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

Representations through more general functions

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:

Relations to inverse 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 :

Representations through other complete elliptic integrals

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