Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

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.

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