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The complete $p$-elliptic integrals are generalizations of the completeelliptic 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 thearithmetic-geometric mean. This is a $\pi_p$-version of the celebrated formulaof $\pi$, independently proved by Salamin and Brent in 1976.

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