Asymptotic expansions of the complete elliptic integrals about unitary modulus

Power series in the complementary modulus for the first and second complete elliptic integrals are deduced in terms of binomial series, by exploiting a suitable decomposition of the integration domain. This approach appears to be straightforward, with respect to the standard one. However, despite the procedure is simple, it needs some non-trivial results about binomial series proved in the appendix. Numerical performances of the expansions are also discussed and compared with existing alternative expansions.