On Laplace equation solution in orthogonal similar oblate spheroidal coordinates

TL;DR

This paper derives the exterior solution of the azimuthally symmetric Laplace equation in the orthogonal similar oblate spheroidal coordinates, providing formulas and transformations useful for physics applications involving spheroidal geometries.

Contribution

It presents the first derivation of the exterior Laplace solution in SOS coordinates and establishes formulas, transformations, and relations for spheroidal harmonic functions.

Findings

Derived the exterior solution of Laplace equation in SOS coordinates.

Established transformation formulas between different SOS coordinate systems.

Expressed Legendre polynomials as finite sums of monomials.

Abstract

Orthogonal coordinate systems enable expressing the boundary conditions of differential equations in accord with the physical boundaries of the problem. It can significantly simplify calculations. The orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies or particles with the geometry of an oblate spheroid. The interior solution of the Laplace equation in the SOS coordinates was recently found; however, the exterior solution was missing. The exterior solution of the azimuthally symmetric Laplace equation in the SOS coordinates is derived. In the steps leading to this solution, important formulas of the SOS algebra are found. Various forms of the Laplace operator in the SOS coordinates in azimuthally symmetric case are shown. General transformation between two different SOS coordinate systems is derived. It is determined that the SOS harmonics are physically the same as the solid harmonics. Further, a formula expressing any generalized Legendre polynomial as a finite sum of monomials is found. The reported relations have potential application in geophysics, astrophysics, electrostatics and solid state physics (e.g. ferroic inclusions).