A generalization of the Schwarz–Christoffel formula

We obtain a formula for mapping the upper half-plane conformally onto a polygonal region, generalizing the Schwarz–Christoffel formula to the case of a countable set of vertices. We indicate a connection of the construction of this mapping to the solution of the Hilbert boundary value problem with a countable set of discontinuity points of the coefficients and polynomial singularity of the index.