Analytical treatment of the near field term of the Green function of planarly stratified media

We consider an exact evaluation of the asymptotic, small distance contribution to the matrix elements of the planarly layered-medium Green function. The method is applicable to Rao-Wilton-Glisson (RWG) basis functions with supports located on the medium interfaces and the asymptotic contribution is defined as that involving all matrix elements between pairs of basis functions supported on either the same interface or on a pair of interfaces of a single homogeneous material layer. The asymptotic matrix elements, given in terms of quadruple surface integrals with singular integrands, are subsequently converted, by using suitably constructed Laplacian representations of the Green function, to double contour integrals over the perimeters of the surface elements, with simple, nonsingular, smoothly varying integrands. The line integrals can be either evaluated analytically (currently resulting in rather lengthy expressions involving elementary functions), or by means of low order numerical quadratures.