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:
