×

zbMATH — the first resource for mathematics

Elementary acceleration and multisummability. I. (English) Zbl 0748.12005
The paper is devoted to the theory of meromorphic systems of linear differential equations. It is extracted from the contents of a forthcoming book by the same authors. The authors give a systematical exposition of new ideas, trends and results obtained recently by means of efforts of the French analytical school mainly. In the paper are discussed questions of “multisummability” “elementary acceleration”, “Stokes phenomena”, “Galois group” and “classification” for a linear system of meromorphic differential equations. For all questions a detailed list of references is given. It is useful to add to this list the paper by Yu. S. Il’yashenko and A. G. Khovanskij [Funkts. Anal. Prilozh. 24, No. 4, 31-42 (1990; Zbl 0719.34013)].

MSC:
12H05 Differential algebra
34M99 Ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] M. Abramowitz and I. Stegun , Handbook of Mathematical Functions , National Bureau of Standards , U.S.A., 1964 . · Zbl 0171.38503
[2] Airy , On the Intensity of Light in the Neighbourhood of a Caustic , Camb. Phil. Trans. , Vol. VI , p. 379 . · JFM 02.0797.02
[3] D.G. Babbitt and V.S. Varadarajan , Local Moduli for Meromorphic Differential Equations , Astérisque , 1980 , pp. 169 - 170 . MR 1014083 | Zbl 0683.34003 · Zbl 0683.34003
[4] N.G. Bakhoom , Asymptotic Expansions of the Function... , Proc. London Math. Soc. , 20 , 35 , 1933 , pp. 83 - 100 . Zbl 0006.30904 · Zbl 0006.30904
[5] N. Bleistein , R.A. Handelsman , and J.S. Lews , Functions Whose Fourier Transforms Decay at Infinity: an Extension of the Riemann-Lebesgue Lemma , Siam J. Math. Anal. , Vol. 3 , n^\circ 3 , 1972 , pp. 485 - 495 . MR 312143 | Zbl 0239.42027 · Zbl 0239.42027
[6] W. Balser , W. Jurkat and J. Lutz , A General Theory of Invariants For Meromorphic Differential Equations, Part. I , Funkcialaj Ekvacioj , Vol. 22 , 1979 , pp. 197 - 221 . Article | MR 556577 | Zbl 0434.34002 · Zbl 0434.34002
[7] W. Balser , W. Jurkat and D.A. Lutz , A General Theory of Invariants For Meromorphic Differential Equations, Part. II, Proper invariants , Funkcialaj Ekvacioj , Vol. 22 , 1979 , pp. 257 - 283 . Article | MR 577846 | Zbl 0473.34028 · Zbl 0473.34028
[8] G.D. Birkhoff , The Generalized Riemann Problem for Linear Differential Equations , Proc. An. Ac. A. Sci. , t. 49 , 1913 , pp. 521 - 568 . JFM 44.0391.03 · JFM 44.0391.03
[9] E. Borel , Mémoire sur les Séries Divergentes , Annales Sci. E.N.S. , 3 e série, T. 16 , 1899 , pp. 2 - 136 . Numdam | JFM 30.0230.03 · JFM 30.0230.03
[10] E. Borel , Leçons sur les Séries Divergentes , Deuxième édition, 1928 , Gauthier-Villars , Paris . · JFM 54.0223.01
[11] B. Candelpergher , Une introduction à la résurgence , Gazette des Mathématiciens, S.M.F. , n^\circ 42 , October 1989 , pp. 36 - 64 . MR 1029893 | Zbl 0825.30006 · Zbl 0825.30006
[12] D.V. Chudnovsky and G.V. Chudnovsky , Computer assisted number theory , Springer, Lect. Notes Math. , 1240 , 1987 , pp. 1 - 68 . MR 894503 | Zbl 0617.10001 · Zbl 0617.10001
[13] P. Deligne , Équations différentielles à points singuliers réguliers , Springer, Lect. Notes Math. , No. 163 , Springer-Verlag , 1970 . MR 417174 | Zbl 0244.14004 · Zbl 0244.14004
[14] P. Deligne , Catégories Tannakiennes, livre en préparation , preprint 1988 . MR 1106898
[15] P. Deligne , Letters to B. Malgrange , 8 , 1977 and 4 , 1978 .
