Formulas for the approximation of the complete Elliptic Integrals

TL;DR

This paper presents Ramanujan-type formulas for approximating complete elliptic integrals, providing highly accurate evaluations of special constants like (1/4)^2/3, with potential applications in high-precision computations.

Contribution

It introduces new Ramanujan-type formulas for elliptic integrals that enable extremely precise numerical evaluations of related constants.

Findings

Derived formulas achieve about 120 digits of accuracy per term.

Provided explicit formulas for elliptic integrals K and E.

Achieved high-precision evaluation of (1/4)^2/3.

Abstract

In this article we give evaluations of the two complete elliptic integrals $K$ and $E$ in the form of Ramanujans type-$\pi$ formulas. The result is a formula for $\Gamma(1/4)^2\pi^{-3/2}$ with accuracy about 120 digits per term.