Error bounds, duality, and Stokes phenomenon.~I

V. P. Gurariǐ

Algebra i analiz, Tome 21 (2009) no. 6, pp. 80-150.

Résumé

We consider classes of functions uniquely determined by coefficients of their divergent expansions. Approximating a function from 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 improvement in the accuracy of the expansion.

Détail

Keywords: Stokes phenomenon, Poincaré's asymptotic theory, Stokes rays, Airy functions.

Bibliographie

[1] Meyer R. E., “A simple explanation of the Stokes phenomenon”, SIAM Rev., 31:3 (1989), 435–445 | DOI | MR | Zbl

[2] Berry M. V., “Stokes' phenomenon; smoothing a Victorian discontinuity”, Inst. Hautes Études Sci. Publ. Math., 68 (1988), 211–221 | DOI | MR | Zbl

[3] Berry M. V., “Uniform asymptotic smoothing of Stokes's discontinuities”, Proc. Roy. Soc. London Ser. A, 422 (1989), 7–21 | DOI | MR | Zbl

[4] Berry M. V., “Infinitely many Stokes smoothings in the gamma function”, Proc. Roy. Soc. London Ser. A, 434 (1991), 465–472 | DOI | MR | Zbl

[5] Berry M., “Asymptotics, superasymptotics, hyperasymptotics $\dots$”, Asymptotics Beyond All Orders (La Jolla, CA, 1991), NATO Adv. Sci. Inst. Ser. B Phys., 284, Plenum, New York, 1991, 1–14 | MR

[6] Paris R. B., Kaminski D., Asymptotics and Mellin–Barnes integrals, Encyclopedia Math. Appl., 85, Cambridge Univ. Press, Cambridge, 2001 | MR | Zbl

[7] Ramis J.-P., Séries divergentes et théories asymptotiques, Bull. Soc. Math. France, 121, Panoramas et Syntheses, Suppl., 1993, 74 pp. | MR | Zbl

[8] Ramis J.-P., “Stokes phenomenon: historical background”, The Stokes Phenomenon and Hilbert's 16th Problem (Groningen, 1995), eds. B. L. J. Braaksma, G. K. Immink, M. Van der Put, World Sci. Publ., River Edge, NJ, 1996, 1–5 | MR | Zbl

[9] Paris R. B., Wood A. D., “Stokes phenomenon demystified”, Bull. Inst. Math. Appl., 31 (1995), 21–28 | MR | Zbl

[10] Stokes G. G., “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 arrange by J. Larmor, Vol. II, Cambridge Univ. Press, London–New York, 1907, 159–160

[11] Pokrovskii V. L., Khalatnikov I. M., “K voprosu o nadbarernom otrazhenii chastits vysokikh energii”, ZhETF, 40:6 (1961), 1713–1719

[12] Boyd J. P., “The devil's invention: asymptotic, superasymptotic and hyperasymptotic series”, Acta Appl. Math., 56:1 (1999), 1–98 | DOI | MR | Zbl

[13] “Group methods in commutative harmonic analysis”, Encyclopaedia Math. Sci., 25, Springer, Berlin etc., 1998, 1–325 | MR

[14] Watson G. N., “A theory of asymptotic series”, Philos. Trans. Roy. Soc. London Ser. A, 211 (1911), 279–313 | Zbl

[15] Nevanlinna F., Zur Theorie der asymptotischen Potenzreihen, Ann. Acad. Sci. Fenn. Ser. A, 12, no. 3, 1916 | Zbl

[16] Carleman T., Les fonctions quasi-analytiques, Gauthier-Villars, Paris, 1926 | Zbl

[17] Hardy G. H., Divergent series, Clarendon Press, Oxford, 1949 | MR | Zbl

[18] Dingle R. B., “Asymptotic expansions and converging factors. I. General theory and basic converging factors”, Proc. Roy. Soc. London Ser. A, 244 (1958), 456–475 | DOI | MR | Zbl

[19] Dingle R. B., “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 | DOI | MR | Zbl

[20] Dingle R. B., Asymptotic expansions: their derivation and interpretation, Acad. Press, London–New York, 1973 | MR | Zbl

[21] Nevanlinna F., “Zur Theorie der asymptotischen Potenzreihen”, Ann. Acad. Sci. Fenn. Ser. A, 12:3 (1918), 1–81

[22] Bieberbach L., Algemeine Theorie der Functionen komplexer Argumente. Kapital 4, Jahrb. Forts. Math., 46, no. 1, 1916–1918 | Zbl

[23] Sokal A. D., “An improvement of Watson's theorem on Borel summability”, J. Math. Phys., 21:2 (1980), 261–263 | DOI | MR

[24] Rezende Jorge, “A note on Borel summability”, J. Math. Phys., 34:9 (1993), 4330–4339 | DOI | MR | Zbl

[25] Jeffreys H., Asymptotic approximation, Clarendon Press, Oxford, 1962 | MR | Zbl

[26] Olver F. W. J., Asymptotics and special functions, Acad. Press, New York–London, 1974 | MR | Zbl

[27] Gillam D. U. Kh., Gurarii V. P., “O funktsiyakh, odnoznachno opredelyaemykh svoimi asimptoticheskimi razlozheniyami”, Funkts. anal. i ego pril., 40:4 (2006), 33–48 | MR | Zbl

