Schwarz Boundary Value Problems for Polyanalytic Equation in a Sector Ring

摘要

Abstract In this article, we first give a modified Schwarz–Pompeiu formula in a general sector ring with angle ϑ=πα,α≥1/2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$vartheta =frac{pi }{alpha }, alpha ge 1/2$$end{document} by proper conformal mappings, and obtain the solution of the Schwarz problem for the Cauchy–Riemann equation in explicit forms. Furthermore, we construct some integral operators and discuss their properties in detail. Finally, by virtue of these operators, Schwarz problems for an inhomogeneous polyanalytic equation and for a generalized polyanalytic equation are investigated, respectively.