The Schwarz problem in multiply connected domains and the Schottky–Klein prime function

Abstract

This article presents concise integral formulae for the solution of the Schwarz problem (and modified Schwarz problem) in multiply connected circular planar domains. The form of the solution is novel in that the kernels appearing in the integral formula are expressed in terms of the Schottky–Klein prime function on the Schottky double of the multiply connected domain. The well-known Poisson integral formula and Villat formula are retrieved as special cases.