The Schwarz problem for the Cauchy-Riemann equation and the Dirichlet and Neumann problems for the Poisson equation are explicitly solved in a half disc and a half ring of the complex plane. The respective Poisson kernels and the Green and Neumann functions are given.
"Harmonic boundary value problems in half disc and half ring." Funct. Approx. Comment. Math. 40 (2) 251 - 282, June 2009. https://doi.org/10.7169/facm/1246454030