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Peanut harmonic expansion for a fundamental solution of Laplace's equation in flat-ring coordinates

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.