The authors obtain two convergent series expansions for the incomplete elliptic integral of the first kind
valid at any point in the unit square . These expansions are expressed in terms of recursively computed elementary functions. The expansions are truncated after terms and, by expressing the tails as integrals combined with use of bounds for certain hypergeometric functions, explicit bounds for the remainders are obtained.
The truncated expansions yield asymptotic approximations for as and/or approach unity. The approximations also remain valid as the logarithmic singularity is approached in any direction. The first two approximations complete with error bounds are presented explicitly and numerical calculations are given to illustrate their accuracy.