1202.1811 (Howard S. Cohl)

Fourier expansions for a logarithmic fundamental solution of the
polyharmonic equation    [PDF]

Howard S. Cohl

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.

View original: http://arxiv.org/abs/1202.1811