The complete $p$-elliptic integrals and a computation formula of $π_p$ for $p=4$

Abstract

The complete $p$-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function $\sin_pθ$ and its half-period $π_p$. It is shown, only for $p=4$, that the generalized $p$-elliptic integrals yield a computation formula of $π_p$ in terms of the arithmetic-geometric mean. This is a $π_p$-version of the celebrated formula of $π$, independently proved by Salamin and Brent in 1976.

13 pages