.0553
9.4914
0I 1.25:,3
0 1.3147
00 y:!;;
II 0 1.2839
0 1.3515
0 1.5142
1.4277
II 0 1.3115
0 1.3848
00 :2:;
0 1.2032
0)1.2649
0)1.3353
0 1.4160
0 1.5085
0 1.6150
0
0
0
0
I 1.35 i
1.44
1.54
1.66
9
6
9
2
0 1:55:!2 0 1.6130 0 1:6738 0 I 1.7379 0 1.80, 0
I
II 0 1.5886
1.6941
1.8149
2.1153
1.9541
Ii 0 1.8799
2.7195
2.4589
2.2360
2.0446
0 2.1614 Oj2.4576 I Oj2.6278 ( Oj2.8159 ( Oj3.0247 0 3.25 I 2
0 2.3551 0 3.66 7
0 2.5818 0 4.14 5
2; 0 2.8478 II 0 2.3028
3.0803
2.7785
2.5218
3.4366 0 4.70 4
2:9 II 0 3.1612 0 5.38 0
3.0
3.1
1)1.0158
3.5 0I 6.7730 II 0 7.5658 0 I 8.4638 ( Oj9.4818 ( 1)1.0637
3:; 0 9.1860
7.8635 10 1.0340
8.8182 0 9 90213
1 1:1653
II II II ii
2; 1 1.0797
1.2766 1 1.4470
1.2196
; ;.;m;
.
t-l 1 2.2395
2.7041 1 2.5539
3.0927 1 3.5401
2.9150 1 3.33 f 04
4.05
4:2 1 3.2829 1 3.7653 1 4.3219 1 4.96 4
4.5
2;
29"
II II II II
1
1
4.0073
4.9179
1 6.0680
1 7.5270
9.3866
2 1.4831
1.1768
1
1
4.6086
5.6708
1 7.0147
21 1.0904
8.7230
2 1.3703
1.7309
1
1
5.3040
6.5433
1 8.1143
2 1.0115
1.2674
2 1.5964
2.0211
1
1
6.10 5
7.55 0
1 9.39 I 1
2 1.47
1.17 60
2 2.36
1.86 81
Table 19.1
(-1)4.8280 -1)4.2896
(-1)4.3327
-1)3.0003
-1)2.6475
(-1)2.9390 (-1)2.7238 (-1)2.5204 -1)2.1487 -1)1.9797
(-1)1.4798 (-1)1.3487
I -1)1.8774
-1 1.6351
-1 11.4193
(-1)1.2278
I -1 I 1.7240
1.2948
1.4965
(-1)1.1165
2.0 (-1)1.7003
t-21713793
(-2)6.2874 (-2j5.0508
2.5 t-2)8.2754
II
2:9 i-2)4:3157
-2 2.5078
2.0830
1.4189
1.7228
(-2)1.1636
3.5 -3)9.5009
3.6
3.7
3.8
3.9 il-3 8.2868
3.4952
4.3655
5.4288
6.7217
t-314.3344 f-3)3.7425
-3 2.9826
-3 I 2.3663
-3 1.8689
-3 1.4693
0.5 -1)8.4934
0.6 -1 I 8.7302
0;7 -1 9.0186 -1 7.1733
-1 9.3633 -1 17.8124
::9" -1 9.7698 -1)8.5344
-1 I 9.7713
0 1.0488
0 1.1309
0 1.2251
0) 1.3024 ( 0)1.3158 0 1.3330
1.5
1.6 II
( 0)1.3779
0 1.4784
0I 1.4949
0 1.6542
0 1.8373
:*i 0 1.5943
1.7281 0 2.0484
1:9 0 1.8829 0)2.1703 0)2.2926
2.0 t Oj2.0622 ( 012.1689 ( Oj2.28116 0)2.7481
II
0I 3.1169
0 3.5483
0 4.0548
00' 2.6757
3.3676
2.9943
2.4030 0)3.6363 0)4.2741 0 4.6517
2.5 II 0 3.5166 0 4.4944
0 5.1536
;:"7 0 3.9749
4.5165 0 5.9365
0I 6.8696
;:t 0 5.1589
5.9235 0 7.9862
3.0 1 1.0378
1 1.2220
111.4455
I
3.5 1 1.7060
1 2.0373
3:8
3'2 1 1 2.4452
2.9495
3.9 1)3.5756 Ii 1 4.65
6.99i 6
5.69
3.82
3.15' 5
4.3
4.4
4.5
i-7"
4:8
4.9
5.0 ( 2)3.5270 ( 2)4.1331 ( 2)4.8456 ( 2)5.6833 ( 2)6.6688 ( 2)7.8285 ( 2)9.19b8
710 PARABOLIC CYLINDER FUNCTIONS
Table 19.1
-1 I 1.9302
-1 1.6146
-1 1.3490
-1 1.1256
II II II II
-1
-1
-2
-2
1.2931
1.0674
8.8019
7.2491
-2
-2
-2
-2
8.4374
6.8788
5.6025
4.5579
-2
-2
-2
-2
5.3758
4.3316
3.4869
2.8040
-2
-2
-2
-2
3.3518
2.6707
2.1262
1.6910
-2 9.3785 -2 5.9624 -2 3.7035 -2 2.2523 -2 1.3434
-2 I 1.7849
-2 1.4503
II II II
-2
-3
1.0327
8.2953
-3
-3
5.8705
4.6645
-3
-3
3.2833
2.5816
I II
-2 1.1759 -3 6.6500 -3 3.6991 -3 2.0262
-3 9.5127 Ii: ;.;g -3 2.9276 -3 1.5873
-3 7.6780 . -3 2.3122 -3 1.2409
-3 3.3818 -3 1.8222 -4 i 9.6810
-3 2.6869 -3 1.4328 -4 7.5364
-3 2.1296 -3 1.1240 -4 5.8538
2.5
z.7"
2:8
II II
-2 2.3966
-2 1.6441
1.9886
-2 1.3544
-2 1.3223
-2 1.0837
-3 8.8509
-3 7.2040
-3
-3
1.6837
1.3277
-4
-4
8.7960
6.8665
-4 4.5364
-4)3.5071
3.5
3.6
-4
-5
I
II II I II
1.2259
9.3061
-5
-5
5.4198
4.0787
-5
-5
-5
-5
-5
-5
7.3727
5.5875
4.2185
3.1726
9.6913
2.3767
1.7736
-5 3.3191
1; :*:;;:
II-5 1:3920
-5
-6
4.4015
1.0342
7.6538
-5
-5
-6
II-6
-5
1.4817
1.1039
8.1946
6.0609
1.9818
ii -6 4.4663
-6 3.2790
IIII III
-6
-6
-6
-6
-6
-6
6.5617
4.8485
3.5701
2.6194
8.8495
1.9150
1.3949
-7
-7
8.1539
5.8942
-5 7.0352 -5 3.0571 -5 1.3183 -6 5.6428 -6 2.3983 -6 1.0124 -7 4.2455
-5 5.2961 -5 2.2819 -6 9.7593 -6 4.1440 -6 1.7475 -7 7.3205 -7 3.0469
-5 3.9701 -5 1.6964 -6 7.1961 -6 3.0315 -6)1.2685 -7 5.2737 -7 2.1788
5.0 (-5)6.9418 (-5)2.9634 (-5)1.2558 ( -6)5.2847 (-6)2.2089 (-7)9.1724 (-7)3.7849 (-7)1.5523
PARABOLIC CYLINDER FUNCTIONS 711
Table 19.1
I -1 I 1.6118
3.3218
2.4481
0.7999
-10 2.4076
Ii -1 1.0497
7.5647
4.8999
0)5.2778
1) 3.0195
1) 3.7699
1 I 4.7150
I 1) 5.1442
1) 6.4978
1 8.2198
1 5.9076
1 7.4155 i 2 1 1.0415
2) 1.3218
0)6.7480
0 I 7.9725
( 1)3.0364
1) 3.7393
( 1)5.2689
1)6.5656
1)9.3262 ( 2) 1.6806
2) 2.14108
0 9.4452
1 1.1222
1) 1.3374 1) 2.276El ( 1) 3.9709 (
1 j 4.6150
1 5.7092
1) 7.0801 I
1 1 8.1989
2 1.0262
2)1.2873
i( 12 2.73125
2 3.49148
2)4.4794
2 1) 8.8025 (
:2 I(
2:9 I
I (
1) 9.2982
I( ( 3)2.9574 I
2) 9.1055 3) 8.10129
4 1.07’22
4 1 1.42’32
I( ( 3)5.4084 4) 1.1642 I 4 1.89’50
4) 2.53113
i
4 1 6.7384
4 9.1425
5I 1.2450
( 5 1.7018
5)2.3348 I
5.0 ( 3)2.0666 ( 3)4.6909 4) 1.0746 ( 4)2.4833 ( 4)5.7864 ( 5)1.3589 ( 5)3.2156 ( 5)7.6639
712 PARABOLIC CYLINDER FUNCTIONS
Table 19.2
0.51440 0.43707
[(-y 1 [(-f)7
5.0 0.08936 -0.21874 0.53861 0.21827 -0.37095 -0.54818
Values of W( a,z),for integral values of a are from National Physical Laboratory, Tables of Weber parabolic cylinder func-
tions. Computed by Scientific Computing Service Ltd. Mathematical Introduction by J. C. P. Miller. Her Majesty’s
Stationery Office, London, England, 1955 (with permission).
PARABOLI:C CYLINDER FUNCTIONS 713
Table 19.e
1.8377
1.4984
1.2246
1.0035
a.2455
i 1 I 1.4686
1.6117 I 1 \ 3.0749
3.5113
Table 19.2
Table 19.2
[ 1 II 1 II 1 [ 1 II 1 [
(-$)4 (-l)3 (-l)3 (-;)3 (-2)3 c-y
1
PARABOLIC CYLINDER FUNCTIONS 717
Table 19.2
[5316
1 1 1 [(-p5 (-l)5
1 [ (-l)5
1 [ 5316
1 c(-[I71 [(-;P1
718 PARABOLIC CYLINDER FUNCTIONS
Table 19.2
5 W(O.4,-2) W(O.5, -x) W(0.6,-:e) W(O.7, -x) W(O.8,-2) W(O.9, -2) W(l.O,-z)
0.0 0.91553 0.87718 0.84130 0.80879 0.77982 0.75416 0.73148
0.97201 0.93642 0.90331 0.87352 0.84714 0.82396 0.80361
0”:: 1.03235 1.00031 0.97072 0.94433 0.92122 0.90115 0.88375
1.09671 1.06911 1.04386 1.02166 1.00258 0.98636 0.97265
E 1.16520 1.14300 1.12302 1.10591 1.09173 1.08022 1.07106
0.5 1.23789 1.22215 1.20846 1.19746 1.18917 1.18338 1.17975
1.31475 1.30664 1.30040 1.29663 1.29538 1.29644 1.29949
i:; 1.39567 1.39648 1.39896 1.40371 1.41079 1.42000 1.43106
1.48046 1.49158 1.50419 1.51888 1.53574 1.55459 1.57519
E 1.56879 1.59174 1.61602 1.64225 1.67051 1.70068 1.73254
1.0 1.6602 1.6966 1.7343 1.7738 1.8153 1.8586 1.9037
1.1 1.7541 1.8057 1.8586 1.9133 1.9700 2.0286 2.0891
1.2 1.8497 1.9184 1.9884 2.0603 2.1345 2.2107 2.2891
1.9460 2.0337 2.1230 2.2144 2.3083 2.4048 2.5037
::: 2.0418 2.1506 2.2613 2.3746 2.4909 2.6102 2.7327
1.5 2.1358 2.2677 2.4020 2.5397 2.6811 2.8264 2.9756
1.6 2.2263 2.3833 2.5437 2.7083 2.8777 3.0520 3.2316
2.3115 2.4956 2.6843 2.8785 3.0788 3.2856 3.4991
287 2.3891 2.6023 2.8216 3.0480 3.2823 3.5249 3.7762
1:9 2.4570 2.7009 2.9529 3.2141 3.4854 3.7674 4.0605
2.0 2.5125 2.7886 3.0752 3.3737 3.6849 4.0097 4.3487
2.5529 2.8623 3.1853 3.5231 3.8770 4.2479 4.6368
22.: 2.5754 2.9188 3.2793 3.6583 4.0573 4.4775 4.9201
2:3 2.5770 2.9546 3.3532 3.7748 4.2209 4.6931 5.1930
2.4 2.5548 2.9660 3.4030 3.8678 4.3624 4.8889 5.4490
2.5 2.5061 2.9496 3.4241 3.9321 4.4760 5.0582 5.6811
2.4283 2.9018 3.4124 3.9626 4.5555 5.1940 5.8811
22.; 2.3192 2.8196 3.3634 3.9538 4.5944 5.2887 6.0405
218 2.1772 2.7001 3.2734 3.9007 4.5863 5.3346 6.1502
2.9 2.0013 2.5413 3.1389 3.7984 4.5251 5.3240 6.2008
3.0 1.7914 2.3419 2.9573 3.6430 4.4050 5.2495 6.1832
1.5484 2.1015 2.7270 3.4312 4.2211 5.1041 6.0883
;*: 1.2746 1.8213 2.4478 3.1612 3.9697 4.8822 5.9081
3:3 0.9733 1.5038 2.1206 2.8324 3.6486 4.5794 5.6359
3.4 0.6496 1.1529 1.7487 2.4466 3.2576 4.1934 5.2669
3.5 +0.3098 0.7746 1.3369 2.0074 2.7987 3.7241 4.7985
-0.0381 iO.3767 0.8923 1.5210 2.2767 3.1746 4.2315
33:; -0.3848 -0.0314 +0.4244 0.9962 1.6994 2.5511 3.5700
-0.7198 -0.4385 -0.0553 +0.4445 1.0779 1.8636 2.8225
;:: -1.0317 -0.8319 -0.5332 -0.1199 +0.4263 1.1259 2.0016
-1.3084 -1.1977 -0.9940 -0.6804 -0.2378 +0.3558 1.1251
t:l" -1.5382 -1.5216 -1.4209 -1.2184 -0.8941 -0.4249 +0.2152
-1.7095 -1.7893 -1.7966 -1.7136 -1.5199 -1.1915 -0.7013
t *: -1.8124 -1.9871 -2.1039 -2.1453 -2.0907 -1.9160 -1.5936
4:4 -1.8391 -2.1032 -2.3268 -2.4930 -2.5817 -2.5692 -2.4280
-1.7844 -2.1283 -2.4513 -2.7376 -2.9685 -3.1213 -3.1692
2: -1.6469 -2.0567 -2.4668 -2.8632 -3.2291 -3.5437 -3.7818
-1.4292 -1.8870 -2.3670 -2.8579 -3.3452 -3.8110 -4.2326
f-i -1.1387 -1.6231 -2.1513 -2.7153 -3.3040 -3.9027 -4.4924
4:9 -0.7876 -1.2742 -1.8252 -2.4359 -3.0995 -3.8054 -4.5392
5.0 -0.3927
-0.8557 -1.4010 -2.0281 -2.7346 -3.5149 -4.3599
c-p (-;I1 C-f)1
[ 1 [ 1 [ 1 [ 1 [ 1 [ 1 [(-;I3 1
C-f)2 c-y c-y
720 PARABOLIC CYLINDER FUNCTIONS
The functions 8,. 82,83of 19.10 and 19.23 are neededin Darwin’s expansionand also
the function Tof 19.7 and 19.20.
