Laplace series expansion of the internal potential of a homogeneous circular torus

Abstract

The internal potential of a homogeneous circular torus first is represented by a series expansion in spherical functions (Laplace series). Exact analytical formulas for the coefficients of this series are derived and it is shown that they can be expressed through the standard Gauss hypergeometric function depending only on the geometric parameter of the torus. Convergence of the series is proved and the radius of convergence is determined. The relation of the radius with the torus geometrical parameter is found. A special spherical shell, where the problem of expansion of the torus potential should be solved in additional investigations, is detected.