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A fundamental solution of Laplace's equation in three dimensions is expandedin harmonic functions that are separated in parabolic or elliptic cylindercoordinates. There are two expansions in each case which reduce to expansionsof the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, inparabolic 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 ofpartial differential equations on the Euclidean plane.

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