Electromagnetic field in the source region of continuously varying current density

Continuity, analyticity, and the singular points of the vector potential A and the field vectors H, E in a spherical source region $\nu$ are investigated thoroughly for, practically, any continuous current density distribution J in $\nu$. In other words, this is a study of the inhomogeneous Helmholtz equation in $\nu$. Explicit results for A, H, E are obtained by direct integration, extending previous results for constant density in $\nu$ to continuously varying ones. The importance of imposing the Hölder condition on J to insure existence of E and of certain second derivatives of A is explicitly demonstrated by a specific continuous J, violating this condition at a point; it is then seen that E and some second derivatives of A do not exist, tending to infinity, at that point.