An Eloquent Formula for the Perimeter of an Ellipse

Abstract

The values of complete elliptic integrals of the first and the second kind are expressible via power series representations of the hypergeometric function (with corresponding arguments). The complete elliptic integral of the first kind is also known to be eloquently expressible via an arithmetic-geometric mean, whereas (before now) the complete elliptic integral of the second kind has been deprived such an expression (of supreme power and simplicity). With this paper, the quest for a concise formula giving rise to an exact it- erative swiftly convergent method permitting the calculation of the perimeter of an ellipse is over!