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