ECE 6382   z

y
x

z0

Fall 2020
David R. Jackson

Notes 14
The Gamma Function

Notes are from D. R. Wilton, Dept. of ECE

1
 The Gamma Function

 The Gamma function appears in many expressions,

including Bessel functions, etc.

 It generalizes the factorial function n! to non-integer values

and even complex values.

2
 Definition 1

Definition # 1

1  2  3 n
( z )  lim nz, z  0, 1, 2,
n  z ( z  1)( z  2)  ( z  n )

This definition gives the Gamma function a nice property, as shown on

the next slide:

( n )   n  1 !

3
 Definition 1
Factorial property:

1  2  3 n
( z )  lim n z , z  0, 1, 2,
n  z ( z  1)( z  2)  ( z  n )

1  2  3 n nz
( z  1)  lim n z 1  ( z ) lim
n  ( z  1)( z  2)  ( z  n  1) n  ( z  n  1)

 ( z  1)  z( z )

1  2  3 n
Note that (1)  lim n  1, and (2)  1  (1)  1,
n  1  2  3 n (n  1)
(3)  2  (2)  2  1, (4)  3  (3)  3  2  1, (5)  4  (4)  4  3  2  1, etc.

Hence (n )   n  1 ! or (n  1)  n !

4
 Definition 2
Definition # 2

 t z 1
( z )   t dt, Re z  0
0
e

This is the Euler-integral form of the definition.

Note:
Leonard Euler
t z 1  t x 1 t iy  t x 1  e 
ln t iy
 t x 1  eiy ln t 
 t z 1  t x 1  x  0 for integral to converge

Note:
Definition 1 is the analytic continuation of definition 2 from the right-half
plane into the entire complex plane (except at the negative integers).
5
 Equivalent Integral Forms

The following three integral definitions are all equivalent :


 t z 1
( z )   t dt,
0
e Re z  0


 s2 2 z 1
( z )  2  e s ds, Re z  0 (let t  s 2 )
0
1 z 1
 1
( z )  0  ln s  ds, Re z  0 (let t  ln  1 / s  )

6
 Equivalence of Definitions 1 and 2
Equivalence of definitions #1 and #2

n
t  t
Use e  lim  1   ,
n 
 n
n n 
 t
Define F ( z, n )    1   t z 1dt;  t dt  2  z 
 t z 1
F ( z, n ) 
 e
0 n n  0

t
Letting w  and integrating by parts n times,
n
    Factor   appearing
   in Definition
  #1    
1
n
F ( z , n )  n z   1  w  w z 1dw  n z
1  2  3  n  1 n 1
z  n 1

0 z ( z  1)( z  2) ( z  n  1) 0  w dw
  
1
Hence lim F ( z, n )  1 ( z ) (Please see next slide.) z n
n 

7
Equivalence of Definitions 1 and 2 (cont.)
Integration by parts development:
1
n
 1  w
z 1
I  w
 dw
0 u
dv
dw

Integrate by parts once:

z 1 1
n w n 1 wz
I   1  w   n  1  w dw
z 0 0 z
1
n 1 wz
 0   n  1  w dw
0 z
1
n n 1
   1  w  w z dw
z0

8
Equivalence of Definitions 1 and 2 (cont.)
1
n
Integrate by parts twice: I   1  w  w z 1
0     dv dw
1 u
n n 1
I    1  w  w z dw
dw

z0
1
z 1
w z 1
1
n n 1 w n n 2
  1  w    n  1  1  w   1 dw
z z 1 0 z 0 z 1
n  n  1 1
n 2
 0
z  z  1   1  w
0
w z 1dw

n  n  1 1
n 2

z  z  1   1  w
0
w z 1dw

After n times:
n  n  1  n  2   3  2 1 1

  1  w wz  n1dw
nn
I
z ( z  1)( z  2) ( z  n  1) 0
9
 Definition 3

Definition # 3

The Weierstrass product form can be shown to be equivalent

to definitions #1 and #2.

1 
 z   nz
 ze   1   e
z
( z ) n 1  n

where   0.5772156619 is the Euler - Mascheroni constant.

10
 Euler Reflection Formula
Euler Reflection Formula


( z ) (1  z ) 
sin  z

Note: We can use this along with definition #2 to find (z) for Re(z) < 0.
 1
( z )  , Re z  0, z  0, 1, 2,
sin  z (1  z )

y
Geometric interpretation of reflection formula:
1 z z The two points are reflections about the x = 1/2 line.

x
x 1/ 2
11
Euler Reflection Formula (cont.)
A special result that occurs frequently is (1/2).

To calculate this, use the reflection formula:


( z )(1  z ) 
sin  z

Set z = 1/2:

(1 / 2)  

12
 Pole Behavior
Simple poles are at n = 0, -1, -2, -3,…

Recall: ( z  1)  z( z ) & (1)  1

Use Simple pole at z = 0

Residue = 1
( z  1)
( z  1)  z( z )  ( z ) 
z

( z  2)
( z  2)   z  1 ( z  1)  ( z  1) 
z 1
( z  2)
 z ( z ) 
z 1 Simple pole at z = -1
Residue = -1
( z  2)
 ( z ) 
z  z  1

13
 Pole Behavior (cont.)

( z  3)
( z  3)   z  2  ( z  2)  ( z  2) 
z2
Simple pole at z = -2
( z  3)
  z  1  z  ( z )  Residue = +1/2
z2
( z  3)
 ( z ) 
z  z  1  z  2 

( z  4)
( z  4)   z  3 ( z  3)  ( z  3) 
z3
( z  4)
  z  2   z  1  z  ( z )  Simple pole at z = -3
z3 Residue = -1/6
( z  4)
 ( z ) 
z  z  1  z  2   z  3

14
 Pole Behavior (cont.)
Residues at Poles

In general, we will have:

Simple pole at z = -n

( z  n  1)
( z ) 
z  z  1  z  2   z  3   z  n 

 1 
Res   n   Lim  z  n   
z  n
 z  z  1  z  2   z  3   z  n  
1

z  z  1  z  2   z  3   z  n  1 z  n Hence
1

 1
n
 n   n  1   3  2   1
Res   n  
 1
n

 n!
 n   n  1   3  2   1
15
 The Gamma Function (cont.)

  z
y
x

z0

 1
n

Note: There are simple poles at z = 0, -1, -2,… Res   n  

n!
16
The Gamma Function (cont.)

(x) and 1 / (x)

Note: (x) never goes to zero.

In fact, 1 / (z) is analytic everywhere.

17
 The Gamma Function (cont.)
Sterling’s formula (asymptotic series for large argument):

 
1 z z  1 1 139 571 
  z   2 z e 1   2
 3
 4
  
z 12 z 288 z 51840 z 2488320
                   z
 w 
Taking the ln of both sides, we also have
1  z   1 1 1 
ln   z   z ln z  z  ln  
      
2  2 3
  12 z 360 z 1260 z
5

w 2 w3
Note : ln  1  w   w   
2 3

Valid for

z  , arg  z   constant
18
The Gamma Function (cont.)
Summary of Factorial Generalization

n !  n  n  1  n  2    3  2   1 Integers


x !    x  1   e  t t x dt Real numbers
0
x  1


z !    z  1   e  t t z dt Complex numbers

0
Re  z   1

19
The Gamma Function (cont.)
Summary of Factorial Generalization (cont.)


z !    z  1   e  t t z dt  Re z  1
0
Complex numbers
+ z  0, 1, 2
 1
( z ) 
sin  z (1  z )

20