On the singular integrals in the source region of electromagnetic fields

Kellogg's classical theory on the singularities of integrals expressing field quantities, like potential phi or force F, in the source region V of static (gravitational or electrostatic) fields is extended to electromagnetic fields. Beyond the similarities some additional features of the latter fields are pointed out. It is shown that this classical approach leads via unambiguous and elegant steps, mainly by direct integration, to explicit results for the field vectors A, H, E of certain particular source regions and source distributions; it is further shown that these results, obtained by the author in several of his previous publications, satisfy the inhomogeneous Maxwell's equations in the source region V and the proper continuity or discontinuity conditions at the boundary of V. Said in another way, this amounts to a more or less complete study of the inhomogeneous Helmholtz equation in V. The special care with which multidimensional improper integrals should be handled to avoid errors is illustrated by means of single, double (surface) and triple (volume) integrals of specific, simple functions that can be evaluated explicitly.