Complete \(p\)-elliptic integrals and a computation formula of \(\pi_p\) for \(p=4\)

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Abstract: The complete

p

-elliptic integrals are generalizations of the complete elliptic integrals by the generalized trigonometric function

s i n_{p} h e t a

and its half-period

p i_{p}

. It is shown, only for

p = 4

, that the generalized

p

-elliptic integrals yield a computation formula of

p i_{p}

in terms of the arithmetic-geometric mean. This is a

p i_{p}

-version of the celebrated formula of

p i

, independently proved by Salamin and Brent in 1976.

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