Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendreʹs first incomplete elliptic integral F ( λ , k ) in terms of recursively computed elementary functions. Both expansions are valid at every point of the unit square 0 < λ , k < 1 . Truncated expansions yield asymptotic approximations for F ( λ , k ) as λ and/or k tend to unity, including the case when logarithmic singularity λ = k = 1 is approached from any direction. Explicit error bounds are given at every order of approximation. For the readerʹs 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.