Two Boundary-Value Problems for the Cauchy–Riemann Equation in a Sector

By reflections, we obtain the Schwarz–Poisson formula in a sector with angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta=\frac{\pi}{n},\,n\in \mathbb{N}}$$\end{document} , which is a generalization of the corresponding result obtained by Begehr and Vaitekhovich (Funct Approx 40(2):251–282, 2009). Especially, boundary behaviors at corner points are discussed in detail. Then we consider the Schwarz and Dirichlet boundary-value problems (BVPs) for the Cauchy–Riemann equation, and expressions of solution and the condition of solvability are explicitly obtained.