$1 93 z 81 62 7
E 192 T 81 92 7
0.57390 0.00000 0.00000 9.7471 8.2546 5.3521
::1" 0.64640 0.01513 0.08015 10.0537 8.5530 5.4803
0.72261 0.04341 0.16185 10.3652 8.8564 5.6092
:*: 0.80265 0.08086 0.24502 10.6817 9.1649
1:4 0.88666 0.12617 0.32964 11.0031 9.4784 :*. 'BEi
0.97473 0.17866 0.41566 11.3295 9.7970 5.9996
::2 1.06696 0.23786 0.50304 11.6608 10.1207 6.1310
1.16344 0.30347 0.59175 11.9970 10.4494 6.2631
:*i 1.26422 0.37527 0.68175 12.3382 10.7832 6.3958
1:9 1.36937 0.45309 0.77300 12.6843 11.1220 6.5290
1.47894 0.53679 0.86549 13.0354 11.4659 6.6629
21" 1.59299 0.62626 0.95917 13.3914 il.8148 6.7974
1.71155 0.72142 13.7524 12.1688 6.9325
z 1.83466 0.82220 :- l":tt:: 14.1183 12.5278 7.0682
2:4 1.96236 0.92853 1:24716 14.4892 12.8919 7.2045
2.5 2.09467 1.04036 1.34539 14.8651 7.3414
2.23163 1.15764 1.44470 15.2459 :t 26%
::7" 2.37325 1.28034 1.54506 15.6316 14:0144 :* E9'
2.51956 1.40843 1.64646 16.0223 14.3987 717555
2: 2.67058 1.54187 1.74888 16.4180 14.7880 7.8947
3. 0 2.82632 1.68063 1.85229 16.8186 15.1823 8.0344
2.98681 1.82470 1.95669 17.2242 15.5817 a.1747
::: 1.97406 2.06206 17.6348 15.9861 a.3155
:- :::o": 2.12867 2.16837 18.0503 16.3956 8.4569
33:; 3149688 2.28853 2.27562 18.4708 16.8101 8.5989
3.5 3.67648 2.45363 18.8962 a.7413
3.86089 2.62394 :* i9":;; 19.3266 ::- ;:2 8.8844
333 4.05011 2.79946 2:60281 19.7620 la:0838 9.0279
4.24416 2.98017 2.71365 20.2024 18.5184 9.1720
::: 4.44305 3.16606 2.82536 20.6477 18.9581 9.3166
4. 0 4.64678 3.35712 2.93791 21.0980 19.4028 9.4617
4.85537 3.55335 3.05131 21.5532 19.8525 9.6074
::: 5.06880 3.75474 3.16554 22.0135 20.3073 9.7535
5.28711 3.96127 3.28058 22.4787 20.7671 9.9002
44:: 5.51028 4.17295 3.39643 22.9488 21.2319 10.0474
4. 5 5.73833 23.4240 21.7017 10.1951
5.97126 44.22 :* Z~ 23.9041 22.1766 10.3433
44:: 6.20908 4183875 3174872 24.3892 22.6565 10.4920
6.45178 5.07093 3.86770 24.8792 23.1414 10.6411
::9" 6.69938 5.30822 3.98743 25.3742 23.6314 10.7908
5. 0 6.95188 5.55062 4.10792 25.8742 24.1264 10.9410
[ 5W.3 [ '-;'"] (-y
. . . . _ [ . .. I
When interpolating for 92and 83for 6near unity, it is better to interpolate for + and then
use
ti2 = f r312 or a3 = i (-.)3/Z.
20. Mathieu Functions
GERTRUDE BLANCH 1
Contents
Page
Mathematical Praperties . . . . . . . . . . . . . . . . . . . . 722
20.1. Mathieu’s Equation . . . . . . . . . . . . . . . . . 722
20.2. Determination of Characteristic Values . . . . . . . . . 722
20.3. Floquet’s Theorem and Its Consequences . . . . . . . . 727
20.4. Other Solutions of Mathieu’s Equation . . . . . . . . . 730
20.5. Properties of Orthogonality and Normalization . . . . . . 732
20.6. Solutions of Mathieu’s Modified Equation for Integral v . . 732
20.7. Representations by Integrals and Some Integral Equations . 735
20.8. Other Properties . . . . . . . . . . . . . . . . . . . 738
20.9. Asymptotic Representations . . . . . . . . . . . . . . 740
20.10. Comparative Notations . . . . . . . . . . . . . . . . 744
References . . . . . . . . . . . . . . . . . . . . . . . . . . 745
b, d(O, a), se, (ljr *)p 84 ($ cz), (Jn>f go,4d, (4P)If,, I(a)
q=O(5)25, 8D or S
721
20. Mathieu Functions
Mathematical Properties
20.1.5
where B, can be taken as zero. If the above is
substituted into 20.1.1 one obtains
where c is a constant.
20.2.2
Again, from the fact that G=c and that u, v
are independent variables, one sets $i2 [(a--m2)A,-q(A,-2+A,+2)1 ~09 m2 ’
20.1.6
+m$-I [(a-m2)Bm-q(Bm-2+Bm+2)] sin mz=O
* a=d2f A+ (k2Lc) p2co& 2u
du2f 2 A+,,, B-,=0 m>o
722 *seepage
n.
MATHIEU FUNCTIONS ~723
Equation 20.2.2 can be reduced to one of four 20.2.18 K=Goc, for odd solutions of perio$ R,
simpler types, given in 20.2.3 nnd 20.2.4 below along with 20.2.15 I
These three-term recurrence relations amon the
20.2.3 yo=~oA2,+, ~0s @m+p)z, p=O or 1
coefficients indicate that every G, can be devel ped
into two types of continued fractions. 1 hus
20.2.4 yl=go Bz,+n sin (2m+p)z, p==O or 1 20.2.15 is equivalent to
20.2.19 )
If p=O, the solution is of period S; if p= 1, the
solution is of period 2x.
Recurrence Relations Among the Coefficients I
20.2.20
Even solutions of period ?r: I
Gm+2=Vm- UC,,,
20.2.5 aAo-qAz=O
1
= v m--vm-,-l- . * * vo;+‘pl (mtZ3)
20.2.6 (a--4)Ar-q(2A,+A,) =0 VffI-2-
where I
20.2.7 (u-m2)A,-q(A,~2+Am+2) =0 (mL3)
pl=d=O; (po=2, if G,,+2=A21/A21-2
Even solutions of period 27r:
q,=d=+q,=O, if G,,,+z=B2,/B2,-2
20.2.8 ’ (a-lb%-&%+&=Oj
a=-1; cpo=d=l, if G,,,+~=A2a+JA~r--l ~
along with 20.2.7 for m 2 3. cPr=d=&=l, if Gn+2=&a+J&a-1 I
Odd solutions of period ?r: The four choices of the parameters (pl, (p, d
20.2.9 correspond to the four types of solutions 20. .3-
(a-4)B,--qB,=O
20.2.4. Hereafter, it will be convenient to ep-
* 20.2.10 (a-m2)B,--(B,_,+B,+,)=O (m>3! arate the characteristic values a into two m i jor
subsets:
Odd solutions of period 2~
u=u,, associated with even periodic solutions
20.2.11 (a-l)&+q(B,-Bd=o,
a = b,, associated with odd periodic solution
along wit,11 20.2.10 for m > 3.
If 20.2.19 is suitably combined with 20.2.13-20. .18
Let there result four types of continued fractions, the
20.2.12 Ge,=A,IA,+ Go,=B,IB,-2; roots of which yield the required character’ 2 tic
values
G,,,=Ge, or Go,,, when the same operations apply 2 1 1
to both, and no ambiguity is likely to arise. 20.2.21 vcl _-__v,- v,- -ve- . . . =0 Roots:1 az,
Further let
20.2.22
20.2.13 V,= (a-m2>/q.
_--- 1 1 1
VI-1 v,- v,- Jr,- * * * =0 Roots: uC,+1
Equations 20.2.5-20.2.7 are equivalent to
16690684Olq8
-458647142400+ ' ' *
32 - / a3(-*)=g+92-93+13p4 5q6
16 64 20480+16384
b&d
28 - 1961 p6 609q'
-23592960+104857600+ ..'
24 -
b,(p)=16+~-~+27~l~o~o~o+ ...
37qe
+891813888+ ' ' *
5861633q'
b&)=36+2+ 187q4
70 43904000-92935987200000+. **
6743617q'
a&)=36+2+ 187q4
70 43904000+92935987200000+ ' *'
20.2.26
a, (59+7) q4
br ="+2(:-1)a+ 32(~~-1)~(r~-4)
-24L
(9r4+58++29) $
FIGURE 20.1. Characteristic Values a,, b, 1,=0,1(1)5 +64(+1)"(+4)(+9)+ ' ' '
MATHIEU FUNCTIONS p25
The above expansion is not limited to integral
se2(2,q)=sin%-- &g+n’(e!.&eg9+. ..
values of r, and it is a very good approximation
for r of the form n++ where n is a.n integer. In
case of integral values of r=n, the series holds 20.2.28
only up to terms not involving r2-n2 in the
denominator. Subsequent terms must be derived
specially (as shown by Mathieu). Mulholland
cdz,n>=cos
se,k
d ~1
4(r+l)
(rz--p(r/2)) --p
1
cos (r+2) 2-q E
[
IE
I .4
Ii
IC
.a
.E
.4
.2
-.2 -
C
-.4 -
-.2
:6-
:4
-6 -
55
-1.0 -
where
FIGURE 20.4. Even Periodic Mathieu Functions, Orders O-5 33 410 405
q= 10. d2=w++-g
MATHIEU FUNCTIONS t727
Solutions having the property 20.3.4 will hereafter Then it can be shown that
be termed Floquet solutions. Whenever F.(z) 20.3.10 cos w-y/1(7r)=O
and Fy(--z) are linearly independent, the general
solution of 20.3.1 can be put into the form 20.3.11 cos nv-1-2~; 6) yz (‘;-)=o ;
20.3.5 y=AFv(z)+BFv(-z)
Thus v may be obtained from
If AB#O, the above solution will not be a Floquet
solution. It will be seen later, from the method y1 (r) or from a knowledge of both
for determining v when a and q are given, that For numerical purposes 20.3.11 may be
there is some ambiguity in the definition of v; desirable because of the shorter range of inte
namely, v can be replaced by v+2k, where k is an tion, and hence the lesser accumulation of rou d-
arbitrary integer. This is as it should be, since off errors. Either V, -v, or f v+2k (k an a Fbi-
the addition of the factor exp (2ikz) in 20.3.2 still trary integer) can be taken as the solutioni of
leaves a periodic function of period r for the 20.3.11. Once v has been fixed, the coefficiebts
coefficient of exp ivz. of 20.3.8 can be determined, except for an arbitr ry
It turns out that when a belongs to the set of multiplier which is independent of z. it
characteristic values a, and b, of 20.2, then v is The characteristic exponent can also be
zero or an integer. It is convenient to associate puted from a continued fraction, in a
V=T with a,(q), and v=--T with b,(q); see [20.36]. analogous to developments in 20.2, if a sufficie
In the special case when v is an integer, F,(z) is close first approximation to v is available.
728 MATHIEU FUNCTIONS
1.8
I.6
-- 29 FIGIJRE~ 20.8
0
2.0
1.6
1.2
m-2q FIGURE 120.9
0
I.0
.6
I
--+
1.7 I.6 1.9 21) 2.1 2.2 2.3 2.4 2.5 2.6 2.7 28 2.9 4
3.1 3.2 3.3 3.4 3.5 Expansions for Small q ([20.36] chapter 2)
II I I I I I I I1
If v, p are fixed:
20.3.15
c5v2+v q4
z+
a=V2+2(~f-1) 32(~~-l)~(v~--4)
1.6
(9v4f5sv2+29)q6
+64(v2-1)5(v2-4)(v2-g)+ . . . (Vf1J2J3).
---da 20.3.18
20.4.12 20.5.4
2A”,+A;+ . . . =A;+A:+ ...
=B;L+E+ . . . =B2,+Ba,+ . . . =l.
It can be verified from 20.4.8 and 20.4.12 that 20.5.5
2r
AZ=- 1
2*
ce2,(z, q)dz; Ai=: ce,(z, q) cos nzdz
20.4.13 M$-&)=E:(0))
0
= s0
20.5.3 s,” [ce,(z, q)lzdz=Jozr [se&, q)]%z=?r Solutions of the first kind
20.6.1
For integral values of v the summation in
20.3.8 reduces to the simpler forms 20.2.3-20.2.4; Ce2r+r(z, a) =ce2t+p(iz, q>
on account of 20.5.3, the coefficients A,,, and B,
(for all orders r) have the property ==po A;;$; (q) cash (2k+p)z
associated with a,
MATHIEU FUNCTIONS 733
ce2,
20.6.3 Ce2k a) = &, z. (- 1)‘LA2,J&fi cash 4 = ““‘$j *) p0 A2JJ2,(2,iTj sinh z)
I
cezr+l
20.6.4 Ce2r+l(z, a> = fiAf,+~ go (--~>“+‘&+~z~+ICW? cosh4
ce2,+l@, cl)
= &-;,+‘- coth zgO (2k+1)A2k+IJ2t+l(2fi sinh z)
20.6.8
20.6.10
where
ul=fiee-g, ua=&e’, Bijl++pP,A::fj’#O, p=O, 1.
See 20.4.7 for definition of Zg’ (x).
Solutions 20.6.7-20.6.10 converge for all values of z, when q#O. If j=2, 3, 4 the logarithmic terms
entering into the Bessel functions Y,(u2) must be defined, to make the functions single-valued. This
can be accomplished as follows:
.DeCne (as in [20.58])
I
20.6.11 ln (fie”>=ln t&)+2
See [20.15] and [20.36], section 2.75 for derivation.
734 MATHIEU FUNCTIONS
Other Expressions for the Radial Functions (Valid Over More Limited Regions)
/y $, hld; (z,q)
3-
2-
I-
-3-
cl:2 25
FIGURE 20.12. Derivative of the Radial Mathieu Function
of the First Kind. FIGURE 20.14. Radial Mathieu Function of th,e Third Kind.
(From J. C. Wiltse and M. J. King, Derivatives, zeros, and other data per‘ (From J. C. Wiltse and M. J. King, Values of the Mathieu functions, The
taining to Mathieu functions, The Johns Hopkins Univ. Radiation Lab- Johns Hopkins Univ. Radiation Laboratory Tech. Rept. AF-53, 1958,
oratory Tech. Rept. AF-57, 1955, with permission) with permission)
MATHIEU FUNCTIONS 735
If j= 1, A@;:, and Ms!‘!,,, p=O, 1 are solutions 20.6.19
of the first kind, proportional to Celr+p and Sezl+P,
Mc:‘(-z, q>=-Mc:“(z, q)-2f,.,McI”(z, IJ)
respectively.
Thus 20.6.20
20.6.15 Ms:‘(-zz, q>=MsP’(z, q)-2f,,,M$)(z, a)
?r
ce2, ipz ce240, a> where
( >
Ce24z, a>= MczY’(z, n>
(-l)‘&’ 20.6.21
n- fe, ,= --Mcs2’(0, q)/McS”(O, q>
ce&+l Z’P ce2t+1@,a>
( >
Cb+l(z, n>= (-l)‘+l,/‘ijA;‘+l M&‘+,(z, a)
See [20.58].
4, (0, abei, 3 p In particular the above equations can be ushd to
( >
Sdz, n>= (-l)‘&’ Ms2 (2, n> extend solutions of 20.6.12-20.6.13 when g&<O.
For although the latter converge for 9&<0,
s&+l(Q, dse2,+l Ej p provided only jcosh zl>l, they do not reprkt
( > the same functions as 20.6.9-20.6.10.
Se2,+l(z, n) = (-l)‘&jBy+ M&)+1(2, a>
20.7. Representations by Integrals and Some
The Mathieu-Hankel functions are Integral Equations
20.6.16 Let
Mi3’(z, q)=Mp’(z, q)+iMp’(z, q) 20.7.1 G(u) =$ K(u, t)V(t)dt I
Mt”(z, q)=Mt”(z, a>--iMY’(z, q) ‘c
M$“=Mc;” orMs’?‘. be defined for u in a domain U and let the co tour
C belong to the region T of the complex t- 4lane,
From 20.6.7-20.6.11 and the known properties
with t=yo as the starting point of the co tour
of Bessel functions one obtains
and t=y, as its end-point. The kernel K & , t)
20.6.17 and the function V(t) satisfy 20.7.3 and1 the
M%,(z+im, a> hypotheses in 20.7.2.
=(-l)"p[ME+,(z, ~)+24W+,(z, n>l 20.7.2 K(u, t) and its first two partial derivqtives
with respect to u and t are continuous for t pn C
Mi?+,(z+in~, n> dV
=(-l)“p[W3!,,(~, a>- 2nM6::&, a>1 and u in U; V and - are continuous in t.
dt
M!?+,(z+in~, n> 20.7.3
= (--l)nPIW:,(~, c-d+‘JnM:?+&, n>l
where M=Mc
equations.
or Ms throughout any of the above C~V-d~K]l’=o;~+(a--Zgcos
70
2t)V+o.