[28] The confluent hypergeometric function with special emphasis on its applications, Springer Tracts in Natural Philos., 15, Springer-Verlag New York Inc., New York, 1969 | MR | MR | Zbl

[29] Gillam D. W. H., Gurarii V. P., Parametric representations in the theory of Laplace–Mellin transforms, in preparation

[30] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia Math. Appl., 71, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[31] Abramowitz M., Stegun I. (eds.), Handbook of mathematical functions with formulas, graphs and mathematical tables, Dover Publ., Inc., New York, 1992 | MR

[32] Copson E. T., An introduction to the theory of a complex variable, Oxford Univ. Press, 1962

[33] Luke Y. L., The special functions and their approximations, vol. 1, Math. Sci. Engrg., 53, Acad. Press, New York–London, 1969 | Zbl

[34] Hermite Ch., “Sur une application du theoreme de M. Mittag-Leffler, dans la theorie des fonctions”, J. Crelle, 92 (1882), 145–155 | DOI

[35] Lindelöf E., Le calcul des résidus et ses applications a la théorie des fonctions, Gauthier-Villars, Paris, 1905 | Zbl

[36] Nielsen N., Handbuch der Theorie der Gammafunction, Teubner, Leipzig, 1906 | Zbl

[37] Olde Daalhuis A. B., “Hyperasymptotic expansions of confluent hypergeometric functions”, IMA J. Appl. Math., 49 (1992), 203–216 | DOI | MR | Zbl

[38] Poincaré H., “Sur les intégrales irrégulières. Des équations linéaires”, Acta Math., 8:1 (1886), 295–344 | DOI | MR

[39] Olver F. W. J., “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., 2:2 (1965), 225–243 | DOI | MR | Zbl

[40] Olver F. W. J., Stenger F., “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., 2:2 (1965), 244–249 | DOI | MR | Zbl

[41] Stenger F., “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

[42] Stenger F., “Error bounds for asymptotic solutions of differential equations. II. The general case”, J. Res. Nat. Bur. Standards Sect. B, 70B (1966), 187–210 | MR

[43] Boyd W. G. C., “Gamma function asymptotics by an extension of the method of steepest descents”, Proc. Roy. Soc. London Ser. A, 447 (1994), 609–630 | DOI | MR | Zbl

[44] Berry M. V., Howls C. J., “Hyperasymptotics”, Proc. Roy. Soc. London Ser. A, 430 (1990), 653–668 | DOI | MR | Zbl

[45] Berry M. V., Howls C. J., “Hyperasymptotics for integrals with saddles”, Proc. Roy. Soc. London Ser. A, 434 (1991), 657–675 | DOI | MR | Zbl

[46] Borwein J. M., Corless R. M., “Emerging tools for experimental mathematics”, Amer. Math. Monthly, 106 (1999), 889–909 | DOI | MR | Zbl

[47] Comtet L., Advanced combinatorics. The art of finite and infinite expansions, D. Reidel Publ. Co., Dordrecht, 1974 | MR | Zbl

[48] Gurarii V. P., Matsaev V. I., “Mnozhiteli Stoksa dlya sistem lineinykh obyknovennykh differentsialnykh uravnenii pervogo poryadka”, Dokl. AN SSSR, 280:2 (1985), 272–276 | MR

[49] Gurarii V. P., Matsaev V. I., “The generalized Borel transform and Stokes multipliers”, Teor. i mat. fiz., 100:2 (1994), 173–182 | MR | Zbl

[50] Gurarii V. P., Steiner J., Katsnelson V., Matsaev V., “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., 33, Kluwer Acad. Publ., Dordrecht, 2001 | MR | Zbl

[51] Gurarii V. P., Lucy D., “The Stokes structure and connection coefficients for the Airy equation”, Mat. Fiz. Anal. Geom., 10:3 (2003), 385–411 | MR | Zbl

[52] Stiltjes T. J., “Recherches sur quelques séries semi-convergentes”, Ann. Sci. Ecole. Norm. Sup. (3), 3 (1886), 201–258 | MR

[53] Stiltjes T. J., “Sur le développement de $\log\Gamma(a)$”, J. Math. (4), 5 (1889), 425–444

[54] de Bruijn N. G., Asymptotic methods in analysis, Bibliotheca Math., 4, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1958 | MR | Zbl

[55] Riekstynsh E. Ya., Otsenki ostatkov v asimptoticheskikh razlozheniyakh, Zinatne, Riga, 1986 | MR

[56] Wang Z. X., Guo D. R., Special functions, World Sci. Publ. Co., Inc., Teaneck, NJ, 1989 | MR | Zbl

[57] van der Corput J. G., “Asymptotics. II. Elementary methods”, Nederl. Akad. Wetensch. Proc. Ser. A, 58 (1955), 139–150 | MR | Zbl

[58] Spira R., “Calculation of the gamma function by Stirling's formula”, Math. Comp., 25 (1971), 317–322 | DOI | MR | Zbl

[59] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F., Higher transcendental functions, McGraw-Hill Book Co., Inc., New York etc., 1953 | Zbl

[60] Gillam D. W. H., Gurarii V. P., A dual monodromy: the mechanism behind the Stokes phenomenon, Preprint, Monash Univ., 2009 (to appear)