Weierstrass Elliptic Function

The Weierstrass elliptic function, denoted by the symbol wp and named after Karl Weierstrass, is an elliptic function, meaning a doubly periodic meromorphic function, that takes a particularly simple form among such functions. Together with its derivative, the wp function can be used to parameterize elliptic curves, and it generates the entire field of elliptic functions belonging to a given period lattice, in a role comparable to how sine and cosine parameterize the circle. Because the function is invertible, it can be used to solve certain nonlinear differential equations whose only movable singularities are poles, a property known as the Painlevé property. It arose from the effort to parameterize cubic curves that, unlike conics, cannot be parameterized by rational functions, and it remains central to the classical theory of elliptic functions in complex analysis.