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The complete $p$-elliptic integrals and a computation formula of $π_p$ for $p=4$ (1503.02394v1)
Published 9 Mar 2015 in math.CA, math.AP, and math.NT
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.
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