| 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: | 
