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Examples of automatic asymptotic expansions

Author:

Bruno SalvyAuthors Info & Claims

ACM SIGSAM Bulletin, Volume 25, Issue 2

Pages 4 - 17

https://doi.org/10.1145/122520.122521

Published: 01 April 1991 Publication History

Abstract

We describe the current state of a Maple library, gdev, designed to perform asymptotic expansions for a large class of expressions. Many examples are provided, along with a short sketch of the underlying principles. At the time when this report is written, a striking feature of these examples is that none of them can be computed directly with any of today's most widespread symbolic computation systems (Macsyma, Mathematica, Maple or Scratchpad II).

References

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Abramowitz, M., and Stegun, I. A. Handbook of Mathematical Functions. Dover, 1973.

Digital Library

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Darboux, G. M&eacute;moire sur l'approximation des fonctions de tr&egrave;s grands nombres, et sur une classe &eacute;tendue de d&eacute;veloppements en s&eacute;rie. Journal de Math&eacute;matiques Pures et Appliqu&eacute;es (F&eacute;vrier 1878), 5--56, 377--416.

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De Bruijn, N. G. Asymptotic Methods in Analysis, third ed. North Holland, 1958. Reprinted by Dover, 1981.

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Erd&eacute;lyi, A., and Wyman, M. The asymptotic evaluation of certain integrals. Archive for Rational Mechanics and Analysis 14 (1963), 217--260.

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Hayman, W. K. A generalization of Stirling's formula. Journal f&uuml;r die reine und angewandte Mathematik 196 (1956), 67--95.

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Salvy, B. Fonctions g&eacute;n&eacute;ratrices et asymptotique automatique. Research Report 967, Institut National de Recherche en Informatique et en Automatique, 1989.

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Schmidt, H. Beitr&auml;ge zu einer Theorie der allgemeinen asymptotischen Darstellungen. Mathematische Annalen 113 (1936), 629--656.

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Stanley, R. P. Generating functions. In Studies in Combinatorics, M.A.A. Studies in Mathematics, Vol. 17. (1978), G.-C. Rota, Ed., The Mathematical Association of America, pp. 100--141.

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Cited By

Enting IClisby N(2011)Series Analysis of a Kosterlitz-Thouless Transition: The 6-State Planar Potts ModelJournal of Statistical Physics10.1007/s10955-011-0322-8145:3(696-712)Online publication date: 9-Sep-2011
https://doi.org/10.1007/s10955-011-0322-8
van Rensburg EPrellberg TRechnitzer A(2008)Partially directed paths in a wedgeJournal of Combinatorial Theory Series A10.1016/j.jcta.2007.08.003115:4(623-650)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.1016/j.jcta.2007.08.003
Zenine NBoukraa SHassani SMaillard J(2006)Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising modelJournal of Physics: Conference Series10.1088/1742-6596/42/1/02542(281-299)Online publication date: 14-Jun-2006
https://doi.org/10.1088/1742-6596/42/1/025
Show More Cited By

Index Terms

Examples of automatic asymptotic expansions
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Computer algebra systems
      1. Special-purpose algebraic systems
    2. Representation of mathematical objects
2. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Generating functions

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Published In

cover image ACM SIGSAM Bulletin

ACM SIGSAM Bulletin Volume 25, Issue 2

April 1991

33 pages

ISSN:0163-5824

DOI:10.1145/122520

Editor:
Franz Winkler
Johannes Kepler Univ., Linz, Austria

Issue’s Table of Contents

Copyright © 1991 Author.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 April 1991

Published in SIGSAM Volume 25, Issue 2

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Citations

Cited By

Enting IClisby N(2011)Series Analysis of a Kosterlitz-Thouless Transition: The 6-State Planar Potts ModelJournal of Statistical Physics10.1007/s10955-011-0322-8145:3(696-712)Online publication date: 9-Sep-2011
https://doi.org/10.1007/s10955-011-0322-8
van Rensburg EPrellberg TRechnitzer A(2008)Partially directed paths in a wedgeJournal of Combinatorial Theory Series A10.1016/j.jcta.2007.08.003115:4(623-650)Online publication date: 1-May-2008
https://dl.acm.org/doi/10.1016/j.jcta.2007.08.003
Zenine NBoukraa SHassani SMaillard J(2006)Beyond series expansions: mathematical structures for the susceptibility of the square lattice Ising modelJournal of Physics: Conference Series10.1088/1742-6596/42/1/02542(281-299)Online publication date: 14-Jun-2006
https://doi.org/10.1088/1742-6596/42/1/025
Zenine NBoukraa SHassani SMaillard J(2005) Square lattice Ising model susceptibility: connection matrices and singular behaviour of χ (3) and χ (4) Journal of Physics A: Mathematical and General10.1088/0305-4470/38/43/00438:43(9439-9474)Online publication date: 12-Oct-2005
https://doi.org/10.1088/0305-4470/38/43/004
Krattenthaler CMüller T(2004)Equations in finite semigroupsJournal of Combinatorial Theory Series A10.1016/j.jcta.2003.12.001105:2(291-334)Online publication date: 1-Feb-2004
https://dl.acm.org/doi/10.1016/j.jcta.2003.12.001
Van der Hoeven J(2002)Relax, but don't be too lazyJournal of Symbolic Computation10.1006/jsco.2002.056234:6(479-542)Online publication date: 1-Dec-2002
https://dl.acm.org/doi/10.1006/jsco.2002.0562
Salvy BShackell J(1999)Symbolic AsymptoticsJournal of Symbolic Computation10.1006/jsco.1999.028127:6(543-563)Online publication date: 1-Jun-1999
https://dl.acm.org/doi/10.1006/jsco.1999.0281
Gruntz D(1993)A new algorithm for computing asymptotic seriesProceedings of the 1993 international symposium on Symbolic and algebraic computation10.1145/164081.164136(239-244)Online publication date: 1-Aug-1993
https://dl.acm.org/doi/10.1145/164081.164136

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