Strongly and uniformly convergent Green's function expansions

Abstract

The convergence of Green's function expansions used in the exact analytical treatment of problems involving boundaries of different shapes is a property crucial in obtaining their solution. Existing expansions in most cases suffer from two serious setbacks: they do not converge uniformly in their region of validity, exhibiting a slow and conditional convergence near the source (singular) point and, even worse, they change expression when the field point moves past the source point. For such reasons they are unsuited for the solution of singular integral equations, in which values of the Green's function G at the source point do appear inside the integral. These inadequacies are met head-on by extracting the singular behavior in a closed-form term. Additional simple terms are also extracted to improve the convergence of the expansion of the remaining, non-singular part of G. The so-obtained new eigenfunction expansions for G converge uniformly over the whole region of their validity and very strongly (exponentially) near the source point. They are particularly suited for the solution of singular integral equations by the Carleman-Vekua method, otherwise known as the method of regularization by solving the dominant equation. These new expansions can be further subjected to a Watson transformation yielding a third expansion exhibiting strong convergence in regions where the convergence of the preceding series weakens, and vise versa. All these considerations are illustrated in this paper by means of a two-dimensional harmonic Green's function of a line source inside a rectangular shield, which is useful in a variety of shieldedline configurations. Extensions to different dielectric sublayers, to wave (Helmholtz) Green's functions, etc., are also discussed.