Poisson-Schwarz Integral Formulae

Abstract

A typical example of boundary value problems is to find a function Ф such that the Laplace equation ΔФ = 0 is satisfied in a domain D and Ф itself or its normal derivative δФ/δn assumes specified values on the boundary. Many problems of physics and engineering can be reduced to this problem. In two-dimensional problems, an equivalent is to find an analytic function F(z) regular in D such that Re F(z) or Im F(z) assumes specified values on the boundary. When D is a circle or a halfplane, formulae to express the solution are known and are called the Poisson-Schwarz integral formulae. In this chapter, we discuss these formulae and related facts from the viewpoint of hyperfunction theory. As an example of their application we deal with integral equations related to the Hilbert transforms.