[16] P. Deligne , Letters to J. P. Ramis , 1 , 1986 and 2 , 1986 .
[17] P. Deligne and J.S. Milne , Tannakian Categories , Springer, Lect. Notes Math. , No. 900 , Springer-Verlag , 1980 . Zbl 0477.14004 · Zbl 0477.14004
[18] R.B. Dingle , Asymptotic Expansions: their Derivation and Interpretation , Academic Press , 1973 . MR 499926 | Zbl 0279.41030 · Zbl 0279.41030
[19] A. Duval , Étude asymptotique d’une intégrale analogue à la fonction ”\Gamma modifiée”, Équations Différentielles et systèmes de Pfaff dans le champ complexe-II , R. GÉRARD and J. P. RAMIS Eds., Springer, Lect. Notes Math. , No. 1015 , 1983 , pp. 50 - 63 . MR 766242 | Zbl 0541.41035 · Zbl 0541.41035
[20] A. Duval and M. Loday-Richaud , A propos de l’algorithme de Kovacic , Preprint, Orsay , 1989 .
[21] A. Duval and C. Mitschi , Matrices de Stokes et groupes de Galois des équations hypergéométriques confluentes généralisées , Pac. J. Math. , Vol. 138 , 1 , 1989 , pp. 25 - 56 . Article | MR 992173 | Zbl 0705.34068 · Zbl 0705.34068
[22] J. Ecalle , Les Fonctions Résurgentes , T. I , Publications Mathématiques d’Orsay , 1981 . Article
[23] J. Ecalle , Les Fonctions Résurgentes , T. II , Publications Mathématiques d’Orsay , 1981 . Article
[24] J. Ecalle , Les Fonctions Résurgentes , T. III , Publications Mathématiques d’Orsay , 1985 . Article | Zbl 0602.30029 · Zbl 0602.30029
[25] J. Ecalle , L’accélération des fonctions résurgentes , manuscrit, 1987 .
[26] J. Ecalle , Introduction à l’accélération et à ses applications , book submited to Travaux en Cours, 1990 .
[27] J. Ecalle , Finitude des cycles limites et accélérosommabilité de l’application de retour, proceedings of a Luminy Conference , (to appear), 1990 . MR 1094378 | Zbl 0729.34016 · Zbl 0729.34016
[28] J. Ecalle , La conjecture de Dulac : preuve constructive , book submited to Travaux en Cours, 1990 .
[29] J. Ecalle , Résurgence et accélération. Applications à l’analyse et à la géométrie , books in preparation, 1991 .
