Determining Formulas for the Approximation of the Complete Elliptic Integrals

Abstract

This is a book chapter of an article published first time in the journal IMF in 2012. In section 2 of the chapter, we give evaluations of Legendre functions, which are quite general. As a consequence in section 3, we give the two complete elliptic integrals K and E in the form of Ramanujan’s type-1/ formulas. These formulas, consit of a new way, to write K and E other form the traditional one. As main application, in section 4 a formula for the constant \(\Gamma\)\((1/4)^2\) \(\pi^{-3/2}\), with accuracy about 120 digits per term is given.