Eigenfunction expansions for a fundamental solution of Laplace's equation on $\R^3$ in parabolic and elliptic cylinder coordinates

TL;DR

This paper develops eigenfunction expansions of the fundamental solution to Laplace's equation in three dimensions using parabolic and elliptic cylinder coordinates, involving Bessel functions and harmonic functions.

Contribution

It introduces new eigenfunction expansions of Laplace's fundamental solution in specialized coordinate systems, linking Bessel functions with harmonic functions in these coordinates.

Findings

Two types of expansions using Bessel functions $J_0(kr)$ and $K_0(kr)$

Expansions reduce to known functions in special cases

Provides tools for solving PDEs in parabolic and elliptic cylinders

Abstract

A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane.