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