Connections within the group of Jacobi functions and with other elliptic functions

The twelve Jacobi functions , , , , , , , , , , , and can be represented through the Weierstrass sigma functions:

The twelve Jacobi functions , , , , , , , , , , , and can be represented through the Weierstrass function:

The twelve Jacobi functions , , , , , , , , , , , and can be represented through the elliptic theta functions:

The twelve Jacobi functions , , , , , , , , , , , and can be represented through the Neville theta functions:

The Jacobi functions , , , , , , , , , , , , and are connected with the corresponding inverse functions by the following formulas:

By definition, the three basic Jacobi functions have the following representations through the amplitude function :

The other nine Jacobi functions can be easily expressed through the three basic Jacobi functions , , and and, consequently, they can also be represented through the amplitude function .

The twelve Jacobi functions , , , , , , , , , , , and are interconnected by formulas that include rational functions, simple powers, and arithmetical operations from other Jacobi functions. These formulas can be divided into the following eleven groups:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are:

Representations of through other Jacobi functions are: