Series expansions for the third incomplete elliptic integral via partial fraction decompositions

We find convergent double series expansions for Legendreʹs third incomplete elliptic integral valid in overlapping subdomains of the unit square. Truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity ( 1 , 1 ) if one of the variables approaches this point faster than the other. Each approximation is accompanied by an error bound.