[30] I. Kaplansky , An introduction to Differential Algebra , Hermann , Paris , 1957 . MR 93654 | Zbl 0083.03301 · Zbl 0083.03301
[31] H. Faxen , Expansion in series of the integral, ... , Ark. Math. Astronom. Fys. , Vol. 15 , n^\circ 13 , 1921 , pp. 1 - 57 . JFM 48.0353.01 · JFM 48.0353.01
[32] W. Feller , An Introduction to Probability Theory and its Applications , Vol. II , John Wiley . MR 270403 · Zbl 0077.12201
[33] G.M. Hardy and J.E. Littlewood , Some Problems of ”Partitio Numerorum”; I. A New Solutions of Waring’s Problem , Wiss. Nachrichten Math-Phys. Klasse , 1920 . Article | JFM 47.0114.02 · JFM 47.0114.02
[34] W.B. Jurkat , Meromorphe Differentialgleichungen , Springer Lect. Notes Math. , No. 637 , Springer-Verlag Berlin , 1978 . MR 494886 | Zbl 0408.34004 · Zbl 0408.34004
[35] N.M. Katz , On the Calculation of Some Differential Galois Groups , Invent. Math. , Vol. 87 , 1987 . MR 862711 | Zbl 0609.12025 · Zbl 0609.12025
[36] N.M. Katz , An Overview of Deligne’s Work on Hilbert’s Twenty-First Problem , Mathematical Developments Arising From Hilbert Problems, Am. Math. Soc. , Providence , 1976 . Zbl 0347.14010 · Zbl 0347.14010
[37] N.M. Katz , Exponential Sums and Differential Equations , book to appear, preprint, spring, 1989 . MR 1081536
[38] N.M. Katz , Exponential Sums Over Finite Fields and Differential Equations Over the Complex Numbers: Some Interactions , A.M.S. Winter meeting, 1989 . MR 1032857
[39] E.R. Kolchin , Differential Algebra and Algebraic Groups , Academic Press , New York , 1973 . MR 568864 | Zbl 0264.12102 · Zbl 0264.12102
[40] J. Kovacic , An Algorithm for Solving Second Order Linear Homogeneous Differential Equations , J. Symb. Comp. , Vol. 1 , 1986 . MR 839134 | Zbl 0603.68035 · Zbl 0603.68035
[41] E. Leroy , Sur les séries divergentes et les fonctions définies par un développement de Taylor , Ann. Fac. Université de Toulouse , 1900 , p. 317 - 430 . Numdam | JFM 31.0256.01 · JFM 31.0256.01
[42] C.H. Lin , Phragmén-Lindelof theorem in a cohomological form , University of Minnesota Math . reports, 1982 , p. 81 - 151 ; Proc. Am. Math. Soc. , Vol. 89 , 4 , 1983 , pp. 589 - 597 . MR 718979 | Zbl 0575.30002 · Zbl 0575.30002
[43] M. Loday-Richaud , Thèse , en préparation. · Zbl 0523.34006
[44] M. Loday-Richaud , Calcul des invariants de Birkhoff des systèmes d’ordre deux , Funkcialaj Ekvacioj (to appear). Article | Zbl 0719.34016 · Zbl 0719.34016
[45] M. Loday-Richaud , Introduction à la multisommabilité , Gazette des Mathématiciens, S.M.F. , April 1990 . MR 1060425 · Zbl 0722.34005
[46] Y.L. Luke , The Special Functions and their Approximations , Vol. 1 , Academic Press , 1969 . Zbl 0193.01701 · Zbl 0193.01701
[47] W. Magnus , F. Oberhettinger and R.P. Soni , Formulas and Theorems for the Special Functions of Mathematical Physics , Springer-Verlag , Berlin , 1966 . MR 232968 | Zbl 0143.08502 · Zbl 0143.08502
[48] B. Malgrange , Sur les points singuliers des équations différentielles , Enseign. Math. , Vol. 20 , 1974 , pp. 147 - 176 . MR 368074 | Zbl 0299.34011 · Zbl 0299.34011
[49] B. Malgrange , Sur la reduction formelle des équations à singularités irrégulières , preprint, Grenoble , 1979 .
[50] B. Malgrange , Remarques sur les équations différentielles à points singuliers , in ” Équations Différentielles et Systèmes de Pfaff dans le champ complexe ”, R. GÉRARD and J. P. RAMIS Eds., Springer , Lect. Notes Math. , No. 712 , Springer-Verlag , 1979 . Zbl 0423.32014 · Zbl 0423.32014
[51] B. Malgrange , La classification des connexions irrégulières à une variable, ”Mathématique et Physique” , Sém. École Normale Sup. , 1979 - 1982 , Birkhäuser , 1983 . MR 728430 | Zbl 0531.58015 · Zbl 0531.58015
[52] B. Malgrange , book in preparation, Birkhäuser .
[53] B. Malgrange , Modules microdifférentiels et classes de Gevrey . Zbl 0468.46026 · Zbl 0468.46026
[54] B. Malgrange , Letter to J. P. Ramis , 2 , 1985 .
