Fourier series for the Legendre elliptic integrals

Abstract

Simple Fourier series in terms of the amplitude φ are derived for the common Legendre elliptic integrals. The coefficients in the various series contain Legendre functions of the second kind and half-integral degree, also known as toroidal functions. For the Legendre elliptic integrals of the first and second kinds, F(φ, k) and E(φ, k), repectively, the expansions given are simple sine series in the amplitude φ and an additional aperiodic term proportional to φ. These series are valid for φ∈ℝ. The complete elliptic integral of the third kind Π(α2, k) can be expressed in terms of Heuman’s lambda function Λ0(β, k) and the Jacobi zeta function Z(β, k), which in turn can be expressed in terms of the integrals of the first and second kinds. This enables simple sine series in terms of β to be derived for Λ0(β, k) and Z(β, k). The series for Λ0(β, k) has an aperiodic term in β but the series for Z(β, k) does not. The various series are obtained by first expanding the delta amplitude and Δ−1(φ, k) as cosine series in the amplitude φ and integrating term by term with repect to φ. The recurrence relation for the Legendre functions is frequently used to simplify or rearrange the various series.