On the longitudinal component of the Green's function dyadic

Discussion of the divergenceless property of the right-hand side of the dyadic-wave equation for the Green's dyad has centered on the inconsistency of expanding the Green's dyad only in terms of transverse-wave functions. By including the longitudinal functions in the Ohm-Rayleigh expansion of the dyad, a simple closed form expression for the longitudinal component is derived which yields the expected singular quasi-static field. The result is verified in a coordinate independent manner with the aid of the Helmholtz theorem.