[55] B. Malgrange , Introduction aux travaux de J. Ecalle , Enseign. Math. , Vol. 31 , 1985 , pp. 261 - 282 . MR 819354 | Zbl 0601.58043 · Zbl 0601.58043
[56] B. Malgrange , Équations différentielles linéaires et transformation de Fourier : une introduction , Conférences à l’I.M.P.A., Rio de Janeiro , 1988 . · Zbl 1202.34001
[57] B. Malgrange , Sur le théorème de Maillet , Asymptotic Anal. , Vol. 2 , 1989 , pp. 1 - 4 . MR 991413 | Zbl 0693.34004 · Zbl 0693.34004
[58] J. Martinet and J.P. Ramis , Problèmes de modules pour des équations différentielles non linéaires du premier ordre , Publ. Math. I.H.E.S. , Vol. 55 , 1982 , pp. 64 - 164 . Numdam | MR 672182 | Zbl 0546.58038 · Zbl 0546.58038
[59] J. Martinet and J.P. Ramis , Théorie de Galois Différentielle et Resommation , Computer Algebra and Differential Equations , E. TOURNIER Ed., Academic Press , 1989 , pp. 117 - 214 . MR 1038060 | Zbl 0722.12007 · Zbl 0722.12007
[60] J. Martinet and J.P. Ramis , Théorie de Cauchy Sauvage , book in preparation.
[61] R.E. Meyer , A simple explanation of Stokes phenomenon , Siam. Rev. , Vol. 31 , 1989 , pp. 435 - 445 . MR 1012299 | Zbl 0679.34068 · Zbl 0679.34068
[62] C. Mitschi , Groupes de Galois différentiels et G-fonctions , Thèse , Strasbourg , 1989 . · Zbl 0695.12017
[63] C. Mitschi , Groupes de Galois différentiels de certaines équations hypergéométriques généralisées d’ordre 7 , preprint, I.R.M.A. , Strasbourg , 1989 . MR 1006733
[64] C. Mitschi , Computing Galois differential groups of generalized hypergeometric differential equations , Proceedings CADE 2 , 1990 , Academic Press , (to appear). MR 1115233 · Zbl 0883.12004
[65] Nevanlinna , Zur Theorie der Asymptotischen Potenzreihen , Ann. Acad. Sci. Fennicae , ser. A , From XII, 1919 , pp. 1 - 81 . JFM 46.1463.01 · JFM 46.1463.01
[66] F.W.J. Olver , Asymptotics and Special Functions , Academic Press , 1974 . MR 435697 | Zbl 0303.41035 · Zbl 0303.41035
[67] H. Poincaré , Sur les groupes des équations linéaires , Acta. Math. , Vol. 5 , 1884 , pp. 240 - 278 . JFM 16.0252.01 · JFM 16.0252.01
[68] J.P. Ramis , Devissage Gevrey , Astérisque , Vol. 59 - 60 , 1978 , pp. 173 - 204 . MR 542737 | Zbl 0409.34018 · Zbl 0409.34018
[69] J.P. Ramis , Les séries k-sommables et leurs applications , Analysis, Microlocal Calculus and Relativistic Quantum Theory, Proceedings ”Les Houches” , 1979 , Springer , Lect. Notes Phys. , No. 126 , 1980 , pp. 178 - 199 . MR 579749 · Zbl 1251.32008
[70] J.P. Ramis , Phénomène de Stokes et filtration Gevrey sur le groupe de Picard-Vessiot , C. R. Acad. Sci. Paris , T. 301 , 1985 , pp. 165 - 167 . MR 801953 | Zbl 0593.12015 · Zbl 0593.12015
[71] J.P. Ramis , Phénomène de Stokes et resommation , C. R. Acad. Sci. Paris , T. 301 , 1985 , pp. 99 - 102 . MR 799602 | Zbl 0582.34006 · Zbl 0582.34006
[72] J.P. Ramis , Filtration Gevrey sur le groupe de Picard-Vessiot d’une équation différentielle irrégulière , Preprint Instituto de Matematica Pura e Aplicada, I.M.P.A. , Rio de Janeiro , Vol. 45 , 1985 , pp. 1 - 38 . MR 801953
[73] J.P. Ramis , Irregular Connections, wild \pi 1 and confluence , Proceedings of a conference at Katata (Japan) , 1987 , Taniguchi Fundation (to appear).
