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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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