Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

TL;DR

This paper develops two methods to derive Fourier expansions of a logarithmic fundamental solution for the polyharmonic equation in even-dimensional spaces, enhancing analytical tools for such equations.

Contribution

It introduces algebraic and derivative-based approaches to obtain Fourier expansions of the logarithmic fundamental solution, providing new analytical techniques.

Findings

Derived Fourier expansion using polynomial construction.

Expressed Fourier series with associated Legendre functions.

Compared and validated two different expansion methods.

Abstract

In even-dimensional Euclidean space for integer powers of the Laplacian greater than or equal to the dimension divided by two, a fundamental solution for the polyharmonic equation has logarithmic behavior. We give two approaches for developing a Fourier expansion of this logarithmic fundamental solution. The first approach is algebraic and relies upon the construction of two-parameter polynomials. We describe some of the properties of these polynomials, and use them to derive the Fourier expansion for a logarithmic fundamental solution of the polyharmonic equation. The second approach depends on the computation of parameter derivatives of Fourier series for a power-law fundamental solution of the polyharmonic equation. The resulting Fourier series is given in terms of sums over associated Legendre functions of the first kind. We conclude by comparing the two approaches and giving the azimuthal Fourier series for a logarithmic fundamental solution of the polyharmonic equation in rotationally-invariant coordinate systems.