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EllipticPi






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Elliptic Integrals > EllipticPi[n,m] > Introduction to the complete elliptic integrals





Connections within the group of complete elliptic integrals and with other function groups


All complete elliptic integrals , , and can be represented through more general functions. Through the Gauss hypergeometric function:

Through the Meijer G function:

Through the hypergeometric Appell function of two variables:

Through the hypergeometric function of two variables:

Through the incomplete elliptic integrals:

Through the elliptic theta functions:

Through the arithmetic geometric mean:

Through the Jacobi elliptic functions:

Through the Weierstrass elliptic functions and inverse elliptic nome :

Through the Legendre and functions:

The complete elliptic integral is related to Jacobi amplitude by the following formula, which demonstrates that Jacobi amplitude is the some kind of inverse function to the elliptic integral :

All complete elliptic integrals , , and can be represented through other complete elliptic integrals by the following formulas:





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