Definitions of Weierstrass functions and inverses

The Weierstrass elliptic function , its derivative , the Weierstrass sigma function , associated Weierstrass sigma functions , Weierstrass zeta function , inverse elliptic Weierstrass function , and generalized inverse Weierstrass function are defined by the following formulas:

The function is the unique value of for which and . For the existence of , the values and must be related by .

The previous nine functions are typically called Weierstrass elliptic functions. The last two functions are called inverse elliptic Weierstrass functions.

Despite the commonly used naming convention, only the Weierstrass function and its derivative are elliptic functions because only these functions are doubly periodic. The other Weierstrass functions , , and are not elliptic functions because they are only quasi‐periodic functions with respect to . But historically they are also placed into the class of elliptic functions.

The Weierstrass half‐periods and the invariants , the Weierstrass function values at half-periods , and the Weierstrass zeta function values at half-periods are defined by the following formulas. The description of the Weierstrass functions follows the notations used throughout. The left‐hand sides indicate that and are either independent variables or depend on and , or vice versa:

is the Klein invariant modular function, is the Weierstrass elliptic function, and denotes the Weierstrass zeta function.