Kernels KI(z, t) and Kz(a, t) where F”(t) is the Floquet solution 20.3.8. The
path C is chosen so that G(z, t, a) exists, and
20.7.6 Kl(z, t)=Z$‘)(u)[M(z, t)]-““, @z>O)
20.7.2, 20.7.3 are satisfied. Then it may be
where verified that K3(z, t, a), considered as a function
of z and t, satisfies 20.7.4; also, considered as a
20.7.7 u=1/2q(cosh 2z+cos 2t) function of a and t, K3 satisfies 20.7.5. Conse-
20.7.8 M(z, t)=cosh (z+it)/cosh (z-a) quently G(z, p, a)=Y(z, &/(a, a), where Y and y
satisfy 20.1.2 and 20.1.1, respectively.
To make M-i” single-valued, define Choice of Path C. Three paths will be defined:
20.7.9 20.7.16
cash (z+&r) =ek cash z
Path C3: from -d,+im to d2--ia, dI, dz real
cash (z-i?r)=e+cosh z
M(z, 0) = 1 -dl<arg [&{ cash (z+b) f 1 }I<*-do
[M(z, 7r)]+=e-iv*M(z, 0) -d2<arg [&{ cash (z-k) f 1 }]<?r-d2
Let
20.7.17
20.7.10 G(z, p)=;f: j-Or Kl(u,t)Fv(t)dt, (9 z>O)
Path C4: from d2--ico to 2~+i~ -d,
where F,(t) is defined in 20.3.8. It may be verified
(same dI, d2 as in 20.7.16)
that K,F, satisfies 20.7.3, K satisfies 20.7.2 and
20.7.4. Hence G is a solution of 20.1.2 (with z 20.7.18
replacing u). It can be shown that KI may be re-
placed by the more general function 4;
20.7.11 F&)Mi(z, ql=k e2’AwF,(t)dt j=3,4
SFci
Kz(z, t)=Z’?+zs(u)[M(z, t)]-+Af’, s any integer.
where M?(z, q) is also given by 20.4.12.
See 20.4.7 for definition of Z(!)+,,(u).
20.7.19 Path C.i: from -d,+ia to 2r-dl+im
From the known expansions for El,)+.,,(u) when
8,s is large and positive it may be verified that 4”;
M!“(z, q)=
See [20.36], section 2.68.
If Y is an integer the paths can be simplified;
for in that case F,(t) is periodic and the integrals
exist when the path is taken from 0 to 2~. Still
further simplifications are possible, if z is also real.
where MY (z, q) is given by 20.4.12, s=O, 1, . . ., The following are among the more important
cza#O, and F”(t) is the Floquet solution, 20.3.8. integral representations for the periodic functions
Kernel K&, t, a) ce,(z, n), se,(z, n) and for the associated radial
solutions.
20.7.13 K&z, t, a)=e2+
where Let r=2s+p, p=O or 1
2 R -2
20.7.24 ~~=;ce~~ 2’ p I&(q); p=Op,=;- cei,+l ;, p /,iijAF+l (q) if p=l, for functions ce,(z, *)
( > ( >
CT,=: cezs(O,q)/&(q) if p=O; c,,=4 ce2s+l(0, q)/A:“+‘(q), if p-l; associated with functions ce,(jz, q)
R
Mf$)(z,q)=* J’ &$j)(u)ce,,(t,
*)&;M&)+l(z,*)=(-1)‘8d
,A;‘+’
‘Osh’ J” zP’(u)‘OSt 032.+~(t,~&& U
0 0
Msil,)+l(z, q)=(-1)r84j
=B27+1
1
sinh z
S 0
a W(U) sin t w,+l(t,
U
ddt
In the above the j-convention of 20.4.7 applies and the functions MC, Ms are defined in 20/.5.1-
20.5.4. (These solutions are normalized so that they approach the corresponding Bessel-Hankel
functions as 9?2+ 0 .)
Other Integrals for Mc!“(a, q) and Msj”(s, q)
2
20.7.30 ~~~f)+,(Z, q)=; sin (2rq sinh z sin t)se2,+l(t, q)dt
4 J-(-l)’ Z
20.7.31 Jw+,(z, a)=-
= s&+l(O, a> ST 0
sinh z sin t cos (2Jq cash z cos t)se2,+l(t, q)dt
a t) t
20.7.32 M&‘(z, q)=z 4j ;e;;;‘d,, S 0
sin (2Jq cash z cos [sinh z sin ae2, (t, n>ldt
20.7.33
S’ ’ sm (2& sinh z sin t)[cosh z cos t ses,(t, q)ldt
738 MATHIEU FUNCTIONS
(-lp(ip+‘JTi
20.7.37 se:& q)M+)(z, q)= 2”e2idw g se,(t, q)dt
a s0
r=2s+p
20.7.38 MC?’ (2, a>=-Yr m sin 2 &j cash z cash t+p;) McS”(t, q)dt
s0 (
yT =2ce28
MsY(z,
a>
=Yr m t
20.7.39
S 0
sinh z sinh 24 cash z cash t-p Ms:“(t, q)dt
a0
-yT= ;I p /@rlP, if p=O x= -4se2,+1
( > ( >
y,= -2ce2,(+r,
s(
q)/rAi8, if p=O
cos 24 cash z cash t-p ;) Mcl”(t,
yr=2ce;s+1(4u, q)/qiTjA?+‘,
q)dt
if p=l
20.7.41 Ms?' (2, n>=-rr m sin 24 cash z cash t+p t) sinh z sinh t Ms’:’ (t, q)dt
s0 (
Y,=-4sei,($u, q)& ?rBa2*,if p=6 ~,=4se2,+1(&r, ~)/TB~~‘+~, if p=l
20.8.8 For M:“(z, -q), j=2, 4, one may use the dbfini-
lez,(z, n,=& (-l)“+‘Azt[l~-~(ul)Z+~(uz)
tions
h2,+l(z, n)=po (-- l)“+*B2,+,[1,-,(2(,)I,+,+,(~,) M=Mc or Ms;for real z, p, M?’ (z, -q)l
+I~~+l(a)l~-,(zL2)1/~2~+~ are in general complex if j=2,4.
Zeros of the Functions Xoor Real Values of 4.~
102,+1(~, q,=$ (-1)“+“A2~+~[~~--s(u~)~~+s+~(u2)
-In+r+~(u~)~~-,(~z)I/A2a+,
See [20.36], section 2.8 for further results.
20.8.9 Zeros of ce,(s, q) and se,(z, q), Mc!"(5, q), Ms!')@, q).
20.8.12
+2p-a] D,,,+( m-i) [16q(l-~~)-%fi urnID,-1
+4q(2m-3) (2m-~)~(l--a2)D,,+~=0
20.9.3
c&+1 ;I P =(-l)‘+lgc,z,+l(q)A:‘+’ (!-I> &I
( > Mcl”‘(% a>
(-l)‘Ms’:‘(z, q)
n-
se&
(
2’ q =(-1)‘g0,2r(q)B~‘(p).q
> -\i
; e--t[2+m z-++] o dm
- ?r+q”4(coshz-u) 4 go [4i ,@(cosh z-u)]”
n-
se2,+1 5’ P =(-1>‘g~.2r+l(dB1 2r+1(!d &7
( > d-I=d-z=O; d,=l, and
MATHIEU FUNCTIONS 741
20.9.4 20.9.9
(m+l)dm+l+[(m+;)l+(m+;) &&a
Fo(Z)-l+-w-
! S&j Gosh2z
+2*-+,+(m-;) [16p(l-u*)+8i~am]d,-,
__ 1 w4+c%6w~~lo5
+4q(2m-3)(2m-l)(l--a2)d,-2=0. +2048p [
- (w5+14w3+33w) i
In the above + 16384~~‘~
l cosh2 z
I
-2?r<arg fi cash z<?r
lcosh z-uj>lufl(, Sz>O,
- (zw5+124w3+1122w)+3w5+290w3+1627p
cosh4 z coshe z
,
1 ..
+
20.9.14 20.926
41
(2~+3)J7~+4W-. ..
64& 1024q
r’+2P-121rZ-122r-84 -?4
See 20.9.23-20.9.24 for expressions relating to
ce,(O, q) and se:(O, q). When ]cos z]>dw/p’,
20.9.11-20.9.12 are useful. The approximations
+ 2048q + ii&@+-
j ** 1
j2=216+5r4-416?-629T1-1162r-476
become poorer as r increases.
Expansions in Terms of Parabolic Cylinder Functions It should be noted that 20.9.15 is also valid as an
approximation for se,+l (2, q), but 20.9.16 may give
(Good for angles close to $s, for large values of *
slightly better results. See [20.4.]
p, especially when ]cos s]<2t/q’.) Due to Sips
[20.44-20.461. Explicit Expansions for Orders 0, 1, to Terms in q-*I*
(9 Lw34
20.9.15 4x, n>-CWX4+%(41 20.9.21 For r=O:
20.9.16
Z,,- D,,-
se,+~(x,n)-fCK&)--Z~(41 sin 5, a=2pf co9 5.
1
Let Dk=Dk(a)=(-l)“e’a”s e-+u2. - (-$+gp&)+ ...
+64p
20.9.17
++y@*]+ .. .
20.9.18 1
- (-g+;-2&)+. ..
1
Z,(a)-@
C
-4 ’ Dr+2
-9 Drm2
1 +64p
MC ,(I) (2 9 A)
A4d2)(Z,
I g) id2; , (z, A)
Ad2+z,
r 9) Ma;2’(z, A)
dMs
‘I’
;i;
-MC
(2,
q) ,I(n (0 g)/ikfc ,I(I’ (0 g) --Mc~” (0, A)/bfe .(” (0, A) &eNotea.
[gMI’:’
(2’9)
1r. I
NOTE: 1. The conversion factors A’ and Br are tabulated in I20.581 along with the coefficients.
Seme es this volume
[20&l] F. W. Sch&fke, Uber die Stabilitiitskarte der. [20.56] E. L. Ince, Tables of the elliptic cylinder functions,
Mathieuschen Differentialgleichung, Math. Proc. Roy. Sot. Edinburgh 52, 355-423; Zeros
Nachr. 4, 175-183 (1950). and turning points, 52,424-433 (1932). Charac-
[29.41] F. W. Schiifke, Das Additions theorem der teristic values aa, al, . . ., as, bl, bl, . . ., be,
Mathieuschen Fusktionen, Math. Z. 58,436-447 and coefficients for 0=0(1)10(2)20(4)40; 7D.
(1953). Aiso ce,(z, 0), se&r, e), @=O(l)lO, z=O”(lo)900;
[20.42] F. W. Schiifke, Eine Methode zur Berechnung des 5D, corresponding to characteristic values in
charakteristischen Exponenten einer Hillschen the tables. a,=be,-2q; b,=bo,--29; e=q.
Differentialgleichung, Z. Angew. Math. Mech. [20.57] E. T. Kirkpatrick, Tables of values of the modified
33, 279-280 (1953). Mathieu function, Math. Comp. 14, 70 (1960).
(20.431 B. Sieger, Die Beugung einer ebenen elektrischen r=0(1)5, r=1(1)6;
C4u, d, fh(u, d,
Welle .an einem Schirm von elliptischem u=.l(.l)l; q=1(1)20.
Querschnitt, Ann. Physik. 4, 27, 626-664 (1908). [20.58] National Bureau of Standards, Tables relating to
120.441 R. Sips, Representation asymptotique des fonctions
Mathieu functions (Columbia Univ. Press, New
de Mathieu et des fonctions d’onde spheroidales, York, N.Y., 1951). Characteristic values be,(s),
Trans. Amer. Math. Sot. 66, 93-134 (1949). ho,(s) for 0 5s <lOO, along with @*, interpolable
[20.45] R. Sips, Repr&entation asymptotique des fonctions
to SD; coefficients Der(s), Do&) and conversion
de Mathieu et des fonctions spheroidales II, factors for cc,(q), se,(q), same range, without
Trans. Amer. Math. Sot. 90, 2, 340-368 (1959). differences but interpolable to 9D with Lagran-
[20.46] R. Sips, Recherches sur les fonctions de Mathieu,
gian formulas of order 7. “Joining factors”
Bull. Soo. Roy. Sci. Liege 22, 341355, 374-387, &g,,,, sfrgB.,,, a%. ,, srjo,r along with 62*; inter-
444-455, 530-540 (1953); 23, 37-47, 90-103
polable to 8s.
(1954).
[20.47] M. J. 0. Strutt, Die Hiiische Differentialgleichung [20.59] J. A. Stratton, P. M. Morse, L. J. Chu and R. A.
Hutner, Elliptic cylinder and spheroidal wave
im komplexen Gebiet, Nieuw. Arch. Wisk. 18
31-55 (1935). functions (John Wiley & Sons, Inc., New York,
N.Y., 1941). Theory and tables for bo, bl, bz,
[20.48] M. J. 0. Strutt, Lambche, Mathieusche und
bs, br, b;, b;, b;, b;, and coefficients for Ser(s, x)
verwandte Funktionen in Physik und Technik, and So&, z) for c=O(.2)4.4 and .5(1)4.5; mostly
Ergeb. Math. Grensgeb. 1, 199323 (1932). 55; c=2qf, b,=a,+2q, b:=b,+2q.
[20.49] M. J. 0. Strutt, On Hiil’s problems with complex
120.601 T. Tamir, Characteristic exponents of Mathieu *
parameters and a real periodic function, Proc.
Roy. Sot. Edinburgh Sect. A 62,278-296 (1948). equations, Math. Comp. 16, 77 (1962). The
Floquet exponent Y, of the first three stable
L20.501 E. T. Whittaker, On functions associated with regions; namely r=O, 1, 2; q=.1(.1)2.5; a=r
elliptic cylinders in harmonic analysis, Proc. (.l)r+l, 5D.
Intl, Congr. Math. Cambr. 1, 366 (1912).
[20.61] J. C. Wiltse and M. J. Ring, Values of the Mathieu
[20.51] E. T. Whittaker, On the general solution of
functions, The Johns Hopkins Univ. Radiation
Mathieu’s equation, Proc. Edinburgh Math. Sot.
Laboratory Technical Report AF-53, Baltimore,
32, 75-80 (1914).
Md. (1958). (Notation of [20.58] used:
[20.52] E. T. Whittaker and G. N. Watson, A course of
IX&J, q)/A, se&, q)/B for 12 values of q between
modern analysis, 4th ed. (Cambridge Univ. Press, .25 and 10 and - from 8 to 14 values of v; dir/2
Cambridge, England, 1952). Mc$(u, q), 442 M&u, q), j=l, 2 for 6 to 8
Tables values of a between .25 and 10 and about 20
values of i, r=O, 1, 2; 48 McP’(- 1~1, q),
L20.531 G. Blanch and I. Rhodes, Table of characteristic d?r/2 Me?(- ]u], q), r=O, 1, 2 for about 9 values
values of Mathieu’s equation for large values of of u and q, 2 to 4 D in all.
the parameter, J. Washington Acad. Sci. 45, 6,
166-196 (1955). Be,(t)=a,(q)+2q-2(2r+l)fi [20.62] J. C. Wiltse and M. J. Ring, Derivatives, seros,
Bo,(t)=&(q)+2q-2(2r-l)fit=1/2Jq, r=O(l) and other data pertaining to Mathieu functions,
15, O<t<.l, with 62, a’*; SD (about); inter- The Johns Hopkins Univ. Radiation Laboratory
polable. Technical Report AF-57, Baltimore, Md.
[20.54] J. G. Brainerd, H. J. Gray, and R. Merwin, Solu (1958).
tion of the Mathieu equation, Am. Inst. Elec. [20.63] S. J. Zaroodny, An elementary review of the
Engrs. 67 (1948). Characteristic exponent over Mathieu-Hill equation of real variable based on
a wide range. p, M for e=l(l)lO; k=.l(.l)l, numerical solutions, Ballistic Research Labora-
5D; g(l), h(t) for t=0(.1)3.1, x, 5D; e=l(l)lO, tory Memorandum Report 878, Aberdeen
k=.l(.l)l, where g(t), h(l) are solutions of
y”+a(l+k cos t)y=O, with g(O)=h’(O)=l, Proving Ground, Md. (1955). Chart of the
g’(0) =h(O) =o, CO8 2?rp=2g(?r)h’(*) - 1, M= characteristic exponent.
[-s(~)g’(~)/h(~)h’(~)ll’*. See also [20.18]. It contains, among other
[20.55] J. G. Brainerd and C. N. Weygandt, Solutions of tabulations, values of a,, b, and coefficients for
Mathieu’s equation, Phil. Mag. 30, 458-477 ce,(z, q), se,@, q), q=40(20)100(50)200; 5D,
(1940). rs2.
