Fourier series for the Legendre elliptic integrals.

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(α 2 , k ) can be expressed in terms of Heuman’s lambda function Λ 0 (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 Λ 0 (β, k ) and Z (β, k ). The series for Λ 0 (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. [ABSTRACT FROM AUTHOR]