Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity

TL;DR

The paper develops convergent series expansions and bounds for Legendre's incomplete elliptic integral of the second kind near its logarithmic singularity, providing accurate approximations and error estimates.

Contribution

It introduces two new series expansions for the elliptic integral that converge everywhere in the unit square and offers explicit error bounds and inequalities.

Findings

Series expansions converge at all points in the unit square.

Explicit two-sided error bounds are provided for approximations.

Numerical examples demonstrate high accuracy of the expansions.

Abstract

We find two series expansions for Legendre's second incomplete elliptic integral $E(\lambda, k)$ in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the $(\lambda, k)$ plane. Partial sums of the proposed expansions form a sequence of approximations to $E(\lambda,k)$ which are asymptotic when $\lambda$ and/or $k$ tend to unity, including when both approach the logarithmic singularity $\lambda=k=1$ from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for $E(\lambda, k)$. For the reader's convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.