EVEN SOLUTIONS
7 9 “F ce,(O,
q) ce,W, 4) (4d%, Ad
0 0.00000 000 -1)7.07106 781 (-1) 7.07106 78 (-1)7.97884 56
i - 5.80004 602 -2)4.48001 817 1.33484 87 1.97009 00
10 - 13.93697 996 (-3)7.62651 757 1.46866 05 2.40237 95
15 - 22.51303 776 (-3)1.93250 832 1.55010 82 2.68433 53
20 - 31.31339 007 -4)6.03743 829 1.60989 09 2.90011 25
25 - 40.25677 955 I -4)2.15863 018 1.65751 03 3.07743 91
2 0 4.00000 000 1.00000 000 -1.00000 00
7.44910 974 (-1 7.35294 308 (-1 -7.24488 15
1: 7.71736 985 (-1 2.45888 349 C-1 -,';', ;;
5.07798 320 (-2 I 7.87928 278
:i t 1.15428 288 (-2)2.86489 431 -1.07529 32 4.29953 32
25 - 3.52216 473 (-2)1.15128 663 -1.11627 90 1.11858 69
10 0 100.00000 000 1.00000 000 23 I 2.30433 72
5 100.12636 922 1.02599 503 23 2.31909 77
10 100.50677 002 1.05381 599 23 2.36418 54
15 101.14520 345 1.08410 631 23 2.44213 04
102.04891 602 1.11778 863
z 103.23020 480 1.15623 992 12)1.40118 52
ODD SOLUTIONS
4(0, u) se,(%,59
+ 1.00000 000 1.00000 00 1.00000 00 1.59576 91 2.54647 91
- 5.79008 060 (-1 1.74675 40 1.33743 39 2.27041 76
- 13.93655 248 -2 4.40225 66 1.46875 57 2.63262 99 I-- 3)2.21737
2)3*74062 82
88
- 22.51300 350 I -2 i 1.39251 35 1.55011 51 2.88561 87 (- 4)2.15798 83
- 31.31338 617 (-3)5.07788 49 1.60989 16 3.08411 21 (- 4)2.82474 71
- 40.25671 898 (-3)2.04435 94 1.65751 04 3.24945 50 (- 6)4.53098 74
25.00000 000 5.00000 00 1.00000 00 3)9.80440 55 (
5 Li 25.51081 605 4.33957 00 -1) 9.06077 93 4 1.14793 21 (
10 26.76642 636 3.40722 68 -1) 8.46038 43 4j 1.52179 77 ( 8)5.46799 57
27.96788 060 2.41166 65 -1) 8.37949 34 4)2.20680 20 I 8)4.26215
8)5.27524 66
17
:05 28.46822 133 1.56889 69 -1) 8.63543 12
25 28.06276 590 (-1)9.64071 62 -1) 8.99268 33 I 4 3.27551
4 4.76476
12
62 ( 8)2.94147 89
15 0 225.00000 000 1 1.50000 00 -1.00000 00
5
10
225.05581 248
225.22335 698
I 1 I 1.48287
1 1.46498
89
60
(-1 -9.88960
(-1 -9.78142
70
35
I(1919)3.73437
19)3.78055
3.83604
81
49
43
225.50295 624 1 1.44630 01 -1 -9.67513 70 (19 3.90140 52
:i 225.89515 341 1 I 1.42679
1 46 I -1 1 -9.57045 25 19 i 3.97732 29
25 226.40072 004 ( 1)1.40643 73 (-l)-9.46708 70 I 19)4.06462 83 ( 40)2.19249 18
b,+Zq- (4r-2)@
cl-*\? 1 2 5 10 15 <q>
0.16 -0.25532 994 -1.30027 164 -11.53046 855 -51.32546 875 - 55.93485 112 39
0.12 -0.25393 098 -1.28658 971 -11.12574 983 -56.10964 961 -108.31442 060 69
0.08 -0.25257 851 -1.27371 191 -10.78895 146 -51.15347 975 -132.59692 424 156
0.04 -0.25126 918 -1.26154 161 -10.50135 748 -47.72149 533 -114.76358 461 625
0.00 -0.25000 000 -1.25000 000 -10.25000 000 -45.25000 000 -105.25000 000 m
For go,Tand f,, r see20.8.12.
< q> = nearestinteger to 4.
MATHIEU FUNCTIONS
4
r/=5
7rA7 0 2 10 ?ll’,T 1 5 15
0 +0.54061 2446 +0.43873 7166 +O.OOOOO 1679 1 co.76246 3686 +0.07768 5798 0.00000 0000
2 -0.62711 5414 +0.65364 0260 +0.00003 3619 3 -0.63159 6319 +0.30375 1030 +o.ooooo 0002
4 +0.14792 7090 -0.42657 8935 +0.00064 2987 5 +0.13968 4806 +0.92772 8396 +O.OOOOO 0106
6 -0.01784 8061 +0.07588 5673 +0.01078 4807 9' -0 01491 5596 -0 20170 6148 +O.OOOOO 4227
8 +0.00128 2863 -0.00674 1769 +0.13767 5121 +0'00094 4842 +0'01827 4579 +0.00014 8749
10 -0.00006 0723 +0.00036 4942 +0.98395 5640 11 -0:00003 9702 -0:OOOSS 9038 +0.00428 1393
12 +o.ooooo 2028 -0.00001 3376 -0.11280 6780 13 +O.OOOOO 1189 +0.00003 3457 +0.08895 2014
14 -0.00000 0050 +O.OOOOO 0355 +0.00589 2962 15 -0.00000 0027 -LX00000 0839 +0.99297 4092
16 +O.OOOOO 0001 -0.00000 0007 -0.00018 9166 17 +O.OOOOO 0001 +O.OOOOO 0016 -0.07786 7946
+0.00286 6409
:: -0.00000
+O.OOOOO 0071
4226 :;) -0.00006 6394
22 +O.OOOOO 0001 23 +O.OOOOO 1092
25 -0.00000 0014
q=25
77’ 1 0 2 10 d,r 1 5 15
0 +0.42974 1038 +0.33086 5777 +0.00502 6361 1 to.39125 2265 +0.65659 0398 +O.OOOOO 4658
2 -0.69199 9610 -0;04661 4551 co.02075 4891 3 -0.74048 2467 +0.36900 8820 +0.00003 7337
4 +0.36554 4890 -0.64770 5862 +0.07232 7761 5 +0.50665 3803 -0.19827 8625 +0.00032 0026
6 -0.13057 5523 +0.55239 9372 +0.23161 1726 7 -0.19814 2336 -0.48837 4067 +0.00254 0806
8 +0.03274 5863 -0.22557 4897 +0.55052 4391 9 +0.05064 0536 +0.37311 2810 +0.01770 9603
10 -0.00598 3606 +0.05685 2843 +0.63227 5658 11 -0.00910 8920 -0.12278 1866 +0.10045 8755
12 +0.00082 3792 -0.00984 6277 -0.46882 9197 13 +0.00121 2864 +0.02445 3933 +0.40582 7402
14 -0.00008 7961 +0.00124 8919 +0.13228 7155 15 -0.00012 4121 -0.00335 1335 +0.83133 2650
16 +O.OOOOO 7466 -0.00012 1205 -0.02206 0893 17 +0.00001 0053 +0.00033 9214 -0.35924 8831
18 -0.00000 0514 +O;OOOOO 9296 +0.00252 2374 19 -0.00000 0660 -0.00002 6552 +0.06821 6074
20 +O.OOOOO 0029 -0.00000 0578 -0.00021 3672 21 +O.OOOOO 0036 +o.ooooo 1661 -0.00802 4550
22 -0.00000 0001 +o.ooooo 0030 +0.00001 4078 23 -0.00000 0002 -0.00000 0085 +0.00066 6432
24 -0.00000 0001 -0.00000 0746 25 +o;ooooo 0004 -0.00004 1930
26 +O.OOOOO 0032 +O.OOOOO 2090
28 -0.00000 0001 -0.00000 0085
31 +o.ooooo 0003
9=5
2 10 TV\?. 1 5 15
+0.93342 94'42 +0.00003 3444 +0.05038 2462 0.00000 0000
-0.35480 3915 +0.00064 2976 : +0.94001
-0.33654 9024
1963 +0.29736 5513 +o.ooooo 0002
+0.05296 3730 +0.01078 4807 5 +0.05547 7529 +0.93156 6997 +O.OOOOO 0106
-0.00429 5885 to.13767 5120 7 -0.00508 9553 -0.20219 3638 +O;OOOOO 4227
+0.00021 9797 +0.98395 5640 9 +0.00029 3879 +0.01830 5721 +0.00014 8749
-0.00000 7752 -0.11280 6780 :: +O.OOOOO
-0.00001 1602
0332 -0.00096 0277 tO.00428 1392
+b.ooooo 0200 to.00589 2962 +0.00003 3493 +0.08895 2014
-0.00000 0004 -0.00018 9166 15 -0.00000 0007 -0.00000 0842 +0.99297 4092
+O.OOOOO 4227 17 +o.ooooo 0017 -0.07786 7946
-0.00000 0070 +0.00286 6409
+o.ooooo 0001 :: -0.00006 6394
+o.ooooo 1093
:: -0.00000 0013
q=25
?n~s 2 10 m/r 1 5 15
2 +0.65743 9912 +0.01800 3596 +0.30117 4196 +o.ooooo 3717
-0.66571 9990 +0.07145 6762 : +0.81398
-0.52931 3846
0219 +0.62719 8468 +0.00003 7227
2 +0.33621 0033 to.23131 0990 5 +0.22890 0813 +0.17707 1306 +0.00032 0013
8 -0.10507 3258 +0.55054 4783 ; +0.01453
-0.06818 2972
0886 -0.60550 5349 +0.00254 0804
10 to.02236 2380 +0.63250 8750 +0.33003 2984 +0.01770 9603
12 -0.00344 2304 -0.46893 3949 11 -0.00229 5765 -0.09333 5984 +0.10045 8755
+0.00040 0182 +0;13230 9765 13 +0.00027 7422 +0.01694 2545 +0.40582 7403
:: -0.00003 6315 -0.02206 3990 15 -0.00002 6336 -0.00217 7430 +0.83133 2650
18 +O.OOOOO 2640 +0.00252 2676 +0.00021 0135 -0.35924 8830
20 -0.00000 0157 -0;00021 3694 :< +O.OOOOO
-0.00000 0126
2009 -0.0Ou01 5851 +0.06821 6074
22 +O.OOOOO 0008 +0.00001 4079 ;: +o.ooooo 0007 +O.OOOOO 0962 -0.00802 4551
24 -0.00000 0746 -0.00000 0048 +0.00066 6432
26 +o.ooooo 0033 +o.ooooo 0002 -0.00004 1930
:: +O.OOOOO 2090
For -1, and II,,, see 20.2.3-20.2.11 -0.00000 0086
+o.ooooo 0003
Compiled from National Bureau of Standards, Tables relating to Mathieu functions, Columbia Univ.
Press, New York, N.Y., 1951 (with permission).
21. Spheroidal Wave Functions
ARNOLD N. LOWAN 1
Contents
Page
Mathematical Properties. ................... 752
21.1. Definition of Elliptical Coordinates .......... 752
21.2. Definition of Prolate Spheroidal Coordinates ...... 752
21.3. Definition of Oblate Spheroidal Coordinates. ...... 752
21.4. Laplacian in Spheroidal Coordinates .......... 752
21.5. Wave Equation iu Prolate and Oblate Spheroidal Coordinates 752
21.6. Differential Equations for Radial and Angular Spheroidal
Wave Functions .................. 753
21.7. Prolate Angular Functions .............. 753
21.8. Oblate Angular Functions .............. 756
21.9. Radial Spheroidal Wave Functions .......... 756
21.10. Joining Factors for Prolate Spheroidal Wave Functions . 757
21.11. Notation ...................... 758
r1 and rz are the distances to the foci of a family where 4,~ and 4 are oblate spheroidal coordinates.
of confocal ellipses and hyperbolas ; 2f is the dis-
Relations Between Cartesian qnd Oblate Spheroidal
tance between foci. Coordinates
es- f 21.3.2
21.1.2 a=ft, b=jm, a
a=sen-&major axis; b=semi-minor axis; e=ec-
21.4. Laplacian in Spheroidal Coordinates
centricity.
Equation of Family of Confocal Ellipses 21.4.1
21.1.3 $+&=f
21.1.4 e-L=fa
q2 l--$
(---1<?<l)
Relations Between Cartesian and Elliptical Coordinates
752
SPHEROIDAL WAVE FlJNC.?l’IONS 753
Wave Equation in Oblate Spheroidal Coordinates (21.6.3 may be obtained from 21.6.1 by the
21.5.2 transformations [*fit, c++ic; 21.6.4 may be
obtainedfrom21.6.2 by the transformation c-+~ ic.)
21.7. Prolate Angular Functions
21.7.1
21.5.2 may be obtained from 21.5.1 by the =Prolate angular function of the first kind
transformations 21.7.2
p++i[, c++c.
21.6. Differential Equations for Radial and
Angular Prolate Spheroidal Wave Functions
=Prolate angular function of the second kind
If in 21.5.1 we put
(E?(v) and R‘(d are associated Legendre
functions of the first and second hinds respectively.
However, for -1Izj1, ~(z)=(l-z2)“‘2d”P,(z)/
then the “radial solution” R,,(c, 5) and the
dz” (see 8.6.6). The summation is extended over
“angular solution” &,,(c, 7) satisfy the differential
even values or odd values of T.)
equations
21.6.1 Recurrence Relations Between the Coefficients
21.7.3
(2m+k+2)(2m+k+l)c2
21.6.2 ak=(2m+2k+3)(2mf2kf5)
he,=(m+k) (m+k+l)
2(m+k)(m+k+l)-2m2-1 c2
’ (2m+2k-1)(2m+2k+3)
$ [ Q”+U$ Gnn(c,
E)] k(k-1)(2m+k)(2m+k-l)c’
~‘(2m+2k-l)2(2m+2k+1)(2m+2k-3)
(k2 2)
Y?= (m+k) (m+k+l)
+4cfl-(zm+2k~Y~$+2k+31 1(k>
0)
(The choice of r in 21.7.4 is arbitrary.)
754 SPHEROIDAL WAVE FUNCTIONS
&=n(n+l)
1J2m--1Pm+l)
(27+-l) (%+3) 1
I,=-(n--m+l)(n--m+2)(~+m+l)(n+m+2)+(7a--m-l)(n--m)~~+m-l)(n+m)
2(2n+l) (2n+3)3(2n+5) 2(2n-3)(2n-l)3(2n+l)
ze= (4d- 1)
(n-m+l)(n-m+2)(n+m+l)(n+m+2)~
(Zn-1) (2n+l) (2n+3)‘(2n+5) (2n+7)
(VP-m-l)(n-m)(n+m-l)(n+m)
(2n--5)(2n-3)(2n-l)6(2n+l)(2n+3) 1
ls=2(4mp-1)2A+~ B+; C+; D
(?--m-l)(n-m)(n+m-l>(n+m) (~-m+1)(7a--m+2)(n+m+l)(~+m+2)
A=(2n-5)2(2~-3)(2~-l)7(2n+l)(2n+3)2- (2n-l)2(2n+1)(2n+3)7(2n+5)(2n+7)2
1
-z & (63$+4940q4+43327tf+22470)
[
T(T-1)(2m+T)(2m+r-1)c’
+29s951q)+g (355n”+15o5p)-~]+o(c-7 fl~=(2m+2r-l)y2m~+2r+l)(2m.t27--3)
p=2(n-m)+1 (f 22)
SPHEROIDAL WAVE FUNCTIONS 755
Evaluation of Coeflicienta 21.7.15
Step 1. Calculate NT’s from
21.7.8
Ny+,=+bn.-g CT221
I (n-m) even
N?=rT-Ln; Ny=ry’-L,, 21.7.16
= C-1)
?(n+m+l)!
2”-” (y-1) ! (“+;+1> ! (n-m) Odd
Step 2. Calculate ratios $ and & from
2r P+l
21.7.17
and the formula for NT in 21.7.7.
The coefficients dy” are determined to within &f&, 9) = 0 -112>‘~mn(c, 7) (c-- >
the arbitrary factor do for r even and d, for T odd.
The choice of these factors depends on the normal-
um&keJ5m Wl,&) 1-n-m
ization scheme adopted.
