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We find two series expansions for Legendre's second incomplete ellipticintegral $E(\lambda, k)$ 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 ofapproximations 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 ateach approximation order. These bounds yield a sequence of increasingly preciseasymptotically correct two-sided inequalities for $E(\lambda, k)$. For thereader's convenience we further present explicit expressions for low-orderapproximations and numerical examples to illustrate their accuracy. Ourderivations are based on series rearrangements, hypergeometric summationalgorithms and extensive use of the properties of the generalizedhypergeometric functions including some recent inequalities.

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