Applying Lauricella's function to construct conformal mapping of polygons' exteriors

We consider the problem of calculating the parameters of the Christoffel–Schwarz integral implementing a conformal mapping of the upper half-plane onto a polygon containing an interior point at infinity. The paper proposes a solution to this problem based on new formulas for the analytic continuation of the Lauricella function $F_D^{(N)}$– the hypergeometric function of $N$ complex variables. A set of new identities and continuation formulas for this function is obtained, aimed at calculating the parameters of the Christoffel–Schwarz integral in the “crowding” situation. Representations via the Lauricella function are found for the Christoffel–Schwarz integral that are convenient for calculating such a conformal mapping.