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Eigenfunction expansions for a fundamental solution of Laplace's equation on R3 in parabolic and elliptic cylinder coordinates

Published

Author(s)

Howard S. Cohl, Hans Volkmer

Abstract

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
Volume
45

Keywords

Fundamental solution, Laplace equation, Parabolic cylinder coordinates, Elliptic cylinder coordinates, Parabolic cylinder harmonics, Confluent hypergeometric functions, Mathieu functions

Citation

Cohl, H. and Volkmer, H. (2012), Eigenfunction expansions for a fundamental solution of Laplace's equation on R<sup>3</sup> in parabolic and elliptic cylinder coordinates, Journal of Physics A: Mathematical and Theoretical, [online], https://doi.org/10.1088/1751-8113/45/35/355204 (Accessed April 18, 2024)
Created August 14, 2012, Updated June 2, 2021