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 three-dimensional Euclidean space. This is a double series products functions that are harmonic interior and exterior coordinate surfaces which peanut shaped orthogonal to flat-rings. These internal external expressed terms Lamé—Wangerin functions. Using solution, we addition theorem azimuthal Fourier component odd-half-integer degree Legendre function second kind as infinite also integral identities over product three In limiting case obtain spherical coordinates.