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

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation