  We propose a powerful approach to solve Laplace's equation for point sources
near a spherical object. The central new idea is to use prolate spheroidal
solid harmonics, which are separable solutions of Laplace's equation in
spheroidal coordinates, instead of the more natural spherical solid harmonics.
We motivate this choice and show that the resulting series expansions converge
much faster. This improvement is discussed in terms of the singularity of the
solution and its analytic continuation. The benefits of this approach are
illustrated for a specific example: the calculation of modified decay rates of
light emitters close to nanostructures in the long-wavelength approximation. We
expect the general approach to be applicable with similar benefits to a variety
of other contexts, from other geometries to other equations of mathematical
physics.
