Geodesic
Lauricella Function and the Conformal Mapping of Polygons
Matematičeskie zametki, Tome 112 (2022) no. 4, pp. 500-520

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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.
Keywords: Schwarz–Christoffel integral, hypergeometric functions of many variables, analytic continuation, crowding. 
@article{MZM_2022_112_4_a2,
     author = {S. I. Bezrodnykh},
     title = {Lauricella {Function} and the {Conformal} {Mapping} of {Polygons}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {500--520},
     publisher = {mathdoc},
     volume = {112},
     number = {4},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_4_a2/}
}
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S. I. Bezrodnykh. Lauricella Function and the Conformal Mapping of Polygons. Matematičeskie zametki, Tome 112 (2022) no. 4, pp. 500-520. http://geodesic.mathdoc.fr/item/MZM_2022_112_4_a2/