K-space properties of classical single- and double-layer potentials and their derivatives

Abstract

Single- and double-layer potentials are well known functions used in many areas of mathematical physics. One extensive example is their use in the formulation of scattering problems as boundary integral equations. Their properties are summarized in several textbooks. Here we derive these properties in a different way using the Weyl representation for the Green's function and some elementary ideas from complex variables and distribution theory. In addition we provide k-space representations for the functions and their derivatives. The latter have not appeared in the literature and are useful for formulating scattering integral equations in Fourier transform space.