# Definition:Weierstrass's Elliptic Function

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## 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 $\C \setminus \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$

- Results about
**Weierstrass's Elliptic Function**can be found**here**.

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

## Sources

- 1920: E.T. Whittaker and G.N. Watson:
*A Course of Modern Analysis*(3rd ed.): $20.2$: The construction of an elliptic function. Definition of $\map \wp z$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Weierstrass elliptic function**