Normalization of Angular Functions where the D,(z)‘s are the parabolic cylinder func-
Meixner- SchZke Scheme tions (see chapter 19).
21.7.11
S-1l [Sm&, d12&=&
Stratton-Morse-Chu-Little-Corbat6
$$$
Scheme
C, (T+2m)! d =(n+m)! and the H,(z) are the Hermite polynomials (see
21.7.12 chapter 22). (For tables of &,/hi see [21.4].)
r=o, 1 T! ’ (n-m)!
21.7.20 ~n=-2-gp[33p4+114p2+37-2m2(23q2+25)
(n-m) odd + 13m*]
,?y= -2-10[63q6+340q4+239q2+ 14
1
___~ m (2m+2’)!(--T)k(m+T+$@ -10m2(10p4+23q2+3)+m4(39q2-18)-2me]
C’m;‘2mk!(m+k)! 2 (a?)!
(n-m) even ~‘“=v(v+m)a,‘+(v+l)(v+m+l)a:’
Expansion 21.7.2 ultimately leads to where the L$““’(2) are Laguerre polynomials (see
chapter 22) and
21.7.21
f&g a,f l( m, n>cmk
21.8.1 x,.=~(-1)$kc’”
v=f (n-m) for (n-m) even; 21.9.2 R7%(C, s> =E~(c, 8 +W~(c, s>
21.10.1
Kg;(c) =
S~~(c,I)=K~~(C)R~~(C,5‘)
(n-m) odd
References
[21.1] M. Abramowitz, Asymptotic expansion of spheroi- [21.7] P. M. Morse and H. Feshbach, Methods of
dal wave functions, J. Math. Phys. 28, 195-199 theoretical physics (McGraw-Hill Book Co.,
(1949). Inc., New York, N.Y., 1953).
[21.2] G. Blanch, On the computation of Mathieu func- [21.8] L. Page, The electrical oscillations of a prolate
tions, J. Math. Phys. 25, l-20 (1946). spheroid, Phys. Rev. 65, 98-117 (1944).
[21.3] C. J. Bouwkamp, Theoretische en numerieke [21.9] J. A. Stratton, P. M. Morse, L. J. Chu and R. A.
behandeling van de buiging door en ronde Hutner, Elliptic cylinder and spheroidal wave
opening, Diss. Groningen, Groningen-Batavia, functions (John Wiley & Sons, Inc., New York,
(1941). N.Y., 1941).
[21.4] C. Flammer, Spheroidal wave functions (Stanford [21.10] J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C.
Univ. Press, Stanford, Calif., 1957). Little, and F. J. Corbat6, Spheroidal wave
[21.5] A. Leitner and R. D. Spence, The oblate spheroidal functions (John Wiley & Sons, Inc., New York,
wave functions, J. Franklin Inst. 249, 299-321 N.Y., 1956).
(1950).
[21.6] J, Meixner and F. W. Schafke, Mathieusche
Funktionen und Sphiiroidfunktionen (Springer-
Verlag, Berlin, Gottingen, Heidelberg, Germany,
1954).
760 SPHEROIDAL WAVE FUNCTIONS
Table 21.1 EIGENVALUES-PROLATE AND OBLATE
PROLATE
k(c)--m(mf1) *
G/n 0 1 2 3 4
0.000000 2.000000 6.000000 12.000000 20.000000
f 2.593084 6.533471 12.514462 20.508274
x'K40 3.172127 7.084258 13.035830 21.020137
32 0:879933 3.736869 7.649317 13.564354 21.535636
4 1.127734 4.287128 8.225713 14.100203 221054829
1.357336 4.822809 8.810735 14.643458 22.577779
1.571155 5.343903 9.401958 15.194110 23.104553
1.771183 5.850492 9.997251 15.752059 23.635223
1.959206 6.342739 10.594773 16.317122 24.169860
2.136732 6.820888 11.192938 16.889030 24.708534
10 2.305040 7.285254 11.790394 17.467444 25.251312
2.465217 7.736212 12.385986 18.051962 25.798254
:: 2.618185 8.174189 12.978730 18.642128 26.349411
2.764731 8.599648 13.567791 19.237446 26.904827
:: 2.905523 9.013085 14.152458 19.837389 27.464530
3.041137 9.415010 14.732130 20.441413 28.028539
3.172067 9.805943 15.306299 21.048960 28.596854
[ 1
c-t)3
[ 1
(-92
[t-32
1 [(-;I91 [(-p1
c-%0&1
c-l\n 0 1 2 3 4
0.25 0.793016 2.451485 3.826574 5.26224 7.14921
0.24 0.802442 2.477117 3.858771 5.25133 7.05054
0.811763 2.503218 3.895890 5.25040 6.96237
od:: 0.820971 2.529593 5.26046 6.88638
0:21 0.830059 2.556036 EZ
. 5.28251 6.82460
0.20 0.839025 2.582340 5.31747 6.77941
0.19 0.847869 2.608310 5.36610 6.75360
0.18 0.856592 2.633778 6.75030
0.17 0.865200 2.658616 E: 6.77286
0.16 0.873698 2.682743 5:59516 6.82451
0.15 0.882095 2.706127 4.324653 5.69566 6.90779
0.14 0.890399 2.728784 4.381878 5.80359 7.02356
0.13 0.898617 2.750762 4.436798 5.91452 7;16962
0.12 0.906758 4.489168 6.02383 7.33916
0.11 0.914827 %19?1
. 4.539096 6.12806 7.52035
0.10 0.922830 2.813346 4.586895 6.22577 7.69932
0.09 0.930772 2.833316 4.632927 6.31730 7.86638
0.08 0.938657 2.852927 4.677506 6.40385 8.01951
0.07 0.946487 2.872213 6.48655 8.16148
0.06 0.954267 2.891203 i%slet;
. 6.56618 8.29538
0.05 0.961998 2.909920 4.804519 6.64326 8.42315
0.04 0.969683 2.928382 4.845033 6.71812 8.54594
0.03 0.977324 2.946608 4.884779 6.79104 8.66452
0.02 0.984923 2.964611 4.923820 6.86221 8.77945
0.01 0.992481 2.982404 4.962212 6.93182 8.89116
0.00 1.000000 3.000000 5.000000 7.00000 9.00000
[: 1
(342
[ (-yJ1 [(-$I6
1 [c--y
1 [ 1
$314
SPHEROIDAL WAVE FUNCTIONS 761
EIGENVALUES-PROLATE AND OBLATE Table 21.1
OBLATE
k,,(-ic)-m(mf1) *
X04- ic)
G\n 0 1 ‘-I
3 4
0.000000 2.000000 6.OO;OOO 12.000000 20.000000
10 -0.348602 1.393206 5.486800 11.492120 19.495276
-0.729391 0.773097 4.996484 10.990438 18.994079
f -1.144328 +0.140119 4.531027 10.494512 18.496395
4 -1.594493 -0.505243 4.091509 10.003863 18.002228
-2.079934
-2.599668
-3.151841
-1.162477
-1.831050
-2.510421
z%~~
2:923796
9.517982
9.036338
8.558395
17.511597
17.024540
16.541110
-3.733981 -3.200049 2.578730 8.083615 16.061382
-4.343292 -3.899400 2.251269 7.611465 15.585448
-4.976895 -4.607952 1.938419 7.141427 15.113424
-5.632021 -5.325200 1;637277 6.673001 141645441
-6.306116 -6.050659 1.345136 6.205705 14.181652
-6.996903 -6.783867 1.059541 5.739084 13;722230
-7.702385 -7.524384 0.778305 5.272706 13.267364
15 -8.420841 -8.271795 0.499495 4.806165 12.817261
16 -9.025710 0;221407 4.339082 121372144
[:t-y1
-9.150793
II 1
(y)4
c 1
c-y
[ 1
(-;I3
[ 1
6gW
c-2[Xon( GC)]
c-L\n 0 1 2 3 4
0.25 -0.571924 -0.564106 +0.013837 0.271192 0.77325
0.24 -0.585248 -0.579552 -0.009136 0.213225 0.67822
0.23 -0.599067 -0.595037 -0.031481 0.157464 0.58772
0.22 -0.613349 -0.610591 -0.053477 0.103825 0.50191
0.21 -0.628058 -0.626242 -0.075480 0.052196 0.42099
0.20 -0.643161 -0.642016 -0.097943 +0.002437 0.34521
0.19 -0.658625 -0.657938 -0.121428 -0.045635 0.27490
0.18 -0.674418 -0.674031 -0.146603 -0.092251 0.21043
0.17 -0.690515 -0.690310 -0.174201 -0.137692 0.15215
0.16 -0.706891 -0.706792 -0.204894 -0.182301 0.10020
0.15 -0.723530 -0.723486 -0.239109 -0.226469 0.05428
-0.740416 -0.740399 -0.276886 -0.270627 +0.01332
if:; -0.757541 -0.757535 -0.317881 -0.315206 -0.02476
0:12 -0.774896 -0.774894 -0.361548 -0.360594 -0.06337
0.11 -0.792476 -0.792476 -0.407352 -0.407081 -0.10723
0.10 -0.810279 -0.810279 -0.454896 -0.454839 -0.16065
-0.828301 -0.828301 -0.503937 -0.503928
KG -0.846539 -0.846539 -0.554337 -0.554337 :g;;;;
0:07 -0.864992 -0.664992 -0.606021 -0.606021 -0:37117
0.06 -0.883657 -0.883657 -0.658931 -0.658931 -0.45125
0.05 -0.902532 -0.902532 -0.713025 -0.713025
0.04 -0.921616 -0.921616 -0.768262 -0.768262 1:;;;;;
0.03 -0.940906 -0.940906 -0.824608 -0.824608 -0:71218
0.02 -0.960402 -0.960402 -0.882031 -0.882031 -0.80533
0.01 -0.980100 -0.980100 -0.940503 -0.940503 -0I90131
0.00 -1.000000 -1.000000 -1.000000 -1.000000 -1.00000
[ t-f)6
1 [ (-$14
1 c 1
(-S4)3
c(-y 1
762 SPHEROIDAL WAVE FUNCTIONS
10 0.000000
0.195548
4.000000
4.424699
10.000000
10.467915
18.000000
18.481696
28.000000
28.488065
s
4
0.382655
0.561975
0.734111
4.841718
5.251162
5.653149
10.937881
11.409266
11.881493
18.965685
19.451871
19.940143
28.977891
29.469456
29.962738
0.899615 6.047807 12.354034 20.430382 30.457716
1.058995 6.435272 12.826413 20.922458 30.954363
1.212711 6.815691 13.298196 21.416235 31.452653
1.361183 7.189213 13.768997 21.911569 31.952557
1.504795 7.555998 14.238466 22.408312 32.454044
1.643895 7.916206 14.706292 22.906311 32.957080
1110 1.778798 8.270004 15.172199 23.405410 33.461629
12 1.909792 8.617558 15.635940 23.905451 33.967652
13 16.097297 24.406277 34.475109
14 9%t:
. %XE!
. 16.556078 24.907729 34.983956
15 2.281832 9.624450 17.012115 25.409649 35.494147
16 2.399593 9.948719 17.465260 25.911881
IIc-p1 [1(-$)l1
36.005634
[L 1
(-;I4
*
[ 1
t-y3
c 1
(-y
c-'IX&)-21
c-l\n 1 2 3 4 5
0.25 0.599898 2.487179 4.366315 6.47797 9.00140
0.24 0.613295 2.491544 4.338520 6.38296 8.80891
0.23 0.627023 2.497852 4.315609 6.29522 8.62445
0.22 0.641073 2.506130 4.297923 6.21556 8.44916
0.21 0.655431 2.516383 4.285792 6.14494 8.28436
0.20 0.670084 2.528591 4.279522 6.08438 8.13163
0.19 0.685014 2.542705 4.279366 6.03498 7.99282
0.18 0.700204 2.558644 4.285495 5.99788 7.87010
0.17 0.715632 4.297965 5.97420 7.76598
0.16 0.731281 ;*z:192
. 4.316672 5.96496 7.68328
0.15 0.747129 2.616135 4.341320 5.97090 7.62508
0.14 0.763159 2.637968 4.371397 5.99230 7.59446
0.13 0.779353 4.406191 6.02874 7.59407
0.12 0.795696 I'%69 4.444844 6.07889 7.62539
0.11 0.812174 2:708934 4.486445 6.14051 7.68773
0.10 0.828776 2.733891 4.530151 6.21063 7.77728
0.09 0.845493 2.759305 4.575277 6.28624 7.88714
0.08 2.785099 4.621329 6.36482 8.00897
0.07 %% 4.667984 6.44473 8.13579
0.06 0:896251 %%%E
. 4.715031 6.52505 8.26355
0.05 0.913352 2.864224 4.762333 6.60532 8.39048
0.04 0.930535 2.891056 4.809790 6.68528 8.51592
0.03 0.947796 2.918069 4.857332 6.76480 8.63963
0.02 0.965129 2.945243 4.904906 6.84378 8.76153
0.01 0.982531 2.972558 4.952472 6.92219 8.88164
0.00 1.000000 3.000000
1(-;I8
5.000000 7.00000 9.00000
[ c-p4
1 [ 1
c-y
1 [(-;I2
1 [ (-?)4
1
*See page x1.
SPHEROIDAL WAVE FUNCTIONS
x,,(-ic)--77)2(772+1) *
xI,~-ic)-2 *
c2\n 1 2 3 4 5
0.000000 4.000000 10.000000 18.000000 28.000000
10 -0.204695 9.534818 17.520683 27.513713
-0.419293 mi1 9.073104 17.043817 27.029223
f -0.644596 2:678958 8.615640 16.569461 26.546548
4 -0.881446 2.222747 8.163245 16.097655 26.065706
-1.130712 1.758534 7.716768 15.628426 25.586715
-1.393280 1.286300 7.277072 15.161786 25.109592
-1.670028 0.806045 6.845015 14.697727 24.634357
-1.961809 +0.317782 6.421425 14.236229 24.161031
-2.269420 -0.178458 6.007074 13.777252 23.689634
-2.593577 -0.682630 5.602649 13.320743 23.220190
-2.934882 -1.194673 5.208724 12.866634 22.752726
-3.293803 -1.714511 4.825732 12.414640 22.287271
-3.670646 -2.242055 4.453947 11.965266 21.823856
-4.065548 -2.777205 4.093464 11.517803 21.362516
-4.478470 -3.319848 3.744202 11.072331 20.903290
[Ic-y1
-4.909200 -3.869861 3.405903 10.628718 20.446222
[ 1
C-$)2
[(--p’1 c 1
c-t)3
1 1
C-2)3
*
c-l\n 1 2 3 4 5
0.25 -0.306825 -0.241866 0.21286 0.66429 1.2778
0.24 -0.318148 -0.266693 0.17062 0.57759 1.1420
0.23 -0.330984 -0.291340 0.13125 0.49460 1.0120
0.22 -0.345469 -0.315894 0.09476 0.41533 0.8879
0.21 -0.361702 -0.340450 0.06107 0.33974 0.7697
0.20 -0.379735 -0.365113 0.03001 0.26779 0.6575
0.19 -0.399564 -0.389998 +0.00127 0.19942 0.5515
0.18 -0.421125 -0.415222 -0.02563 0.13449 0.4520
0.17 -0.444308 -0.440907 -0.05142 0.07282 0.3591
0.16 -0.468974 -0.467166 -0.07710 +0.01411 0.2735
0.15 -0.494976 -0.494104 -0.10406 -0.04205 0.1958
0.14 -0.522180 -0.521805 -0.13412 -0.09625 0.1271
0.13 -0.550474 -0.550335 -0.16924 -0.14929 0.0680
0.12 -0.579775 -0.579732 -0.21076 -0.20210 +0.0183
0.11 -0.610027 -0.610016 -0.25868 -0.25572 -0.0250
0.10 -0.641193 -0.641191 -0.31185 -0.31111 -0.0685
0.09 -0.673251 -0.673251 -0.36901 -0.36888 -0.1219
0.08 -0.706186 -0.706186 -0.42934 -0.42932 -0.1907
0.07 -0.739985 -0.739985 -0.49242 -0.49242 -0.2714
0.06 -0.774638 -0.774638 -0.55807 -0.55807 -0.3598
0.05 -0.810135 -0.810135 -0.62616 -0.62616 -0.4542
0.04 -0.846468 -0.846468 -0.69657 -0.69657 -0.5540
0.03 -0.883628 -0.883628 -0.76923 -0.76923 -0.6588
0.02 -0.921608 -0.921608 -0.84406 -0.84406 -0.7682
0.01 -0.960401 -0.960401 -0.92100 -0.92100 -0.8820
0.00 -1.000000 -1.000000 -1.00000 -1.0000
cc-y1 cc-y1
-1.00000
c 1
c-y
c 1
C-l)5
II 1
(-72
c-‘Ix,“(c) -61 *
c-l\n 2 3 4 5 6
0.25 0.475965 2.703239 5.073371 7.74906 10.8360
0.24 0.489447 2.683149 4.994116 7.58138 10.5536
0.23 0.503526 2.665356 4.919290 7.41971 10.2781
0.22 0.518220 2.650003 4.849313 7.26479 10.0103
0.21 0.533551 2.637236 4.784640 7.11743 9.7512
0.20 0.549534 2.627196 4.725757 6.97858 9.5023
0.19 0.566185 2.620017 4.673177 6.84931 9.2649
0.18 0.583513 2.615819 4.627427 6.73081 9.0409
0.17 0.601526 2.614701 4.589031 6.62442 8.8323
0.16 0.620224 2.616735 4.558480 6.53155 8.6417
0.15 0.639604 2.621954 4.536196 6.45371 8.4718
0.14 0.659659 2.630349 4.522485 6.39236 8.3260
0.13 0.680376 2.641862 4.517479 6.34878 8.2078
0.12 0.701737 2.656384 4.521086 6.32389 8.1208
0.11 0.723722 2.673764 4.532956 6.31794 8.0678
0.10 0.746308 2.693817 4.552484 6.33030 8.0507
0.09 0.769471 2.716339 4.578871 6.35935 8.0688
0.793186 2.741120 4.611219 6.40263 8.1184
F% 0.817429 2.767960 4.648642 6.45738 8.1932
0:06 0.842175 2.796673 4.690346 6.52096 8.2864
0.05 0.867402 2.827089 4.735658 6.59127 8.3919
0.04 0.893087 2.859059 4.784022 6.66670 8.5057
0.03 0.919209 2.892449 4.834980 6.74607 8.6249
0.02 0.945747 2.927138 4.888160 6.82849 8.7477
0.01 0.972684 2.963019 4.943252 6.91330 8.8730
0.00 1.000000 3.000000 5.000000 7.00000 9.0000
1 1
C-j'9
[ c-t)4
1 [c-y1 (33’2
[ 1
*See page II.
