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The incomplete elliptic integrals  ,  ,  , and  can be represented through more general functions. Through the hypergeometric Appell  function of two variables:
Through the generalized hypergeometric function of two variables:
Through elliptic theta functions, for example:
Through inverse Jacobi elliptic functions, for example:
Through Weierstrass elliptic functions and inverse elliptic nome  , for example:
Through some elliptic‐type functions, for example:
The incomplete elliptic integrals  ,  ,  , and  can be represented through some related functions, for example:
The incomplete elliptic integral  is related to the Jacobi amplitude by the following formulas, which demonstrate that the Jacobi amplitude is within a restricted domain, the inverse function of elliptic integral  :
The incomplete elliptic integrals  ,  and  can be represented through incomplete elliptic integral  by the following formulas:
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