  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.
