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Lauricella Function and the Conformal Mapping of Polygons

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Abstract

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.

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Funding

This work was supported by the Russian Science Foundation under grant 22-21-00727, https:// rscf.ru/ project/22-21-00727/.

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Authors and Affiliations

  1. Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Moscow, 119333, Russia

    S. I. Bezrodnykh

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  1. S. I. Bezrodnykh
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Correspondence to S. I. Bezrodnykh.

Additional information

Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 500–520 https://doi.org/10.4213/mzm13694.

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Bezrodnykh, S.I. Lauricella Function and the Conformal Mapping of Polygons. Math Notes 112, 505–522 (2022). https://doi.org/10.1134/S0001434622090218

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