Connections within the group of Weierstrass functions and inverses and with other function groups

Representations through more general functions

The Weierstrass elliptic function and its inverse can be represented through the more general hypergeometric Appell function of two variables by the following formulas:

Representations through related equivalent functions

The Weierstrass functions , , , , , , and can be represented through some related equivalent functions, for example, through Jacobi functions:

where is modular lambda function, or through theta functions:

or through elliptic integrals and the inverse elliptic nome:

Relations to inverse functions

The Weierstrass function and its derivative are interconnected with the inverse functions and by the following formulas:

Representations through other Weierstrass functions

Each of the Weierstrass functions , , , , and can be expressed through the other Weierstrass functions using numerous formulas, for example:

Note that the Weierstrass functions , , , , and form a chain with respect to differentiation: