Definition:Weierstrass's Elliptic Function

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


 * $\ds \map \wp {z; \omega_1, \omega_2} = \frac 1 {z^2} + {\sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } } \paren {\frac 1 {\paren {z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }$

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

Also known as
Some sources use the form Weierstrass elliptic function.

Weierstrass's elliptic function is also known as the Weierstrass P-function.

Also see

 * Weierstrass's Elliptic Function Converges Locally Uniformly Absolutely: $\map \wp {z; \omega_1, \omega_2}$ is holomorphic on $\C \setminus \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$