SPHEROIDAL WAVE FUNCTIONS 765
EIGENVALUES-PROLATE AND ORLATE Table 21.1
OBLATE
X,,(-ic)--m(m+l) *
&,a(--id-6 *
G\n 2 3 4 5 6
0.000000 6.000000 14.000000 24.000000 36.000000
10 -0.144837 5.664409 13;597220 23.564371 35.545806
2 -0.293786 5.324253 13.194206 23.129322 35.092330
3 -0.447086 4.979458 12.791168 22.694912 34.639597
4 -0.604989 4.629951 12.388328 22.261201 34.187627
-0.767764 4.275662 11.985928 21.828245 33.736444
-0.935698 3.916525 11.584224 21.396098 33.286069
-1.109090 3.552475 11.183489 20.964812 32.836522
-1.288259 3.183450 10.784014 20.534436 32.387826
-1.473539 2.809393 10.386106 20.105013 31.940000
10 -1.665278 2.430250 9.990084 19.676587 31.493066
-1.863838 2.045970 9.596286 19.249195 31.047043
:: -2.069595 1.656508 9.205059 18.822869 30.601952
-2.282933 1.261822 8.816762 18.397640 30.157814
:i -2.504245 0.861875 8.431761 17.973532 29.714648
15 -2.733927 0.456635 8.050424 17.550565 29.272476
16 -2.972375 0.046076 7.673121 17.128753 28.831317
[C-f)1
1 (-p7
[ 1 [ 1 * II(-pl1
(-:)5
[Ic-y1
~-~[X~,(-ic)-6]
c-l\n 2 3 4 5 6
0.25 -C.185773 +0.002879 0.47957 1.07054 1.8019
0.24 -0.190754 -0.030028 0.41280 0.95365 1.6261
0.23 -0.196680 -0.062228 0.34933 0.84167 1.4577
0.22 -0.203790 -0.093813 0.28933 0.73461 1.2965
0.21 -0.212386 -0.124893 0.23297 0.63251 1.1428
0.20 -0.222841 -0.155607 0.18049 0.53537 0.9964
0.19 -0.235596 -0.186120 0.13215 0.44322 0.8574
0.18 -0.251126 -0.216631 0.08816 0.35607 0.7260
0.17 -0.269873 -0.247375 0.04864 0.27389 0.6022
0.16 -0.292149 -0.278624 +0.01342 0.19662 0.4863
0.15 -0.318047 -0.310677 -0.01813 0.12409 0.3785
0.14 -0.347414 -0.343847 -0.04727 +0.05600 0.2795
0.13 -0.379928 -0.378432 -0.07609 -0.00822 0.1901
0.12 -0.415213 -0.414688 -0.10778 -0.06954 0.1120
0.11 -0.452947 -0.452800 -0.14643 -0.12937 +0.0470
0.10 -0.492902 -0.492871 -0.19508 -0.18959 -0.0051
0.09 -0.534942 -0.534937 -0.25333 -0.25217 -0.0517
0.08 -0.578991 -0.578991 -0.31876 -0.31861 -0.1076
0.07 -0.625006 -0.625006 -0.38955 -0.38955 -0.1844
0.06 -0.672956 -0.672956 -0.46494 -0.46494 -0.2768
0.05 -0.722813 -0.722813 -0.54456 -0.54456 -0.3791
0.04 -0.774556 -0.774556 -0.62821 -0.62821 -0.4895
0.03 -0.828164 -0.828164 -0I71571 -0.71571 -0.6073
0.02 -0.883618 -0.883618 -0.80691 -0.80691 -0.7319
0.01 -0.940902 -0.940902 -0.90171 -0.90171 -0.8629
,1.000000 -1l0000
IIc-p1
0.00 -1.000000 -1.00000 -1.00000
C-i)5
II 1 [C-l’2
1 [ 1
(536
[ 1
(-i)3
111 0.1578 0.3134 0.4643 0.6067 0.7355 0.8450 0.9290 0.9819 1.000
2 0.1194 0.2437 0.3757 0.5149 0.6562 0.7892 0.9000 0.9740 1.000
3 0.0776 0.1654 0.2724 0.4030 0.5546 0.7144 0.8597 0.9627 1.000
0.0449 0.1018 0.1832 0.2994 0.4537 0.6353 0.8150 0.9497 llOO0
54 0.0239 0.0588 0.1179 0.2162 0.3650 0.5602 0.7698 0.9361 1.000
12 1 0.4788 0.9054 1.232 1.417 1.435 1.276 0.9562 0.5119 0
2 0.3896 0.7509 1.052 1.253 1.316 1.212 0.9335 0.5088
0.2780 0.5538 0.8148 1.030 1.149 1.118 0.8992 0.5039 :
i 0.1762 0.3683 0.5813 0.7968 0.9643 1.008 0.8575 0.4979 0
5 0.1011 0.2254 0.3896 0.5906 0.7879 0.8957 0.8127 0.4911 0
13 1 0.9928 1.745 2.075 1.903 1.280 0.3775 -0.5521 -1.244 -1.500
2 0.9559 1.710 2.092 1.998 1.432 0.5298 -0.4541 -1.214 -1.500
0.8745 1.611 2.063 2.097 1.640 0.7606 -0.2972 -1.174 -1.500
i 0.7393 1.418 1.934 2.128 1.841 1.032 -0.0951 -1.097 -1.500
5 0.5662 1.146 1.691 2.047 1.975 1.299 +0.1319 -1.017 -1.500
2 21 0.0844 0.3295 0.7111 1.189 1.710 2.211 2.627 3.000
0.0690 0.2744 0.6092 1.054 1.572 2.101 2.566 %3e 3.000
; 0.0500 0.2051 0.4773 0.8738 1.380 1.944 2.475 2:859 3.000 e
4 0.0328 0.1405 0.3487 0.6876 1.171 1.764 2.367 2.827 3.000
5 0.0198 0.0898 0.2414 0.5212 0.9701 1.580 2.251 2.791 3.000
2 31 0.4222 1.570 3.116 4.596 5.530 5.548 4.501 2.522
0.3597 1.358 2.755 4.175 5.170 5.327 4.417 2.510
32 0.2765 1.070 2.255 3.576 4.641 4.994 4.286 2.491
0.1934 0.7758 1.723 2.909 4.025 4.588 4.122 2.466
: 0.1244 0.5226 1.243 2.269 3.395 4.150 3.936 2.437
From C. Flammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with
permission).
SPHEROIDAL WAVE FUNCTIONS 767
ANGULAR FUN4 ZTIONS-PROLATE AND OBLATE Table 21.2
OBLATE
&m(-k, 7)
m n C\l 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0 01 1.000 1.002 1.007 1.016 1.028 1.044 1.064 1.088 1.115 1.147 1.183
2 1.000 1.008 1.032 1.132 1.210 1.434 1.585 1.767 1.986
1.000 1.022 1.089 :*i;: 1.377 1.617 2.366 2.923 3.648 4.589
: 1.000 1.047 1.191 1:449 1.854 2.452 4.557 6.323 8.837 12.42
5 1.000 1.083 1.341 1.835 2.648 3.952 9.211 14.23 22.11 34.48
011 : 0.1001 0.2034
0.2009 i%: 0.4274
0.4065 0.5128 0.6952
0.6222 0.8530
1.035 0.9760
1.243 1.484
1.105
: i 0:1016
K~ 0.2079 0.3526
0:3273 0.4664 ii-:::;
0:7681 1.096
0.8398 2.195
1.425 3.105
1.842 4.396
2.378
5” 0 0.1032 K::;
. 0.3884 FE:
. 0.9804 1.525 3.684 5.741 8.970
0 21 -0.4863 -0.4450 -0.3757 -0.2779 -0.1507 +0.0070 0.1965 0.4197 0.6784 0.9749
2 -0.4897 -0.4585 -0.4052 -0.3277 -0.2231 -0.0872 +0.0849 0.2999 0.5660 0.8930
-0.4943 -0.4766 -0.4448 -0.3952 -0.3223 -0.2183 -0.0721 +0.1311 0.3845 0.7958
4' -0.4966 -0.4891 -0.3681 -0.2485 -0.0458 0.2868 0.8201
5 1:;;;;
. -0.5234 -0.5495 r:;,'g
. -0.5977
-0.4356 -0.5869 -0.5067 -0.2880 0.1892 1.132
-0.1477 -0.2810 -0.3855 -0.4466 -0.4491 -0.3768 -0.2130 +0.0600 0.4613 1.011
O 3 3: 2f:g
-0.2839
-0.2885
-0.3947
-0.4097
-0.4668
-0.4998
-0.4839
-0.5421
-0.4275
-0.5140
-0.2757
-0.3841
-0.0015
-0.1091
0.4274
0.3711
1.051
1.138
4 -0:1495 -0.4306 -0.6432 -0.5540 -0.2765 0.2912 1.327
5 -0.1504 1;:;;;
. -0.4589 1;;:::
. -0.7489
-0.6270 -0.8356 -0.8080 -0.5447 0.1715 1.723
111 0.9961 0.9838 0.9628 0.8299 0.7506 0.6402 0.4731 0
0.9994 0.9973 0.9923 0.9316
0.9827 i%: 0.9340 0.8802 0.7864 0.6118 0
: 1.006 1;025 1;05sm 1.093 i135 1.172 1.188 1.149 0.9724 0
4 1.020 1.319 1.498 1.708 1.920
5 1.041 :%i
. :*:i:
. 1.776 2.242 2.878 3.642 2.067
4.400 1.950
4.651 i
0.5897 1.113 1.322 :'E 1.247
1.487 0
l 2 2’ 0.5950
0.6043
i%::
.
0.9140
1.153
1.228
1.398
1.541
:-2:
1:837
:%I
2:082 2:200 2.000 :
: 0.6213 0.9640 1.349 I.780 2.250 2.723
5 0.6400 1.040 1.537 2.165 2.947 3.868 3.092
4.786 3.033
5.138 :
131 -1.500 -1.421 -1.189 -0.8136 -0.3165 0.2710
-1.500 -1.431 -0.8941 -0.4427 +0.1060 0.9015
0.7174 1.501 1.946
1.826 1.951
1.988 :
: -1.500 -1.447 -0.6502 -0.1738 +0.3916 El 1.572 1.834. 0
-1.500 -1.467 +%;
rg;. -1.184
-1.024
-1.353 -0.9148 -0.5415 -0.0538 0:5403 1.177 1.619 0
: -1.500 -1.486 -1.198 -0.9435 -0.5506 0.0161 0.7471 1.439 0
2.972 2.889 2.748 1.970 1.585 1.131 0.6041 0
2 2 2’ 2.979
2.992 2.965
2.915 ZE
%4
2:830
2.291
2.425
2.693 %F z1"
1.305
:z: 1:615
%E :
:5 3.013
3.052 x. 3.469
3:111 x. 3.200
4.202
3:157
4.564
2:966
4.746 4:460 3.188 i
2 3. 1 1.486 x 4.115 5.086 5.704 5.877 5.503
1.488 4.180 5.226 5.954 4.477
4.990 2.683
3.077 i
: :-:2 2.996
21943 4.295 5.482 6.413 t-9':: 5.982
6.904
iso9 3.073 4.475
4.738 5.891 7.166 8:132 8.515 6.008
7.857 5.408
3.879 :
: 6.515 8.347 10.07 11.28 11.21 8.354 0
768 SPHEROIDAL WAVE FUNC!l’IONS
R(l)
mn (c’ t) JP)nm (c f E)
0 1
I II
-1 3.249
-1 5.308
-1 5.162
5.786
-1 4.125
-1 3.328
-1 5.311
1: 54'::;
-1 3:137
0 2
0 3
II II
-1 4.413
-2 4.444
1.833
4.954
3.421 -2 5.373
-1 3.976
4.293
3.509
1.947
-1 1.287
II II II
11
-1 12.323
3
4
s -10 -2.077
-4.885
-1.075
-7.294
-6.911 -10 -1.417
-2.874
-7.453
-4.734
-4.585 -10 -1.071
-1.248
-5.480
-3.432
-2.924
12
;
: II II II II
-2
-3 2.378
6.503
-2 4.658
-2 1.322
4.802
-2 9.296
-2 2.012
7.227
-1 1.372
1.960
-2 2.754
9.738
-1 1.798
13
5.
II1;:-:;:
15 $$:: 1: :-:g
.
-1 3:553
-2 7.089
1.108
-1 2.376 1; :-g
.
II
2 2
-2 6.612
1:
-4
-3 1:372
:3';2"
2,566
2 3
From C. F’lammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif., 1957 (with permission).
SPHEROIDAL WAVE FUNCTIONS 769
OBLATE RADIAL FUNCTIONS-FIRST AND SECOND KINDS Table 21.4
R@)
mn(- GC, a)
nk n 0 0.75
0 0 7864
9707
:z:
2189
0356
3758
0 1
0 2
1 1
1 2
1 3
2 2
From C. F’lammer, Spheroidal wave functions. Stanford Univ. Press, Stanford, Calif.,
1957 (with permission).
