Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates

TL;DR

This paper develops a new harmonic expansion for Laplace's fundamental solution in flat-ring coordinates, utilizing Lamé-Wangerin functions, and derives related addition theorems and integral identities.

Contribution

It introduces a novel peanut harmonic expansion in flat-ring coordinates using Lamé-Wangerin functions, extending classical solutions and addition theorems.

Findings

Derived a double series expansion of the fundamental solution in flat-ring coordinates.

Established an addition theorem involving Legendre functions and Lamé-Wangerin functions.

Obtained integral identities for products of Lamé-Wangerin functions.

Abstract

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lam\'e-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lam\'e-Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lam\'e-Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.