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:
