Analytic continuation of the Appell function $F_1$ and integration of the associated system of equations in the logarithmic case

S. I. Bezrodnykh

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 4, pp. 555-587

Résumé

The Appell function $F_1$ (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of $F_1$ are related in a special way. Formulas for the analytic continuation of $F_1$ beyond the unit bicircle are constructed in which $F_1$ is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of $F_1$ in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.

Bibliographie

[1] Regge T., “Algebraic topology methods in the theory of Feynman relativistic amplitudes”, Battelle rencontres (1967), Lectures in Mathematics and Physics, eds. C. M. DeWitt, J. A. Wheeler, W. A. Benjamin, New York, 1968 | MR

[2] Mathai A. M., Saxena R. K., Generalized hypergeometric functions with applications in statistics and physical sciences, Lecture Notes in Mathematics, 348, Springer-Verlag, Berlin–Heidelberg–New York, 1973 | DOI | MR | Zbl

[3] Exton H., Multiple hypergeometric functions and application, J. Willey Sons inc, New York, 1976 | MR

[4] Miller U., Simmetrii i razdelenie peremennykh, Mir, M., 1981

[5] Slavyanov S. Yu., Lay W., Special functions. A unified theory based on singularities, Oxford University Press, Oxford, 2000 | MR | Zbl

[6] Pochhammer L., “Über hypergeometrische Funktionen höherer Ordnungen”, Crelle's Journal, 71 (1870), 316–352 | DOI | MR

[7] Appel P., “Sur les séries hypergéométriques de deux variables et sur des équations diff'erentielles lin'eaires aux d'eriv'ees partielles”, Comptes Rendus, 90 (1880), 296–298

[8] Picard E., “Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques”, Ann. mole Normale Supérieure, 10:2 (1881), 305–322 | DOI | MR

[9] Appel P., “Sur les fonctions hypergeometriques de deux variables”, Journal de mathématiques pures et appliquos, 3e série, 8 (1882), 173–216

[10] Horn J., “Über die konvergenz der hypergeometrischen Reihen zweier und dreier Veränderlichen”, Math. Ann., 34 (1889), 544–600 | DOI | MR

[11] Lauricella G., “Sulle funzioni ipergeometriche a piu variabili”, Rendiconti Circ. Math. Palermo, 7 (1893), 111–158 | DOI

[12] Appel P., Kampe de Feriet J., Fonctions hypergeometriques et hypersphérique, Gauthier-Villars, Paris, 1926

[13] Ore O., “Sur la forme de fonctions hypergeometriques de plusiers variable”, J. Math. Pure et Appl., 9:4 (1930), 311–327

[14] Horn J., “Hypergeometrische Funktionen zweier Veranderlichen”, Math. Ann., 105 (1931), 381–407 | DOI | MR

[15] Erdelyi A., “Hypergeometric functions of two variables”, Acta Mat., 83:131 (1950), 131–164 | DOI | MR | Zbl

[16] Srivastava H. M., “Hypergeometric functions of of three variabless”, Ganita, 15 (1964), 97–108 | MR | Zbl

[17] Olsson O. M., “Integration of the partial differential equations for the hypergeometric function $F_1$ and $F_D$ of two and more variables”, J. Math. Phys., 5:420 (1964), 420–430 | DOI | MR | Zbl

[18] Gindikin S. G., “Analiz v odnorodnykh oblastyakh”, Uspekhi matem. nauk, 19:4(118) (1964), 3–92 | Zbl

[19] Slater L., Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966 | MR | Zbl

[20] Deligne P., Mostow G. D., “Monodromy of hypergeometric functions and nonlattice integral monodromy”, Publ. Math. Inst. Hautes Etudes Sci., 63 (1986), 5–89 | DOI | MR | Zbl

[21] Aomoto K., “On the structure of integrals of power products of linear functions”, Sci. Papers Coll. Gen. Educ. Univ. Tokyo, 27:2 (1977), 49–61 | MR | Zbl

