Interior and exterior spherical basis solutions

(solutions of the Helmholtz-equation in spherical coordinates)

Interior solution

The interior solution must stay finite for all radii r
spherical basis-solution for interior fields

Exterior solution

The exterior solution must fulfill a harder constraint: the Sommerfeld radiation condition. If the function diverges near r=0, this is can be accepted. However, far-enough away, the solutions must decay with 1/r and their phase propagation must accord to the free-field wave length.
spherical basis-solution for radiation