[74] J.P. Ramis , Théorèmes d’indices Gevrey pour les équations différentielles ordinaires , Memoirs of the Am. Math. Soc. , Vol. 296 , 1984 , pp. 1 - 95 . MR 733946 | Zbl 0555.47020 · Zbl 0555.47020
[75] J.P. Ramis , Un problème inverse en théorie de Galois différentielle , Preprint, I.R.M.A. , Strasbourg , 1989 .
[76] J.P. Ramis and Y. Sibuya , Hukuhara’s domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type , Asymptotics , Vol. 2 , 1989 , pp. 39 - 94 . MR 991416 | Zbl 0699.34058 · Zbl 0699.34058
[77] J.P. Ramis and Y. Sibuya , Asymptotic expansions with Gevrey estimates and cohomological methods , book in preparation.
[78] N. Saavedra , Catégories Tannakiennes , Springer , Lecture Notes Math. , No. 265 , 1972 . MR 338002 · Zbl 0241.14008
[79] L. Schlesinger , Handbuch der Theorie der linearen Differentialgleichungen , Teubner , Leipzig , 1895 . JFM 26.0329.01 · JFM 26.0329.01
[80] J.P. Serre , Lie Algebras and Lie Groups, 1964 Lectures Given at Harvard University , W. A. Benjamin Inc. , New York , Amsterdam , 1966 . Zbl 0132.27803 · Zbl 0132.27803
[81] Y. Sibuya , Stokes Phenomena , Bull. Am. Math. Soc. , Vol. 83 , 1977 , pp. 1075 - 1077 . Article | MR 442337 | Zbl 0386.34008 · Zbl 0386.34008
[82] G.G. Stokes , On the Discontinuity of Arbitrary Constants which Appear in Divergent Developments , Trans. Cambridge Phil. Soc. , Vol. X , 1857 , pp. 106 - 128 .
[83] J. Thomann , Resommation des séries formelles solutions d’équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières , Numer. Math. (to appear). Article | MR 1080304 · Zbl 0701.30002
[84] E. Tournier , Solutions formelles d’équations différentielles. Le logiciel de calcul formel : DESIR. Étude théorique et réalisation , Thèse , Grenoble , 1987 .
[85] C. Tretkoff and M. Tretkoff , Solution of the Inverse Problem of Differential Galois Theory in the Classical Case , Am. J. Math. , T. 101 , 1979 , pp. 1327 - 1332 . MR 548884 | Zbl 0423.12021 · Zbl 0423.12021
[86] H.L. Turrittin , Convergent Solutions of Ordinary Linear Homogeneous Differential Equations in the Neighbourhood of an Irregular Singular Point , Acta Math. , Vol. 93 , 1955 , pp. 27 - 66 . MR 68689 | Zbl 0064.33603 · Zbl 0064.33603
[87] W. Wasow , Asymptotic Expansions for Ordinary Differential Equations , Interscience Publishers , New York , 1965 , Reprint Dover Publ. , New York , 1987 . MR 919406 | Zbl 0133.35301 · Zbl 0133.35301
[88] G.N. Watson , A Theory of Asymptotic Series, Philosophical , Trans. R. Soc. London , Ser. A , Vol. CCXI , 1911 , pp. 279 - 313 . JFM 42.0273.01 · JFM 42.0273.01
[89] G.N. Watson , The Characteristics of Asymptotics Series , Quaterly J. Math. , Vol. XLIII , 1912 , pp. 65 - 77 . JFM 42.0273.02 · JFM 42.0273.02
[90] E.T. Whittaker and G.N. Watson , A Course of Modern Analysis , Cambridge University Press , 1902 . MR 1424469 · JFM 33.0390.01
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.