zbMATH — the first resource for mathematics

On the coefficients that arise from Laplace’s method. (English) Zbl 1096.41008
The author studies the extensions to Erdélyi’s theorem on many types of integrals including the Laplace transform. It’s essential in the method of steepest descent. He shows that the coefficients in any particular application and clarify the theoretical basis of Erdélyi’s expansion: it turns out that Faà di Bruno’s formula has always played a central role in it. He studies an example and gives an open problem about this theme.

41A10 Approximation by polynomials
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
[1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth ed., Dover, New York, 1972. · Zbl 0543.33001
[2] Bashkin, E.P.; Wojdylo, J., Dimerization of \({}^3\operatorname{He}\) in \({}^3\operatorname{He}\)-\({}^4\operatorname{He}\) dilute mixtures filling narrow channels, Phys. rev. B, 62, 10, 6614-6628, (2000)
[3] Comtet, L., Advanced combinatorics, the art of finite and infinite expansions, (1974), D. Reidel Dordrecht, Holland
[4] Copson, E.T., An introduction to the theory of functions of a complex variable, (1935), Oxford University Press London · Zbl 0012.16902
[5] Erdélyi, A., Asymptotic expansions, (1956), Dover New York · Zbl 0070.29002
[6] Erdélyi, A., General asymptotic expansions of Laplace integrals, Arch. rational mech. anal., 7, 1-20, (1961) · Zbl 0097.08802
[7] L. Euler, Introductio in Analysin Infinitorum, vol. 1, Lausanne, Bousquet, 1748 (English translation by John Blanton, Introduction to Analysis of the Infinite, Book 1, Springer, New York, 1988). · Zbl 0657.01013
[8] Fabijonas, B., Laplace’s method on a computer algebra system with an application to the real valued modified Bessel functions, J. comput. appl. math., 146, 2, 323-342, (2002) · Zbl 1013.65015
[9] Fabijonas, B.R.; Olver, F.W.J., On the reversion of an asymptotic expansion and the zeros of the Airy functions, SIAM rev., 41, 4, 762-773, (1999) · Zbl 1053.33003
[10] Henrici, P., Automatic computations with power series, J. assoc. comput. Mach., 3, 10-15, (1956)
[11] van der Hoeven, J., Relax, but don’t be too lazy, J. symbolic comput., 34, 479-542, (2002) · Zbl 1011.68189
[12] Johnson, W., The curious history of faà di Bruno’s formula, MAA monthly, 109, 217-234, (2002) · Zbl 1024.01010
[13] D. Knuth, The Art of Computer Programming, third ed., vol. 2, Addison-Wesley, Reading, MA, 1998. · Zbl 0895.65001
[14] J.-L. Lagrange, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries, Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin, vol. 24, 1768, pp. 251-326 (Reprinted, with changes, in: M.J.-A. Serret, Oeuvres de Lagrange publiées par le soins de, vol. III, pp. 4-73, Gauthier-Villars, Paris, 1869, and Georg Olms Verlag, Hildesheime, 1973).
[15] P.-S. Laplace, Théorie Analytique des Probabilités, vol. 1, Courcier, Paris, 1820 (Reprinted in Complete Works, vol. 7, Gauthier-Villars, Paris, 1886).
[16] Morse, P.M.; Feshbach, H., Methods of theoretical physics, part I, (1953), McGraw-Hill New York · Zbl 0051.40603
[17] Olver, F.W.J., Asymptotics and special functions, (1974), Academic Press New York, (Reprinted by A.K. Peters, Wellesley, MA, 1997) · Zbl 0303.41035
[18] Riordan, J., Combinatorial identities, (1968), Wiley New York · Zbl 0194.00502
[19] Temme, N.M., Asymptotic estimates of Stirling numbers, Stud. appl. math., 89, 233-243, (1993) · Zbl 0784.11007
[20] Watson, G.N., Harmonic functions associated with the parabolic cylinder, Proc. London math. soc., 17, 116-148, (1918) · JFM 46.0576.04
[21] J. Wojdylo, Computing the coefficients in Laplace’s method, SIAM Rev., 2006, to be published. · Zbl 1091.41026
[22] Wong, R., Asymptotic approximations of integrals, (1989), Academic Press New York, (Reprinted by SIAM, Philadelphia, 2001) · Zbl 0679.41001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.