Fourier Expansions of Functions of the Distance between Two Points

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Abstract
The expansion of any function f(r) of the distance r between two points is given as a Fourier series. This is a generalization of results given earlier by Ashour for rn and log r. The Fourier‐series expansion for the product rNeiMθ is also given, where now r = (r, θ) denotes the sum of r1 and r2, the coordinate vectors of the two points. This is further generalized for the product of a function f(r) of r and a circular harmonic. Expansions of similar products involving spherical harmonics have been given earlier by Sack. Special cases when f(r) is a Bessel function or a modified Bessel function are considered.