Examples of automatic asymptotic expansions

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).

[1]  M. Wyman.  The Asymptotic Behaviour of the Laurent Coefficients , 1959, Canadian Journal of Mathematics.

[2]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[3]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[4]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[5]  M. Wyman.  The Asymptotic Behavior of the Laurent Coefficients , 1959 .

[6]  B. Harris,et al.  Asymptotic expansions for the coefficients of analytic functions , 1968 .

[7]  N. Bleistein,et al.  Asymptotic Expansions of Integrals , 1975 .

[8]  Philippe Flajolet,et al.  Lambda - Upsilon - Omega: An Assistant Algorithms Analyzer , 1988, AAECC.

[9]  E. Capelas de Oliveira.  On generating functions , 1992 .

[10]  G. Darboux,et al.  Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série. , 1878 .

[11]  B. Salvy.  Fonctions generatrices et asymptotique automatique , 1989 .

[12]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[13]  W. Hayman.  A Generalisation of Stirling's Formula. , 1956 .

[14]  A. Odlyzko,et al.  Asymptotic expansions for the coefficients of analytic generating functions , 1985 .

[15]  Arthur Erdélyi,et al.  The asymptotic evaluation of certain integrals , 1963 .

[16]  Reinwald Jungen.  Sur les séries de Taylor n'ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence , 1931 .

[17]  Daniel Richardson,et al.  Some undecidable problems involving elementary functions of a real variable , 1969, Journal of Symbolic Logic.

[18]  H. Schmidt.  Beiträge zu einer Theorie der allgemeinen asymptotischen Darstellungen , 1937 .