22. Orthogonal Polynomials
URS W. HOCHSTRASSER *
Contents
Page
Mathematical Properties .................... 773
22.1. Definition of Orthogonal Polynomials .......... 773
22.2. Orthogonality Relations ................ 774
22.3. Explicit Expressions .................. 775
22.4. Special Values. .................... 777
22.5. Interrelations ..................... 777
22.6. Differential Equations ................. 781
22.7. Recurrence Relations ................. 782
22.8. Differential Relations ................. 783
22.9. Generating Functions ................. 783
22.10. Integral Representations ............... 784
22.11. Rodrigues’ Formula. ................. 785
22.12. Sum Formulas .................... 785
22.13. Integrals Involving Orthogonal Polynomials ....... 785
22.14. Inequalities ..................... 786
22.15. Liiit Relations ................... 787
22.16. Zeros ........................ 787
22.17. Orthogonal Polynomials of a Discrete Variable ...... 788
Numerical Methods ...................... 788
22.18. Use and Extension of the Tables ............ 788
22.19. Least Square Approximations ............. 790
22.20. Economization of Series ................ 791
References .......................... 792
Table 22.1. Coefficients for the Jacobi Polynomials Ppfl)(z) ..... 793
n=0(1)6
Table 22.2. Coefficients for the Ultraspherical Polynomials C’j? (5) and
forz”inTermsof C:)(z). ................... 794
n=0(1)6
Table 22.3. Coefficients for the Chebyshev Polynomials Z’,(z) and for
s”inTermsof T&r) ..................... 795
n=0(1)12
Table 22.4. Values of the Chebyshev Polynomials T,(z) ....... 795
n=0(1)12,
~=..(..)l. 10D
Table 22.5. Coefficients for the Chebyshev Polynomials U,(z) and for
z”in Terms of U,,,(z) .................... 796
n=0(1)12
Table 22.6. Values of-the Chebyshev Polynomials U,(z) ...... 796
n=0(1)12, x=.2(.2)1, 10D
* Guest Worker, Nationd Bureau of Standards, from The American University. (Pres-
ently, Atomic Energy Commission, Switzerland.)
771
772 ORTHOGONAL POLYNOhUALS
Page
Table 22.7. Coefficients for the Chebyshev Polynomials C,(z) and for
z”inTermsof Cm(z) . . . . . . . . . . . . . . . . . . . . . 797
n=0(1)12
Table 22.8. Coefficients for the Chebyshev Polynomials S,(z) and for
Z* in Terms of S,,,(z) . . . . . . . . . . . . . . . . . . . . . 797
n=0(1)12
Table 22.9. Coefficients for the Legendre Polynomials P,(z) and for 9’
in Terms of P,(z) . . . . . . . . . . . . . . . . . . . . . . 798
n=0(1)12
Table 22.10. Coefficients for the Laguerre Polynomials L,(z) and for 2”
in Terms of L,(z) . . . . . . . . . . . . . . . . . . . . . . 799
T&=0(1)12
Table 22.11. Values of the Laguerre Polynomials L,(s) . . . . . . . 800
n=0(1)12, x=.5, 1, 3, 5, 10, Exact or 10D
Table 22.12. Coefficients for the Hex-mite Polynomials H,,(z) and for 2”
in Terms of H,(z) . . . . . . . . . . . . . . . . . . . . . . 801
T&=0(1)12
Table 22.13. Values of the Hermite Polynomials H,,(z) . . . . . . . 802
n=0(1)12, x=.5, 1, 3, 5, 10, Exact or 11s
22. Orthogonal Polynomials
Mathematical Properties
b
w(~>$(~)j&)dx=0 3
s a
(n#m;n, m=o, 1,2,. . .)
b w(x)j~(x)dx=h,, jn(x)=k,x”+k:x~-‘+ . . .
s@
(T&=0,1,2,. ...)
22.1.3 sz(x>f~+m(x>j~+~nfn=o
/=
where gZ(x), m(x) are independent of 72 and a,, a
constant depending only on n.
Recurrence Relation \
where
22.1.5 \
I-
Rodrigues’ Formula
a=0
n#O
22.2.4 T&l Chebyshev of the
first kind
-1 1 (l---z’)-1 T.(l) = 1
n=O
1-q 29 -4 n#O
22.2.6 cvd4 Chebyshev of the
first kind -2 2
( > C,(2)=2
n=O
n#O
22.2.8 T:(4 Shifted Chebyshev
of the first kind 0 1 (z--z’)-+ Tf(1) = 1
n=O
Shifted Chebyshev *
22.2.9 of the second 0 1 (x-x’)b q(1) =n+ 1
kind
22.2.10 Legendre
(Spherical) -1 1 P.(l)=1
22.2.11 1
Shifted Legendre 0 1
2n+l
*See page 11.
22.2. Orthogonality Relations-Continued
- -
Generalized 0 m e-Q? k =(-1)” r(a+n+ 1)
Laguerre n a>-1
n! n!
* ,-‘?t
22.2.15 He,(z) Hermite --a0 m e,=(-1)” &n!
- -
22.3. Explicit Expressions
f”w=d”mgc.h(z)
-- =
f”(Z) N d. Bn (4 k. Remarks
--
P’“.I @)(2) 1 2n+a+B
22.3.1 n ("lfr") &) (z- I)“-++ l)* a>--I,@>-1
5 ( n >
lYa+n+ 1) n r(a+8+n+m+l) 1 2n+a+8
22.3.2 P-6)
” (J) n (x-l)“@ .jz ( a>-1, s>--1
n!lYa+@+n+ 1) 0m . 2=r(a+m+l) n >
r(q+n) r(p+2n-m) zm-m
22.3.3 &(P, 4, d n
(-‘)“, (3 r(q+n-m)
1 P-q>-1, q>o
r(p+W
22.3.4 (72’ (2)
[I; (-l)n rb+n-74
m!(n-2m)!
(22) “-*- 2" r(a+n)
-arG)
a>-f> a#0
22.3.5 cy (2)
[Iz n (-,)m (n-m-l)!
tn!(n-2m)! (22) --*m n
2"
n#O n#O, CA”(l)= 1
n
22.3.6 T.(z)
5
L-1 3
(- l)m (n-m-
m!(n-2m)!
I)! (22) n-h 2nd
[Iz n (n-m)!
22.3.7 Un(4 t---11* m!(n-2m)! (2x) n--l* 2"
22.3.8 E-‘.(4
[I;zn
C---l)- (;,) (,,,,,) ~“-*“I (2n) !
2n(n!)*
22.3.10 H.(z)
[IT n! (--1)m '
m!(n-2m)!
(2.z)"-+m 2" ser! 22.11
n : Z"-*m
22.3.11 He.64 n! (-- llm m!2m(n-2m)! 1
L-15 -
776 ORTHOOONAL POLYNOMIALS
fs(ms@I 0, Remarka
22.3.12 C”’
I (CO8
e) rbSm)r(a+n-m) a#0
a4.0n
mun-m)![Iya)]’
22.3.13 P,(cose) $ (“m”) (2:r:)
-X
22.3.14 CiO)(c4x3e>=i cosne
22.3.15 T,(cos8)=cos nB
-I
FIQURE 22.3. Jacobi Polynomiala P>@(x), FIQURE 22.4. Gegenbaw (Ultrmpherical) PO&W.+
a=1.5, fl=-.8(.2)0, n=5. mid C$)(x), a=.5, n=2(1)5.
ORTHOGONAL POLYNOMIALS
22.4.7 Lp (2) I
(
n+a
n >
-x+a+ 1
iII (-1)” ~8
cm! n=2m
22.4.8 ~I&) (- l)“H.(Z) 2x
(0, n=2m+l
- -
22.5. Interrelations
Jacobi Polynomiab
22.5.1
ppyz>= r b+a+b+l)
G. a+B+l,B+l,T)
nW+a+B+l) (
22.5.2
Q*(p, *,+n!r(n+p) ~y*--1y22-1)
r(2n+P)
(see [22.21]).
22.5.3
Ultraspherical Polynomials
Chebyshev Polynomiab
L-l.5 22.5.5
FIGURE 22.5. Gegenbauer (Uhasphmhzl) PO&LO-
mials cy(z), a=.2(.2)1, n=5. 22.5.6 T”(z)=u”(z)--zu”-l(4
778 ORTHOGONAL POLYNOMIALS
(am
22.5.26
22.5.11 C,(z)= 2T” (;)=2 T*, (y)
22.5.31 2!!@-
T”(@=r(n+4j pc-t.
z 4) @)
22.5.17 Lph)=(-l)m gm [L+mb$l
(n+l)!fi
22.5.32 Pi** 1)(x)
Hermite Polynomials ua(2)=2r(n+*)
22.5.18 He,(x)=2-“‘*H, z
04
(see [22.201).
22.5.19 H,(~)=2~l*Ele.(xJZ)
Jacobi Polynomials
22.5.20
22.5.21
22.5.35 P.(x)=P$yz)
22.5.37
22.5.38
I2 -
FIWRE 22.7. Ciieby~hu Polynomials U,(x), 22.5.39 W”‘(x) =?&&j JJ2.+1(da
T&=1(1)5.
Hermite Polynomials
22.5.33 T.(x)=; CkO)
(2)
22.540 Ha,(z) = (- 1)m2”m!L$1/2) (z2)
22.5.34 U,(x>=C~‘)(x) 22.5.41 H2,+,(5)= (-l)‘“2k+1m!zLg’2)(z2)
Orthogonal Polynomials as Hypergeometric Functions (see chapter 15)
fn(x>=@(a, 6; c; dx))
For each of the listed polynomials there are numerous other representations in terms of hyper-
geometric functions.
f. (2) d a b C VW
l.
x-l
22.5.44 P$"(x) --A --n-B a+1 z+i
Un+2a) l-2
22.5.46 C"'$8(x) -n n+2a a++
n!l-(2a) -T
22.5.47 1 -n n l-2
T.(x) f -3-
22.5.43 U.(x) n+l -n l-2
n+2 * % 2
22.5.49 P,(x) -n l-x
n+l 1 2
2
22.5.50 P"(X) -n --n -2n 1--z
l-n 1
22.5.51 P,(z) -- n 2 i-n 2
2
22.5.52 Pdx) n CW! n+& 4 29
(- 1) 22qny -n
22.5.57 r(a+i>r(2cY+n) 1
[z (22-q-~ P$q$, (5)
n!Iy242)
(2m+l)!
* Hzm+l(~)=(--l)~ m! 2xM(-m, iJ x2)
P”(X)
H&9
FIGURE 22.10. Hermite Polynomials 7)
FIGURE 22.8. Legendre Polynomials P,(x) ,
n=2(1)5. n=2(1)5.
ORTHOGONAL POLYNOMIALS 781
22.6. Differential Equations
g2(2)y”+g1(x)y’+go(x)y=o
-- __-.---
Y sdz) 91(z)
_-
-+I 8&l
22.6.3 (1 --z);;-(1 Sz) * fy’(2) 1 ID
22.6.5 C”‘(
I =) 1-Z - (2,-l- I)2 n(n+2a)
- T"(Z) ; v,-lb) *
22.6.11 l-22 -3x n*- 1
22.6.17 e-z12z(atI)12q4
(r) 1 0 ___
2n+a+i+1--(r2 __--
2x 4x2 41
1--a*
22.6.18 ,-**/2,“+4L’,“’ (x’) 1 0 4n+2a+3-x*+7
2
22.6.20 e-yH,(s) 1 0 2n+ 1 -x1
22.6.21 He.(x) 1 -X n
-
782 ORTHOGONAL POLYNOMIALS
fn aIn ah ah ah
--
22.7.1 @n+a+Bh 2(n+a)b+i3)
Ghfa+B+2)
22.7.4 T,(x) 1 0 2 1
22.7.5 U,(x) 1 0 2 1
22.7.6 S”(X) 1 0 1 1
22.7.7 cm 1 0 1 1
22.7.8 T.’ (4 1 -2 4 1
22.7.9 Kc4 1 -2 4 1
22.7.13 He64 1 0 2 2n
22.7.14 He,(z) 1 0 1 n
- -
Ql(z)~f.(z)=g*(z)f"(z)+no(z)f"-~(~)
f. 61 B1 go
v
22.8.1 Pp’ (2) (zn+a+B)(l-z*) 7&-B- (2n+a+BM 2(n+a)b+f9)
*
22.8.2 c!=) (2) 1-Z’ --Rz n+2a-1
22.8.3 T&) 1-X’ -nx n
22.8.4 U.(z) l-9 -7l.l n+l
R= Jl-2zz+z2
1 *
22.9.16 LA=’ (4 (x~)-t~e~~.(2(xz)*~~l
IYnSa+l)
1 elZ,-”
22.9.17 H,(z) ii
(-.I)” ez co9 (226) *
22.9.18 Hd4 (2n) !
(a>O)
ORTHOGONAL POLYNOMIALS 785
22.10.12 P,(cos e) =-; * (cos O+i sin e co9 fp)U#~ 22.10~14 ~g)(x)=$ m e-It”+; J,@&)dt
s0
.s 0
22.10.15
22.10. 13 P,(cos e,d * sin (n+wJd4 H,(x)=e’* T srn e-l*t” cos (2&-i,) dt
* n- s 0 (co9 e-cos +)* 0
The polynomials given in the following table are the only orthogonal polynomials which satisfy
this formula.
I f”(Z) I a, I PC4 d4
2n ‘(l--y)“(l+y)Bp:“,8)(y)dy
s
Miscellaneous Sum Formulas (Only a Limited Selection ~~lp_~‘,B+l’(~)-(~~x~~+l(l+x~~+lp~~~l~~+l~~~~
Is Given Here.)
22.12.2 22.13.2
Q2&~=t[l+u*n(41
22.12.3 g; ~*m+1(4=3U*n-1(5)
I =cyp(O)- (l-2yqy(2!)
Tl l-T*n+2(4
22.12.4 isou2"(4' 2(1-x*) 22.13.3 r’ Tn Mdy
JTl (y-x)~=Tu~-l(x)
Ic?~)(x)I 5
( >
“;t’ wag, if q=max (a, /3) 2 -l/2
b+--l,8>-1)
22.13.9
i IP$e)(z’)I -$t if *<--l
8-a
2’ maximum point nearest to -
a+P+l
22.14.2
CA>-2)
22.13.10 (“‘y> (a>01
“P,odt= 1 la%M
[T&l +Tn+l(41
{ Ia@W>l (-f<a<o)
s -1 &3 (n+i)fi
= Lm(t)Ln(x-t)dt
s0 z 22.14.1 IPn(4 II 1 (-l<x<l)
=
S 0
L,+,(t)dt=L,+“(x)---L,+“+l(x)
22.14.8
dp,(xl
----&--
1
Ip(n+l) (-l~s~l)
I I
22.13.15
S 0
’ e-f2~~(t)dt=~n-l(O>-e-Z2~,-~(x)
22.14.9 IPn(x)l 2 2 --i- (---l<dl)
J 7m4&?
22.13.16
S0) 0
z H,(t)dt=
22.14.10
22.13.17
S-m e-f2H2,(tx)dt=fi y (9-l)” ~‘.o-P~-l(,)P~+l(x)<3~n~l)(-15x11)
22.14.11
me-f2tH2,+l(tx)dt=~
22.13.18
Pm+l)!
mrx(x2~l)m 1 -P’,(x)
S-m
Pn2(x)-PAdPn+1(x~2 (2n-l)(n+1)
22.13.19
/rm 1 e-“tnH,(Xt)dt--~!P,(X)
22.14.12 ILd4 I I fP (x2 0)
(-l<x<l)
J-m
22.14.13 1Lp) (x) I 5 LfrTr:.:) ezj2 620, x20)
22.13.20
22.14.14
Sme-r2[H,(t)]2
0
cos (xt)dt=Ji;2n-‘n!e+L,
0
g
ORTHOGONAL POLYNOMIALS 787
22.16. Zeros
For tables of the zeros and associated weight factors necessary for the Gaussian-type quadrature
formulas see chapter 25. All the zeros of the orthogonal polynomials are real, simple and located
in the interior of the interval of orthogonality.
Explicit and Asymptotic Formulas and Inequalities
Notations:
xg)mth zero of fn(x) (xj”)<xP< . . . <x?)) j, %, mth positive zero of the Bessel function Jol(x)
22.16.7 P.(x)
+()(+)
22.17. Orthogonal Polynomials of a Discrete Tw*(xi) is finite. The constant factor which is
Variable
still free in each polynomial when only the orthogo-
In this section some polynomialsf,(x) are listed nality condition is given is defined here by the
which are orthogonal with respect to the scalar explicit representation (which corresponds to the
product Rodrigues’ formula)
0) r(b)r(c+z)r(d+z) x!r(b+x)
Hahn n!
s!r(b+x)r(c)r(d) (x-n)!r(b+z-n)
- - - - -
For a more complete list of the properties of these polynomials see [22.5] and [22.17].
Numerical Methods
Evaluation of an orthogonal polynomial for which the coe&ients are given numerically.
Example 1. Evaluate La(1.5) and its first and second derivative using Table 22.10 and the
Horner scheme.
= -
I -36 450 - 2400 5400 - 4320 720
L -306. 140625
1.5 1. 5 -49.5 523. 125 - 1919. 25 1165. 21875 6- 720
=. 42519 53
L,=889. 3125
1.5 1. 5 -47. 25 452. 250 - 1240. 875 6 720
= 1.23515 625
1 L,,=2 [ - 464.06251
-31.5 301. 50 -827. 250 - 464.0625 (I 720
=-I. 28906 25
- - -
ORTHOGONAL POLYNOMIALS 789
Evaluation of an orthogonal polynomial using the explicit representation when the coe&ients are not
given numerically.
