Formulas for computing Euler-type integrals and their application to the problem of constructing a conformal mapping of polygons

This paper deals with Euler-type integrals and the closely related Lauricella function $F^{(N)}_D$, which is a hypergeometric function of many complex variables $z_1,\dots,z_N$. For $F^{(N)}_D$ new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of $\mathbb{C}^N$, including near hyperplanes of the form $\{z_j=z_l\}$, $j$, $l=\overline{1,N}$, $j\ne l$. The continuation formulas and identities for $F^{(N)}_D$ found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space $\mathbb{C}^N$, including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons.