Singularity in Green's function and its numerical evaluation

The free-space scalar Green's functionghas anR^{-1}singularity, whereRis the distance between the source and observation points. The second derivatives ofghaveR^{-3}singularities, which are not generally integrable over a volume. The derivatives ofgare treated as generalized functions in the manner described by Gel'fand and Shilov, and a new formula is derived that regularizes a divergent convolution integral involving the second derivatives ofg. When the formula is used in the dyadic Green's function formulation for calculating theEfield, all previous results are recovered as special cases. Furthermore, it is demonstrated that the formula is particularly suitable for the numerical evaluation of the field at a source point, because it allows the exclusion of an arbitrary finite region around the singular point from the integration volume. This feature is not shared by any of the previous results on the dyadic Green's function.