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 J0(kr) or K0(kr), r2 = (x-x0)2+(y-y0)2, in parabolic and elliptic cylinder functions. Advantage is taken of the fact that K0(kr) is a fundamental solution and J0(kr) is the Riemann function of partial differential equations on the Euclidean plane.
Citation: Journal of Physics A: Mathematical and Theoretical
Pub Type: Journals
Fundamental solution, Laplace equation, Parabolic cylinder coordinates, Elliptic cylinder coordinates, Parabolic cylinder harmonics, Confluent hypergeometric functions, Mathieu functions