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

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 Lamé—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 Lamé—Wangerin functions. We also derive integral identities over the Legendre function of the second kind for a product of three Lamé—Wangerin functions. In a limiting case we obtain the expansion of the fundamental solution in spherical coordinates.