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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.
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