The Poisson kernel for a doubly connected domain

Abstract

In this work the Dirichlet problem for the Laplace equation in a doubly connected domain is solved. For the solution of the proposed problem, we have the Green harmonic function on this domain, and from this function, the Poisson Kernel is obtained explicitly. With these elements and the properties of the conformal mappings, the integral representation formula that solves the Dirichlet problem for the Laplace equation is obtained. This application to obtain Poisson's Kernel through the theory of conformal transformations makes it possible to solve problems of boundary values on a variety of domains in which the known method of parqueting-reflection is not admissible and the method via the Schwarz problem.