Singularities

Abstract

As Radon (1919) and Carleman (1922) have shown, the method of integral equations can also be used to solve the problem of potential theory when the boundary Γ of a simply connected region D contains corners. In such cases Carleman separates the kernel into two parts, one of which corresponds to the corner singularities, whereas Radon uses the Stieltjes integral equations to solve this problem. We shall derive the analogues of Gershgorin’s integral equation and then obtain Arbenz’s integral equation which uses Radon’s method to determine conformal maps for boundaries with corners and has a unique solution. The cases of interior and exterior mapping functions f(z) and f _E (z) are related to each other through inversion by the relations (7.3.12). We are interested in the behavior of these univalent maps and those of doubly connected regions at singularities on and near the boundary, which are corner-type or pole-type. The nature and location of such singularities are determined.