Error bounds, duality, and the Stokes phenomenon. I

Gurariĭ, V.

doi:10.1090/S1061-0022-2010-01125-7

Error bounds, duality, and the Stokes phenomenon. I
HTML articles powered by AMS MathViewer

by V. P. Gurariĭ

St. Petersburg Math. J. 21 (2010), 903-956

DOI: https://doi.org/10.1090/S1061-0022-2010-01125-7

Published electronically: September 23, 2010

PDF | Request permission

Abstract:

We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function in such a class by partial sums of its expansion, we study how the accuracy changes when we move within a given region of the complex plane. Analysis of these changes allows us to propose a theory of divergent expansions, which includes a duality theorem and the Stokes phenomenon as essential parts. In its turn, this enables us to formulate necessary and sufficient conditions for a particular divergent expansion to encounter the Stokes phenomenon. We derive explicit expressions for the exponentially small terms that appear upon crossing Stokes lines and lead to an improvement in the accuracy of the expansion.

References

R. E. Meyer, A simple explanation of the Stokes phenomenon, SIAM Rev. 31 (1989), no. 3, 435–445. MR 1012299, DOI 10.1137/1031090
M. V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 211–221 (1989). MR 1001456
M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. London Ser. A 422 (1989), no. 1862, 7–21. MR 990851
M. V. Berry, Infinitely many Stokes smoothings in the gamma function, Proc. Roy. Soc. London Ser. A 434 (1991), no. 1891, 465–472. MR 1121933, DOI 10.1098/rspa.1991.0106
Michael Berry, Asymptotics, superasymptotics, hyperasymptotics$\ldots$, Asymptotics beyond all orders (La Jolla, CA, 1991) NATO Adv. Sci. Inst. Ser. B: Phys., vol. 284, Plenum, New York, 1991, pp. 1–14. MR 1210328
R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes integrals, Encyclopedia of Mathematics and its Applications, vol. 85, Cambridge University Press, Cambridge, 2001. MR 1854469, DOI 10.1017/CBO9780511546662
Jean-Pierre Ramis, Séries divergentes et théories asymptotiques, Bull. Soc. Math. France 121 (1993), no. Panoramas et Synthèses, suppl., 74 (French). MR 1272100
Jean-Pierre Ramis, Stokes phenomenon: historical background, The Stokes phenomenon and Hilbert’s 16th problem (Groningen, 1995) World Sci. Publ., River Edge, NJ, 1996, pp. 1–5. MR 1443685
R. B. Paris and A. D. Wood, Stokes phenomenon demystified, Bull. Inst. Math. Appl. 31 (1995), no. 1-2, 21–28. MR 1323501
G. G. Stokes, On the numerical calculation of a class of definite integrals and infinite series, Trans. Camb. Philos. Soc. 9 (1850), 166–187; On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Camb. Philos. Soc. 10 (1857), 106–128; Memoir and scientific correspondence, selected and arranged by J. Larmor, Vol. II, Cambridge Univ. Press, London–New York, 1907, pp. 159–160.
V. L. Pokrovskiĭ and I. M. Khalatnikov, On the problem of above-barrier reflection of high-energy particles, Zh. Èksper. Teoret. Fiz. 40 (1961), no. 6, 1713–1719; English transl., Soviet Phys. JETP 13 (1961), 1207–1210.
John P. Boyd, The devil’s invention: asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math. 56 (1999), no. 1, 1–98. MR 1698036, DOI 10.1023/A:1006145903624
V. P. Gurariĭ, Group methods of commutative harmonic analysis, Current problems in mathematics. Fundamental directions, Vol. 25, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 4–303 (Russian). MR 982753
G. N. Watson, A theory of asymptotic series, Philos. Trans. Roy. Soc. London Ser. A 211 (1911), 279–313.
F. Nevanlinna, Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. Ser. A 12 (1916), no. 1.
T. Carleman, Les fonctions quasi-analytiques, Gauthier-Villars, Paris, 1926.
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
R. B. Dingle, Asymptotic expansions and converging factors. I. General theory and basic converging factors, Proc. Roy. Soc. London Ser. A 244 (1958), 456–475. MR 103373, DOI 10.1098/rspa.1958.0054
R. B. Dingle, Asymptotic expansions and converging factors. IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions, Proc. Roy. Soc. London Ser. A 249 (1959), 270–283. MR 103376, DOI 10.1098/rspa.1959.0022
R. B. Dingle, Asymptotic expansions: their derivation and interpretation, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1973. MR 0499926
F. Nevanlinna, Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. Ser. A 12 (1918), no. 3, 1–81.
L. Bieberbach, Algemeine Theorie der Funktionen komplexer Argumente. Kapital 4, Jahrb. Forts. Math. 46 (1916–1918), no. 1.
Alan D. Sokal, An improvement of Watson’s theorem on Borel summability, J. Math. Phys. 21 (1980), no. 2, 261–263. MR 558468, DOI 10.1063/1.524408
Jorge Rezende, A note on Borel summability, J. Math. Phys. 34 (1993), no. 9, 4330–4339. MR 1233275, DOI 10.1063/1.530003
Harold Jeffreys, Asymptotic approximations, Clarendon Press, Oxford, 1962. MR 0147821
F. W. J. Olver, Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0435697
