Editing Weierstrass elliptic function (section)

==Definition==
[[File:Weierstrass elliptic function P.png|thumb|200px|Visualization of the <math>\wp</math>-function with invariants <math>g_2=1+i</math> and <math>g_3=2-3i</math> in which white corresponds to a pole, black to a zero.]]

Let <math>\omega_1,\omega_2\in\mathbb{C}</math> be two [[complex number]]s that are [[Linear independence|linearly independent]] over <math>\mathbb{R}</math> and let <math>\Lambda:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2:=\{m\omega_1+n\omega_2: m,n\in\mathbb{Z}\}</math> be the [[period lattice]] generated by those numbers. Then the <math>\wp</math>-function is defined as follows:
:<math>\weierp(z,\omega_1,\omega_2):=\wp(z) = \frac{1}{z^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac 1 {(z-\lambda)^2} - \frac 1 {\lambda^2}\right).</math>
This series converges locally [[Uniform absolute-convergence|uniformly absolutely]] in the [[complex torus]] <math>\mathbb{C} / \Lambda</math>.

It is common to use <math>1</math> and <math>\tau</math> in the [[upper half-plane]] <math>\mathbb{H}:=\{z\in\mathbb{C}:\operatorname{Im}(z) > 0\}</math> as [[Linear_span|generators]] of the [[Lattice_(group)|lattice]]. Dividing by <math display="inline">\omega_1</math> maps the lattice <math>\mathbb{Z}\omega_1+\mathbb{Z}\omega_2</math> isomorphically onto the lattice <math>\mathbb{Z}+\mathbb{Z}\tau</math> with <math display="inline">\tau=\tfrac{\omega_2}{\omega_1}</math>. Because <math>-\tau</math> can be substituted for <math>\tau</math>, without loss of generality we can assume <math>\tau\in\mathbb{H}</math>, and then define <math>\wp(z,\tau) := \wp(z, 1,\tau)</math>.  With that definition, we have <math>\wp(z,\omega_1,\omega_2) = \omega_1^{-2}\wp(z/\omega_1,\omega_2/\omega_1)</math>.