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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity

We find two convergent series expansions for Legendre's first incomplete elliptic integral $F(λ,k)$ 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 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.