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.