If an isolated value of the orthogonal polynomial f*(x) is to be computed, use the proper explicit
expression rewritten in the form
fkd =&(4dx~
and generate a,,(r) recursively, where
a,-l(x)==l -~jCW.Cx) (m=n, n-l, . . ., 2, 1, a,(z)=l).
The d,(x), b,, c,,.f(z) for the polynomials of this chapter are listed in the following table:
=
fn(4 dn(4 bm I G&
p’u,
” 8)
n+a (n-m+l)(a+B+n+m) 2m(a+m)
( >
cl”,’ (-n. $? Z(n-m+l)(a+n+m-1) m(2m- 1)
C’“’
an+1
(-1)” kp .,& P(n-m+l)(a+n+m) m(2m+ 1)
(-1)” 2n
Pl, (n-m+1)(2n+2m-l) m(2m- 1)
-( 4” n >
m 8 7 6 5 4 3 2 1 0
~~*‘2~3’2’(2)=d~,(2)=(3.33847)(6545.533)=21852.07
Evaluation of orthogonal polynomials by means of their recurrence relations
Example3. Compute@)(2.5)for n=2,3,4,5,6.
From Table 22.2 C$)= 1, C?= 1.25 and from 22.7 the recurrence relation is
n 2 3 4 5 6
-___
cq2
n *5) 3. 65625 13.08594 50.87648 207.0649 867.7516
In some applications it is more convenient to use polynomials orthogonal on the interval [0, 11.
One can obtain the new polynomials from the ones given in this chapter by the substitution x=2:- 1.
The coefficients of the new polynomial can be computed from the old by the following recursive scheme,
provided the standardization is not changed. If
then the n: are given recursively by the a,,, through the relations
u~)-~u~-~‘-u~~,; 7n=n-1, n-2, . . ., j; j=O, 1, 2, . . ., n
(-‘)=a,/2,
am m=O, 1, 2, . . ., n
u$)=2&,j=O, 1, 2, . . ., n and u~~,“‘=uZ; m=O, 1, 2, . . ., n.
Example 4. Given Ts(z)=5z-20z3+16z5, find T:(z).
‘m 5 4 3 2 1 0
j
\
-1 f+ &” 0 -lo=&-” 0 2.5=a;-*) 0
0 -16 -4 -1=a;
1 it -64 -4: 5L;
- 192 3;: -4oo=a;
P 1;: -512 1120=a;
256 - 1280=a;
t 512=a;
Hence, Z’,*(z)=512xs-1280x4+1120z3-400x2+50x---1.
Problem: Given a function j(z) (analytically or Example 5. Find a least square polynomial of
in form of a table) in a domain D (which may be
a continuous interval or a set of discrete p~ints).~ degree 5 for j(z)=kx J in the interval 21x15,
Approximate J(Z) by a polynomial F,,(z) of given using the weight function
degree n such that a weighted sum of the squares
of the errors in lJ is least. 1
Solution: Let w(z) 20 be the weight function W(Z)=,&x-2)(5-x)
chosen according to the relative importance of
the errors in different parts of D. Let j,,,(x) be which stresses the importance of the errors at the
orthogonal polynomials in D relative to w(z), i.e. ends of the interval.
Cj,,,, j,,) =0 for m #n, where
Reduction to interval [-1, 11, t=2q
WMf(4!JW~
if D is a continuous interval
w(x(t)>2 1
3 Jl-t2
*
where
am= (j, jmMj~9 &de 2 l -- 1
u”=3?, s -, Jiq
di!
t+3
*f(z) haa to be square integrable, see e.g. (22.17).
(n+l>(N-~)j”+l(~)=(2~+1)(~--x)j”(~)--n(~+~+l)j~-l(x)
Example 6. Approximate in the least square sense the function j(x) given in the following table
by a third degree polynomial.
2 I f(z) I z=-
z-
2
10 fo@)
I
fit23
I
fi@)
I
fz(z)
10
::
.3162
: 2673
2887
0
1
: 1,; -1,; 4
:: : 2357
2500 i :1
-l/1
-1
-7); 1
$
-1
(f t fn>
.271580 .039940 .0043571 .000310
an=(fn,
j(x)-.27158+.03994(3.5-.25x)+.0043571(23.5-3.5x+.125;Ga)+.00031(266-59.8333x
+4.375x2-.10417x33)
j(x) +.59447- .043658x+ .00190092- 9000322921
22.20. lkonomization of Series 1 Then, since IT,(s)l11(-l<x<l)
Problem: Given j(x)= 2 a,x’ in the interval
m=O -j(x) =m.fo b, T,,, (x)
-11x11 and R>O. Findy(x)=g
b,,,xmwith N
m=O within the desired accuracy if
as small as possible, such that /y(x)-j(x) I<R.
Solution: Express j(x) in terms of Chebyshev
polynomials using Table 22.3,
I% lbml<R
w-N+1
References
Texts [22.14] J. Meixner, Orthogonale Polynomeysteme mit
einer besonderen Gestalt der erzeugenden Funk-
[22.1] Bibliography on orthogonal polynomials, Bull. of tion, J. London Math. Sot. 9, 6-13 (1934).
the National Research Council No. 103, Wash- [22.15] G. Sansone, Orthogonal functions, Pure and
ington, D.C. (1940). Applied Mathematics, vol. IX (Interscience
[22,2] P. L. Chebyshev, Sur l’interpolation. Oeuvres, Publishers, New York, N.Y., 1959).
vol. 2, pp. 59-68. [22.16] J. Shohat, ThQrie g&&ale des polynomes ortho-
[22.3] R. Courant and D. Hilbert, Methods of mathe- gonaux de Tchebichef, MBm. Sot. Math. 66
matical physics, vol. 1, ch. 7 (Interscience (Gauthier-Villars, Paris, France, 1934).
Publishers, New York, N.Y., 1953). [22.17] G. Szego, Orthogonal polynomials, Amer. Math.
[22.4] G. Doetsch, Die in der Statistik seltener Ereignisse Sot. Colloquium Publications 23, rev. ed. (1959).
auftretenden Charlierschen Polynome und 122.181 F. G. Tricomi, Vorlesungen tiber Orthogonalreihen,
eine damit zusammenhiingende Differential- chs. 4, 5, 6 (Springer-Verlag, Berlin, Germany,
differenzengleichung, Math. Ann. 109, 257-266 1955).
(1934). Tables
[22.5] A. ErdBlyi et al., Higher transcendental functions,
[22.19] British Association for the Advancement of Science,
vol. 2, ch. 10 (McGraw-Hill Book Co., Inc.,
Legendre Polynomials, Mathematical Tables,
New York, N.Y., 1953).
Part vol. A (Cambridge Univ. Press, Cambridge,
[22.6] L. Gatteschi, Limitazione degli errori nelle formule England, 1946). P,(z), 2=0(.01)6, n=1(1)12,
asintotiche per le funzioni speciali, Rend. Sem. 7-8D.
Mat. Univ. Torina 16,83-94 (1956-57). [22.20] N. R. Jorgensen, Undersbgelser over frekvens-
[22.7] T. L. Geronimus, Teorla ortogonalnikh mnogo- flader og korrelation (Busck, Copenhagen, Den-
chlenov (Moscow, U.S.S.R., 1950). mark, 1916). He,(z), 2=0(.01)4, n=1(1)6,
[22.8] W. Hahn, tuber Grthogonalpolynome, die q- exact.
Differenzengleichungen gentigen, Math. Nachr. [22.21] L. N. Karmazina, Tablitsy polinomov Jacobi
2, 4-34 (1949). (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R.,
[22.9] St. Kaozmarz and H. Steinhaus, Theorie der 1954). C.(p, q, z), z=O(.Ol)l, q=.l(.l)l,
Orthogonalreihen, ch. 4 (Chelsea Publishing Co., p=1.1(.1)3, n=1(1)5, 7D.
New York, N.Y., 1951). [22.22] National Bureau of Standards, Tables of Cheby-
shev polynomials S.(Z) and C,,(r), Applied
[22.10] M. Krawtchouk, Sur une g6nt%lisation des poly-
Math. Series 9 (U.S. Government Printing
nomes d’Hermite, C.R. Aced. Sci. Paris 187,
Office, Washington, D.C., 1952). 2=0(.001)2,
620-622 (1929). n=2(1)12, 12D; Coeflicients for T.(Z), U,,(z),
[22.11] C. Lanczos, Trigonometric interpolation of empir- C.(z), S,(z) for n=0(1)12.
ical and analytical functions, J. Math. Phys. [22.23] J. B. Russel, A table of Hermite functions,
17, 123-199 (1938). J. Math. Phys. 12, 291-297 (1933). eMsa”H.(z),
[22.12] C. Lanczos, Applied analysis (Prentice-Hall, x=0(.04)1(.1)4(.2)7(.5)8, n=O(l)ll, 5D.
Inc., Englewood Cliffs, N.J., 1956). [22.24] N. Wiener, Extrapolation, interpolation and
[22.13] W. Magnus and F. Oberhettinger, Formeln und smoothing of stationary time series (John Wiley
Siitze fiir die speziellen Funktionen der mathe- & Sons, Inc., New York, N.Y., 1949). L,(Z),
mat&hen Physik, ch. 5, 2d ed. (Springer- n=0(1)5, z=0(.01).1(.1)18(.2)20(.5)21(1)26(2)30,
Verlag, Berlin, Germany, 1948). 3-5D.
Coeffieients for the Jacobi Polynom,ials P$@(z) =a;’ 2 c,,,(z- 1)” Table 22.1
In=0
a. (z-1)0 (z- 1)1 (x-l)’ (z- 1)s (z-l)' (x- 1)s (z-l)‘1
p$z* P, 1 1 E
p+ b, 2 2ta+ 1) a+!9+2 4
z: -
p;a. pl 8 4((r+l), 4(a+@+W(a+% (a+8+3)r
$
p+ P, 48 8(a+l), 12(a+8+4(a+% 6(a+8+4Ma+3) (a+8+4), $
pia*8) 384 16(a+ 1)' Wa+8+5)(a+% Wa+8+5h(a+% 8(a+8+5Ma+4 (a+l9+5)4
zi
pia. P, 3840 32(a+ l)s wa+k3+6)(a+2)4 Wa+8+6Ma+3h 4O(a+8+6Ma+4)r Wa+8+6Ma+5) (a+B+Ws 2
p& 8) 46080 64(a+ l)a lQ%a+8+7)b+2)r 240(a+8+7h(a+3)4 lWa+8+7)s(a+4)a Wa+8+7)da+5h Wa+8+7)s(a+6) (a+@+716 2
(m).=m(m+l)(m+2) . . . (m+n-1) $
z:
F:*“(z)=& [(8)s(z-l)~+10(8),(6)(z-l)4+40(8),(5)~(z-l)s+8O(8),(4),(z-l)’+80(8)(3)4(z-l1)+32(2)s]
Table 22.2 Coefficients for the Ultraspherical Polynomials C:‘(Z) and for 2” in terms of C~)(Z)
20 2’ 23 39 z’ 39 I 3Y I
b. 1 2a %a)3 4(& 4(d4
--
Cc!=)
a,
1 1 1 a 3a(a+3) I 15&+4)(a+5) I cc? I
90 ) -15b)a 1 864s 90
Cp I Cp
w34 - W&
I
55s a9
I z” I %’ a9 2, d
I I I
(a)“=a(a+l)(a+2) . * * (a+n-l)
Table 22.3
Coefficients for the Chebyshev Polynomials T,(Z) and for x” in terms of T,(x)
1
T&z) = 329 - 48++18z*- 1 z+=~ [10To+15T~1+6T,+ Tel
Table 22.5
Coefficients for.the Chebyshev Polynomials U,,(s) and for X* in terms of U,(z)
20 xl 9 39 2’ 26 a? 27 zs x9 xl0 2” 2’9
----- ----- ---
b, 1 2 4 8 16 32 64 128 256 512 1024 2048 4096
-- ---~-- -----
CJO 1 1 1 2 5 14 42 132 CL,
-- __~__-____----_______- ---
Ul 2 1 .2 5 14 42 132 Ul
---__-----------_
ua -1 4 1 3 9 28 90 297 ua
-P----P-- -----
ua -4 8 1 4 14 48 165 Ua
-- ---_
U6 1 ---ppppp-p
-12 16 1 5 20 75 275 U,
------~___- ---
US 6 -32 32 1 6 27 110 U5
---- ---_________- -----
us -1 24 -80 64 1 7 35 154 u,
--_____---___--___- -----
U7 -8 80 -192 128 1 8 44 U7
----__---~___--~ ____~
ua 1 -40 240 -448 256 1 9 54 u,
-----P-----P---
us 10 -160 672 - 1024 512 1 10
------------- --us
UlO -1 60 -560 1792 - 2304 1024 1 11 UlO
---__--~ ~~-~~____-
Ull -12 280 - 1792 4608 -5120 2048 1 U11
--Y----P-- ----
Ul2 1 -84 1120 - 5376 11520 - 11264 4096 1 UN
P-----P-- -~-
:
- 0.84600
-0.73600
00000
00000
+0.80000
-0.36000
- 1.08800
00000
00000
00000
+1.20000
+0.44000
-0.67200
00000
00000
00000
+1:56000
+0.89600
00000
00000 4
3”
4 +0.54560 00000 -0.51040 00000 - 1.24640 00000 -0.12640 00000 5
5 +0.95424 00000 $0.67968 00000 -0.82368 00000 - 1.09824 00000 6
x8 20 X’O 2” 2’2
_---------
1 1 1 1 1
-~~.~~
35 126 462 Co
-~~-~
Cl 11 11 I 31 I 10 i -~-~-~
126 462 Cl
cz
-------~-
-2 1 1 4 15 56
-~~~~~
210 792 c2
C3 -3 1 1 5 21 84 330 C3
----- -___- -~~___~~
c4 2 -4 1 1 6 28 120 495 cc
-~~~~
c5*
-___----- 51 I -51 11
-___- 11 -~~~~--
36 165 C5
C6 -2 9 -6 1 1 8 45 220 Co
------~- -~~~~~
c7 -7 14 -7 1 1 9 55 c7
--------~- --~~~-
CS 2 -16 20 -8 1 1 10 66 cs
--------- ----- --------
CO 9 -30 27 -9 1 1 11 CO
-______-~~
- 10 1 1 12 Cl0
-____--~___
-11 1 1 Cl1
-~~~~-
54 -12 1 1 Cl2
---~~-----
28 ii.70 Z’O 2” x’ f
Sl 1 1 2 5 14 42 132 Sr
--------~~~~~
sz -1 1 1 3 9 28 90 297 Sl
----- ----_____---~~-
ss -2 1 1 4 14 48 165 53
------___--~~-~-
s4 1 -3 1 1 5 20 75 275 S4
---------------
S5 3 -4 1 1 6 27 110 S5
---- ---- ~-~~~~~-
S6 -1 6 -5 1 1 7 35 154 Sa
---------P----P
St -4 10 -6 1 1 8 44 St
----P-i--------
88 1 -10 15 -7 1 1 9 54 698
--P------P-----
SO 5 -20 21 -8 1 1 10 SO
so -1 15 -35 28 -9 1 1 11 SIO
-P-P--------- ~-
Sll -6 35 -56 36 -10 1 1 Sll
~--- ----___--~~~~-
Table 22.9 Coefficients for the Legendre Polynomials P,(z) and for 2“ in terms of P,(X)
- - - - --
&I ti 21 21 21~
-. -- ._ _- _. -_ -. __-
1 1 1 1 2 6 479901509 LO
-. -- _- __ _. __ -_ __-
1 1 -1 -1 -4 -13 -5743918200 4
-. -- _- _- __ __ -_ __-
-. -____
2 2
-_
-4
._
1 2
__
18
__ l44I 12001 10600
1 105849 1 1123939 1 13063880~ 183293Q99 1 2m424o99
__
31614105699 L,
__-
-
Lt
-. -~
6 6
_-
-18
.-
9
__
-1 -6
.- -96I -1299 I -14409 I -176400( -2257920 1 -39431920 I -435453999 I -5535212ooo
__
-105360352ooo 4
__-
L4 24 24 -96