Lauricella Function and the Conformal Mapping of Polygons

S. I. Bezrodnykh

Matematičeskie zametki, Tome 112 (2022) no. 4, pp. 500-520.

Résumé

In this paper, some progress has been made in solving the problem of calculating the parameters of the Schwarz–Christoffel integral realizing a conformal mapping of a canonical domain onto a polygon. It is shown that an effective solution of this problem can be found by applying the formulas of analytic continuation of the Lauricella function $F_D^{(N)}$, which is a hypergeometric function of $N$ complex variables. Several new formulas for such a continuation of the function $F_D^{(N)}$ are presented that are oriented to the calculation of the parameters of the Schwarz–Christoffel integral in the “crowding” situation. An example of solving the parameter problem for a complicated polygon is given.

Détail

Keywords: Schwarz–Christoffel integral, hypergeometric functions of many variables, analytic continuation, crowding.

Bibliographie

[1] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1953 | MR

[2] G. M. Goluzin, Geometricheskaya teoriya funktsii kompleksnogo peremennogo, GITTL, M.–L., 1952 | MR

[3] V. Koppenfels, F. Shtalman, Praktika konformnykh otobrazhenii, IL, M., 1963

[4] G. Goluzin, L. Kantorovich, V. Krylov, P. Melentev, M. Muratov, N. Stenin, Konformnoe otobrazhenie odnosvyaznykh i mnogosvyaznykh oblastei, Nauka, L.–M., 1937

[5] L. V. Kantorovich, V. I. Krylov, Priblizhennye metody vysshego analiza, GITTL, L., 1962 | MR

[6] D. Gaier, Konstructive Methoden der konformen Abbildung, Springer-Verlag, Berlin, 1964 | MR

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

[8] R. Menikoff, C. Zemach, “Methods for numerical conformal mapping”, J. Comput. Phys., 36:3 (1980), 366–410 | DOI | MR | Zbl

[9] Numerical Conformal Mapping, ed. L. N. Trefethen, North Holland, Amsterdam, 1986 | MR | Zbl

[10] P. Henrici, Applied and Computational Complex Analysis, Vol. 1–3, John Wiley and Sons, New York, 1991 | MR

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

[12] P. K. Kythe, Computational Conformal Mapping, Birkhäuser, Basel, 1998 | MR | Zbl

[13] L. N. Trefethen, T. A. Driscoll, Schwarz–Christoffel Transformation, Cambridge Univ. Press, Campridge, 2005 | MR

[14] N. Papamichael, N. Stylianopoulos, Numerical Conformal Mapping. Domain Decomposition and the Mapping of Quadrilaterals, World Sci. Publ., Hackensack, 2010 | MR | Zbl

[15] S. I. Bezrodnykh, “Gipergeometricheskaya funktsiya Laurichelly $F_D^{(N)}$, zadacha Rimana–Gilberta i nekotorye prilozheniya”, UMN, 73:6 (444) (2018), 3–94 | DOI | MR | Zbl

[16] S. I. Bezrodnykh, “Analytic continuation of Lauricella's function $F_D^{(N)}$ for large in modulo variables near hyperplanes $\{z_j=z_l\}$”, Integral Transforms Spec. Funct., 33:4 (2021), 276–291 | DOI

[17] S. I. Bezrodnykh, “Analytic continuation of Lauricella's function $F_D^{(N)}$ for variables close to unit near hyperplanes $\{z_j=z_l\}$”, Integral Transforms Spec. Funct., 33:5 (2021), 419–433 | DOI

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

[19] H. Exton, Multiple Hypergeometric Functions and Application, John Willey and Sons, New York, 1976 | MR

[20] K. Iwasaki, H. Kimura, Sh. Shimomura, M. Yoshida, From Gauss to Painlevé. A Modern Theory of Special Functions, Aspects Math., E16, Friedrich Vieweg and Sohn, Braunschweig, 1991 | MR

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

[22] S. I. Bezrodnykh, V. I. Vlasov, “Zadacha Rimana–Gilberta v oblastyakh slozhnoi formy i ee prilozhenie”, Spectral and Evolution Problems, 16:1 (2006), 51–61

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

[24] N. N. Nakipov, S. R. Nasyrov, “Parametricheskii metod nakhozhdeniya aktsessornykh parametrov v obobschennykh integralakh Kristoffelya–Shvartsa”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 158, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2016, 202–220 | MR

[25] C. Zemach, “A conformal map formula for difficult cases”, J. Comput. Appl. Math., 14 (1986), 207–215 | DOI | MR | Zbl

[26] B. C. Krikeles, R. L. Rubin, “On the crowding of parameters associated with Schwarz–Christoffel transformation”, Appl. Math. Comput., 28:4 (1988), 297–308 | MR | Zbl

[27] T. A. Driscoll, “A MATLAB toolbox for Schwarz–Christoffel mapping”, ACM Transactions Math. Soft., 22 (1996), 168–186 | DOI | Zbl

[28] L. Banjai, “Revisiting the crowding phenomenon in Schwarz–Christoffel mapping”, SIAM J. Sci. Comput., 30:2 (2008), 618–636 | DOI | MR | Zbl

[29] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Gipergeometricheskaya funktsiya. Funktsii Lezhandra, Spravochnaya matematicheskaya biblioteka, Nauka, M., 1973 | MR | Zbl

[30] S. I. Bezrodnykh, V. I. Vlasov, “Asimptotika zadachi Rimana–Gilberta dlya modeli magnitnogo peresoedineniya v plazme”, Zh. vychisl. matem. i matem. fiz., 60:11 (2020), 1898–1914 | DOI | Zbl

[31] V. I. Vlasov, Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Dis. dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1990

[32] T. S. O'Connell, P. T. Krein, “A Schwarz–Christoffel-based analytical method for electric machine field analysis”, IEEE Transactions on Energy Conversion, 24:3 (2009) | DOI

[33] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR | Zbl