D-theorem (on regularization): Green´s function-induced distributed elementary sources — First kind

Maxwell´s equations (in isotropic, anisotropic, bianisotropic media) can be split into two sets of complementary systems of partial differential equations: diagonalized- and supplementary forms, D- form and S- form, respectively. Given a boundary value problem, the D-form (diagonalization) with respect to a distinguished direction in space allows directly determining field components transversal to the distinguished direction. The remaining two field components (parallel to the distinguished direction) can be determined a posteriori from the transversal components by utilizing the S-form. Using the D-form, and the resulting integral representations for the transversal components, a novel distributed elementary source has been constructed for self-consistently regularizing dyadic Green´s functions. The results have been firmly established by providing the complete proof of a theorem. Relevant statements are made for free space. However, details of the proof make immediate the validity of the statements for complex media. It is claimed that a self-consistent procedure for the long-standing renormalization problem has been devised, which promises to exert an impact on spectral domain computational electromagnetics.