Lauricella Function and the Conformal Mapping of Polygons

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.