Peanut Harmonic Expansion for a Fundamental Solution of Laplace's Equation in Flat-Ring Coordinates
Hans Volkmer, Lijuan Bi, Howard Cohl
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.
, Bi, L.
and Cohl, H.
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 December 7, 2023)