Definition:Weierstrass's Elliptic Function

Definition
Weierstrass's Elliptic Function is an elliptic function, given for all $z$ (except at $z \in \left\{ {2 m \omega_1 + 2 n \omega_2: \left({n, m}\right) \in \Z^2}\right\}$ where the function has double poles, by Poles of Weierstrass's Elliptic Function) by:


 * $\displaystyle \wp \left({z; \omega_1, \omega_2}\right) = \frac 1 {z^2} + {\sum_{\left({n, m}\right) \mathop \in \Z^2 \setminus \left({0, 0}\right) } } \left({ \frac 1 {\left({z - 2 m \omega_1 - 2 n \omega_2}\right)^2} - \frac 1 {\left({2 m \omega_1 + 2 n \omega_2}\right)^2} }\right)$

where $\omega_1$ and $\omega_2$ are non-zero constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.

Also known as
The Weierstrass elliptic function is also known as the Weierstrass P-function.