Plemelj Relations for a Class of Cauchy Integrals

Abstract

where the density g is a function also of the parameter z; but has been unable to find in the literature a precise derivation of the corresponding Plemelj relations. The purpose of this note is to extend the work of Muskhelishvili in order to establish sufficient conditions for the existence of the principal value, as well as the one-sided continuity, of the Cauchy integral G(z) at a singular point on a smooth contour. Specifically, THEOREM. If the complex density function g(t, z) satisfies a Lipschitz condition in z in the neighborhood of a smooth Jordan curve (contour) C, and satisfies a Lipschitz condition in t on C, then the principal value G(to) of the Cauchy integral