| Summary: | We introduce a new class of solutions to the Laplace equation, dubbed logopoles, and use them to derive a new relation between solutions in prolate spheroidal and spherical coordinates. The main novelty is that it involves spherical harmonics of the second kind, which have rarely been considered in physical problems because they are singular on the entire z axis. Logopoles, in contrast, have a finite line singularity like solid spheroidal harmonics, but are also closely related to solid spherical harmonics and can be viewed as an extension of the standard multipole ladder toward the negative multipolar orders. As part of our derivations, we also found a new integral representation for the spherical harmonics of the second kind in terms of their source distributions. As an example application, we use logopoles to construct a fast converging series solution for the problem of a point charge interaction with a dielectric sphere, which extends the basic image approximations introduced by Kelvin, Kirkwood, and Friedman. We believe these new solutions will prove a fruitful alternative to either spherical or spheroidal harmonics in a wide range of other physical problems.
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