Source-neutral Green's functions for periodic problems in electrostatics, and their equivalents in electromagnetism

We consider source-neutral problems for periodic problems in electrostatics . These may be used to derive an identity due to Lord Rayleigh entirely wit hin the context of the unit cell, and without use of reasoning based on con tributions from charges at an outer boundary of the periodic system, or emp loying conditionally convergent lattice sums. We also consider the equivale nt of source-free Green's functions for the Helmholtz equation, which are G reen's functions with their specular parts removed. We show that, as the wa venumber tends to zero, the non-specular dynamic Green's functions tend wit h no divergences to the source-neutral electrostatic Green's function.