Integral Equation Methods

Abstract

We shall discuss certain integral equations that arise in the problem of computing the function w = f (z) that maps a simply connected region D, with boundary Γ and containing the origin, conformally onto the interior or exterior of the unit circle 1w 1 = 1. In the case when Γ is a Jordan contour, we obtain Fredholm integral equations of the second kind $\phi (s) = \pm \int_{\Gamma } {N(s,t)\phi (t)dt + g(s)}$ where φ(s) known as the boundary correspondence function, is to be determined and N(s, t) is the Neumann kernel. We shall discuss an iterative method for numerical computation of the Lichtenstein—Gershgorin equation and present the case of a degenerate kernel and also of the Szegö kernel. The case when Γ has a corner yields Stieltjes integral equations and is presented in Chapter 12.