D. U. Kh. Gillam and V. P. Gurariĭ, On functions uniquely determined by their asymptotic expansions, Funktsional. Anal. i Prilozhen. 40 (2006), no. 4, 33–48, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 4, 273–284. MR 2307701, DOI 10.1007/s10688-006-0044-x
Herbert Buchholz, Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen, Ergebnisse der angewandten Mathematik. Bd. 2, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). MR 0054783
D. W. H. Gillam and V. P. Gurariĭ, Parametric representations in the theory of Laplace–Mellin transforms (in preparation).
George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
E. T. Copson, An introduction to the theory of a complex variable, Oxford Univ. Press, 1962.
Yudell L. Luke, The special functions and their approximations, Vol. I, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR 0241700
Ch. Hermite, Sur une application du théorème de M. Mittag-Leffler, dans la théorie des fonctions, J. Crelle 92 (1882), 145–155.
Ernst Lindelöf, Le calcul des résidus et ses applications à la théorie des fonctions, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1989 (French). Reprint of the 1905 original. MR 1189798
Niels Nielsen, Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten, Chelsea Publishing Co., New York, 1965 (German). MR 0185152
A. B. Olde Daalhuis, Hyperasymptotic expansions of confluent hypergeometric functions, IMA J. Appl. Math. 49 (1992), no. 3, 203–216. MR 1198790, DOI 10.1093/imamat/49.3.203
H. Poincaré, Sur les intégrales irrégulières, Acta Math. 8 (1886), no. 1, 295–344 (French). Des équations linéaires. MR 1554701, DOI 10.1007/BF02417092
F. W. J. Olver, On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 225–243. MR 185350
F. W. J. Olver and F. Stenger, Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 244–249. MR 185351
Frank Stenger, Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case, J. Res. Nat. Bur. Standards Sect. B 70B (1966), 167–186. MR 219823
Frank Stenger, Error bounds for asymptotic solutions of differential equations. II. The general case, J. Res. Nat. Bur. Standards Sect. B 70B (1966), 187–210. MR 219824
W. G. C. Boyd, Gamma function asymptotics by an extension of the method of steepest descents, Proc. Roy. Soc. London Ser. A 447 (1994), no. 1931, 609–630. MR 1316625, DOI 10.1098/rspa.1994.0158
M. V. Berry and C. J. Howls, Hyperasymptotics, Proc. Roy. Soc. London Ser. A 430 (1990), no. 1880, 653–668. MR 1070081, DOI 10.1098/rspa.1990.0111
M. V. Berry and C. J. Howls, Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London Ser. A 434 (1991), no. 1892, 657–675. MR 1126872, DOI 10.1098/rspa.1991.0119
Jonathan M. Borwein and Robert M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly 106 (1999), no. 10, 889–909. MR 1732501, DOI 10.2307/2589743
Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
V. P. Gurariĭ and V. I. Matsaev, Stokes multipliers for systems of first-order linear ordinary differential equations, Dokl. Akad. Nauk SSSR 280 (1985), no. 2, 272–276 (Russian). MR 775049
V. P. Gurariĭ and V. I. Matsaev, The generalized Borel transform and Stokes multipliers, Teoret. Mat. Fiz. 100 (1994), no. 2, 173–182 (English, with English and Russian summaries); English transl., Theoret. and Math. Phys. 100 (1994), no. 2, 928–936 (1995). MR 1311191, DOI 10.1007/BF01016755
V. Gurarii, J. Steiner, V. Katsnelson, and V. Matsaev, How to use the Fourier transform in asymptotic analysis, Twentieth century harmonic analysis—a celebration (Il Ciocco, 2000) NATO Sci. Ser. II Math. Phys. Chem., vol. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 387–401. MR 1858792
V. Gurarii and D. Lucy, The Stokes structure and connection coefficients for the Airy equation, Mat. Fiz. Anal. Geom. 10 (2003), no. 3, 385–411. MR 2012270
T.-J. Stieltjes, Recherches sur quelques séries semi-convergentes, Ann. Sci. École Norm. Sup. (3) 3 (1886), 201–258 (French). MR 1508781
—, Sur le développement de $\log \Gamma (a)$, J. Math. (4) 5 (1889), 425–444.
N. G. de Bruijn, Asymptotic methods in analysis, Bibliotheca Mathematica, Vol. IV, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1958. MR 0099564
È. Ya. Riekstyn′sh, Otsenki ostatkov v asimptoticheskikh razlozheniyakh, “Zinatne”, Riga, 1986 (Russian). MR 875418
Z. X. Wang and D. R. Guo, Special functions, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. Translated from the Chinese by Guo and X. J. Xia. MR 1034956, DOI 10.1142/0653
J. G. van der Corput, Asymptotics. II. Elementary methods, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 139–150. MR 0073739
Robert Spira, Calculation of the gamma function by Stirling’s formula, Math. Comp. 25 (1971), 317–322. MR 295539, DOI 10.1090/S0025-5718-1971-0295539-6
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
V. P. Gurariĭ and D. W. H. Gillam, Stokes and Gauss monodromic relations and the duality theorem, Preprint, Monash Univ., 2010 (to appear).