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

Representations through more general functions

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:

Representations through related functions

The incomplete elliptic integrals , , , and can be represented through some related functions, for example:

Relations to inverse functions

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 :

Representations through other incomplete elliptic integrals

The incomplete elliptic integrals , and can be represented through incomplete elliptic integral by the following formulas: