Asymptotic Expansions of Symmetric Standard Elliptic Integrals

Symmetric standard elliptic integrals are considered when one of their parameters is larger than the others. Distributional approach is used for deriving five convergent expansions of these integrals in inverse powers of the respective five possible asymptotic parameters. Four of these expansions involve also a logarithmic term in the asymptotic variable. Coefficients of these expansions are obtained by recurrence. For the first four expansions these coefficients are expressed in terms of elementary functions, whereas coefficients of the fifth expansion involve non-elementary functions. Convergence speed of any of these expansions increases for increasing difference between the asymptotic variable and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation.