On the Dirichlet Boundary Value Problem for the Cauchy-Riemann Equations in the Half Disc

Abstract

In this article, we investigate the Dirichlet boundary value problem for the Cauchy-Riemann equations in the half disc. First, using the technique of parqueting–reflection and the Cauchy-Pompeiu representation formula for a half disc, we obtain an integral representation formula in the half disc. In other words, we construct a unique solution for the Dirichlet boundary value problem. Finally, we solve the Dirichlet boundary value problem for both the homogeneous and the inhomogeneous Cauchy-Riemann equations. In particular, the boundary behaviors at the corner points are considered.