Schwarz problem in lens and lune

Abstract

The Schwarz problem is explicitly solved for the inhomogeneous Cauchy–Riemann equation in a particular circular lens and two related complementary lunes. The solutions are given by the same analytical formula restricted to the respective domain. The parqueting-reflection method is used to construct the Cauchy–Schwarz representation formula leading to the Schwarz and thus the Poisson kernels for the three domains.