logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Eric Delabaere; Jean-Marc Rasoamanana
Sommation effective d’une somme de Borel par séries de factorielles
(Effective Borel-resummation by factorial series)
Annales de l'institut Fourier, 57 no. 2 (2007), p. 421-456, doi: 10.5802/aif.2263
Article PDF | Reviews MR 2310946 | Zbl 1129.30023 | 1 citation in Cedram
Class. Math.: 30E15, 40Gxx
Keywords: Borel-resummation, factorial series.

Résumé - Abstract

In this article, we consider the effective resummation of a Borel sum by its associated factorial series expansion. Our approach provides concrete estimates for the remainder term when truncating this factorial series. We then generalize a theorem of Nevanlinna which gives us the natural framework to extend the factorial series method for Borel-resummable fractional power series expansions.

Bibliography

[1] W. Balser, D. A. Lutz & R. Schäfke, “On the convergence of Borel approximants”, J. Dynam. Control Systems 8 (2002) no. 1, p. 65-92 Article |  MR 1874704 |  Zbl 1029.34074
[2] M. V. Berry & C. J. Howls, “Hyperasymptotics for integrals with saddles”, Proc. Roy. Soc. London Ser. A 434 (1991) no. 1892, p. 657-675 Article |  MR 1126872 |  Zbl 0764.30031
[3] M. Canalis-Durand, “Solutions Gevrey d’équations différentielles singulièrement perturbées”, Thèse d’habilitation à diriger des recherches, 1999
[4] B. Candelpergher, J.-C. Nosmas & F. Pham, Approche de la résurgence, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1993  MR 1250603 |  Zbl 0791.32001
[5] Louis Comtet, Advanced combinatorics, D. Reidel Publishing Co., Dordrecht, 1974, The art of finite and infinite expansions  MR 460128 |  Zbl 0283.05001
[6] E. Delabaere, “Effective resummation methods for an implicit resurgent function”, submitted, 2006 arXiv
[7] E. Delabaere & C. J. Howls, “Global asymptotics for multiple integrals with boundaries”, Duke Math. J. 112 (2002) no. 2, p. 199-264 Article |  MR 1894360 |  Zbl 1060.30049
[8] E. Delabaere & J.-M. Rasoamanana, “Resurgent deformations for an ODE of order 2”, Pacific Journal of Mathematics 223 (2006) no. 1, p. 35-93 Article |  MR 2221018 |  Zbl 05129571
[9] Eric Delabaere, Introduction to the Écalle theory, Computer algebra and differential equations (1992), London Math. Soc. Lecture Note Ser. 193, Cambridge Univ. Press, 1994, p. 59–101  MR 1278057 |  Zbl 0805.40007
[10] Eric Delabaere & Frédéric Pham, “Resurgent methods in semi-classical asymptotics”, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999) no. 1, p. 1-94 Numdam |  MR 1704654 |  Zbl 0977.34053
[11] R. B. Dingle, Asymptotic expansions : their derivation and interpretation, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York,, 1973  MR 499926 |  Zbl 0279.41030
[12] Jean Écalle, Les fonctions résurgentes. Tome I, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81] 5, Université de Paris-Sud Département de Mathématique, Orsay, 1981, Les algèbres de fonctions résurgentes. [The algebras of resurgent functions], With an English foreword Article |  MR 670417 |  Zbl 0499.30034
[13] Jean Écalle, Les fonctions résurgentes. Tome II, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81] 6, Université de Paris-Sud Département de Mathématique, Orsay, 1981, Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration] Article |  MR 670418 |  Zbl 0499.30035
[14] Jean Écalle, Les fonctions résurgentes. Tome III, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay] 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985, L’équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects] Article |  MR 852210 |  Zbl 0602.30029
[15] William Feller, An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons Inc., New York, 1968  MR 228020 |  Zbl 0077.12201
[16] U. Jentschura, “Quantum Electrodynamic, Bound-State Calculations and Large-Order Perturbation Theory”, Habilitation Thesis, Dresden University of Technology, 3rd edition, 2004
[17] Bernard Malgrange, “Sommation des séries divergentes”, Exposition. Math. 13 (1995) no. 2-3, p. 163-222  MR 1346201 |  Zbl 0836.40004
[18] F. Nevanlinna, Zur Theorie der Asymptotischen Potenzreihen, Suomalaisen Tiedeakatemian Kustantama, Helsinki, 1918  JFM 46.1463.01
[19] N.E. Nörlund, Leçons sur les Séries d’Interpolation, Gautier-Villars, Paris, 1926  JFM 52.0301.04
[20] A. B. Olde Daalhuis, “Hyperterminants. I”, J. Comput. Appl. Math. 76 (1996) no. 1-2, p. 255-264 Article |  MR 1423521 |  Zbl 0866.65011
[21] A. B. Olde Daalhuis, “Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one”, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998) no. 1968, p. 1-29 Article |  MR 1632891 |  Zbl 0919.34012
[22] A. B. Olde Daalhuis, “Hyperterminants. II”, J. Comput. Appl. Math. 89 (1998) no. 1, p. 87-95 Article |  MR 1625975 |  Zbl 0910.34014
[23] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Tome I, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Librairie Scientifique et Technique Albert Blanchard, Paris, 1987, Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotiques. [Periodic solutions. Nonexistence of uniform integrals. Asymptotic solutions], Reprint of the 1892 original, With a foreword by J. Kovalevsky, Bibliothèque Scientifique Albert Blanchard. [Albert Blanchard Scientific Library]  MR 926906 |  Zbl 0651.70002
[24] Jean-Pierre Ramis & Reinhard Schäfke, “Gevrey separation of fast and slow variables”, Nonlinearity 9 (1996) no. 2, p. 353-384 Article |  MR 1384480 |  Zbl 0925.70161
[25] B. Simon, “Large orders and summability of eigenvalue perturbation theory : a mathematical overview.”, International Journal of Quantum Chemistry XXI (1982), p. 3-25 Article
[26] G.G. Stokes, “On the Discontinuity of arbitrary constants which appear in divergent developments”, Transactions of the Cambridge Philosophical Society X (1857)
[27] Jean Thomann, “Resommation des series formelles. Solutions d’équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières”, Numer. Math. 58 (1990) no. 5, p. 503-535 Article |  MR 1080304 |  Zbl 0715.30001
[28] Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965  MR 203188 |  Zbl 0133.35301
[29] G.N. Watson, “The transformation of an asymptotic series into a convergent series of inverse factorials”, Cir. Mat. Palermo 34 (1912), p. 41-88 Article |  JFM 43.0314.02
top