Peanut Harmonic Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Coordinates

Published

April 21, 2022

Author(s)

Hans Volkmer, Lijuan Bi, Howard Cohl

Abstract

We derive an expansion for the fundamental solution of Laplace's equation in flat-ring cyclide coordinates in three-dimensional Euclidean space. This expansion is a double series of products of functions that are harmonic in the interior and exterior of coordinate surfaces which are peanut shaped and orthogonal to surfaces which are flat-rings. These internal and external peanut harmonic functions are expressed in terms of Lame-Wangerin functions. Using the expansion for the fundamental solution, we derive an addition theorem for the azimuthal Fourier component in terms of the odd-half-integer degree Legendre function of the second kind as an infinite series in Lame-Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lame-Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.

Citation

Analysis Mathematica

Volume

48

Issue

3

Pub Type

Journals

Download Paper

https://doi.org/10.1007/s10476-022-0175-1

Local Download

Keywords

aplace's equation, fundamental solution, separable curvilinear coordinate system, flat-ring cyclide coordinates, special functions, orthogonal polynomials.

Citation

Volkmer, H. , Bi, L. and Cohl, H. (2022), Peanut Harmonic Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Coordinates, Analysis Mathematica, [online], https://doi.org/10.1007/s10476-022-0175-1, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=934223 (Accessed October 15, 2025)