  We find two convergent series expansions for Legendre's first incomplete
elliptic integral $F(\lambda,k)$ in terms of recursively computed elementary
functions. Both expansions are valid at every point of the unit square
$0<\lambda,k<1$. Truncated expansions yield asymptotic approximations for
$F(\lambda,k)$ as $\lambda$ and/or $k$ tend to unity, including the case when
logarithmic singularity $\lambda=k=1$ is approached from any direction.
Explicit error bounds are given at every order of approximation. For the
reader's convenience we present explicit expressions for low-order
approximations and numerical examples to illustrate their accuracy. Our
derivation is based on rearrangements of some known double series expansions,
hypergeometric summation algorithms and inequalities for hypergeometric
functions.
