Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral F(λ,k)F(λ,k) in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square 0<λ,k<10<λ,k<1. Truncated expansions yield asymptotic approximations for F(λ,k)F(λ,k) as λλ and/or k   tend to unity, including the case when logarithmic singularity λ=k=1λ=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.