The Fourier series expansions of the Legendre incomplete elliptic integrals of the first and second kind

Abstract

Recently, the Fourier series expansions of the Legendre incomplete elliptic integrals F(phi, k) and E(phi, k) of the first and second kind in terms of the amplitude phi were investigated and found in a series of papers. The expansions were derived in several ways, for instance, by using a hypergeometric series approach, and have coefficients involving either the hypergeometric function or the associated Legendre functions of the second kind. In this paper, it is shown that the Fourier series expansions of F(phi, k) and E(phi, k) can be obtained without any difficulty by applying the usual and more familiar Fourier-series technique. Moreover, as an interesting consequence of this approach, both the recently found expansions and the new expansions with coefficients which are solely linear combinations of the complete elliptic integrals of the first and second kind, K(k) and E(k), are obtained in a unified manner. Furthermore, unlike the previously known, the newly established results make it possible to easily compute the Fourier coefficients of F(phi, k) and E(phi, k) analytically.