[22] Gelfand I. M., Graev M. I., Retakh V. S., “Obschie gipergeometricheskie sistemy uravnenii i ryady gipergeometricheskogo tipa”, Uspekhi matem. nauk, 47:4(286) (1992), 3–82 | Zbl

[23] Dwork B., Generalized hypergeometric functions, Clarendon Press, Oxford, 1990 | MR | Zbl

[24] Iwasaki K., Kimura H., Shimomura Sh., Yoshida M., From Gauss to Painleve. A modern theory of special functions, Aspects of mathematics, E16, Friedrich Vieweg Sohn, Braunschweig, 1991 | DOI | MR | Zbl

[25] Aomoto K., Kita M., Theory of hypergeometric functions, Springer monographs in mathematics, Springer, Tokyo–Dordrecht–Heidelberg, 2011 | DOI | MR | Zbl

[26] Sadykov T. M., Tsikh A. K., Gipergeometricheskie i algebraicheskie funktsii mnogikh peremennykh, Nauka, M., 2014

[27] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Nauka, M., 1973

[28] Kummer H. E. E., “Über die hypergeometrische Reihe $1+\frac{\alpha\beta}{1\cdot\gamma}x+\frac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}x^2\dots$”, Journ. für Math., XV (1836), 39–83 ; 127–172 | MR | Zbl | Zbl

[29] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, v. 2, Editorial URSS, M., 2002

[30] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Doktorskaya diss., VTs AN SSSR, M., 1990

[31] Bezrodnykh S. I., “Ob analiticheskom prodolzhenii funktsii Laurichelly”, Mezhdunarodnaya konferentsiya po differentsialnym uravneniyam i dinamicheskim sistemam, Tezisy dokladov (Suzdal, 27 iyunya–2 iyulya 2008 g.), 2008, 34–36

[32] Bezrodnykh S. I., “Formuly analiticheskogo prodolzheniya i sootnosheniya tipa Yakobi dlya funktsii Laurichelly”, Dokl. AN, 467:1 (2016), 7–12 | DOI | Zbl

[33] Marichev O. I., Metod vychisleniya integralov ot spetsialnykh funktsii (teoriya i tablitsy formul), Nauka i tekhnika, Minsk, 1978

[34] Bezrodnykh S. I., “Ob analiticheskom prodolzhenii funktsii Laurichelly”, Matematicheskie zametki, 100:2 (2016), 292–298 | DOI

[35] Vlasov V. I., Skorokhodov S. L., “Metod multipolei dlya zadachi Dirikhle v dvusvyaznykh oblastyakh slozhnoi formy. I. Obschee opisanie metoda”, Zh. vychisl. matem. i matem. fiz., 40:11 (2000), 1633–1647 | Zbl

[36] Trefethen L. N., “Numerical computation of the Schwarz–Christoffel transformation”, SIAM J. Sci. Stat. Comput., 1 (1980), 82–102 | DOI | MR | Zbl

[37] Trefethen L. N., “Numerical construction of comformal maps”, Appendix to: Saff E. B., Snider A. D., Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, Prentice Hall, New York, 1993

[38] Bezrodnykh S. I., Vlasov V. I., “Zadacha Rimana–Gilberta v slozhnoi oblasti dlya modeli magnitnogo peresoedineniya v plazme”, Zh. vychisl. matem. i matem. fiz., 42:3 (2002), 277–312 | Zbl

[39] Trefethen L. N., Driscoll T. A., Schwarz–Christoffel transformation, Cambridge university press, Cambridge, 2005 | MR

[40] Bezrodnykh S. I., Vlasov V. I., “Zadacha Rimana–Gilberta v oblastyakh slozhnoi formy i ee prilozhenie”, Spectral and Evolution Problems, 16 (2006), 51–61

[41] Bogatyrev A. B., “Konformnoe otobrazhenie pryamougolnykh semiugolnikov”, Matem. sb., 203:12 (2012), 35–56 | DOI | Zbl

[42] Bezrodnykh S. I., Vlasov V. I., “Singulyarnaya zadacha Rimana–Gilberta v slozhnykh oblastyakh”, Zh. vychisl. matem. i matem. fiz., 54:12 (2014), 1904–1954 | DOI