The complete $p$-elliptic integrals and a computation formula of ${\pi }_p$ for $p=4$

Abstract

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

13 pages

References in corpus (1)

• A new form of the generalized complete elliptic integrals