Editing Weierstrass elliptic function (section)

== Properties ==
* <math>\wp</math> is a [[meromorphic function]] with a pole of order 2 at each period <math>\lambda</math> in <math>\Lambda</math>.
* <math>\wp</math> is a [[homogeneous function]] in that:
::<math>\wp(\lambda z , \lambda\omega_{1}, \lambda\omega_{2}) = \lambda^{-2} \wp (z, \omega_{1},\omega_{2}).</math> 
* <math>\wp</math> is an even function. That means <math>\wp(z)=\wp(-z)</math> for all <math>z \in \mathbb{C} \setminus \Lambda</math>, which can be seen in the following way:
::<math>\begin{align}
\wp(-z) & =\frac{1}{(-z)^2} + \sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(-z-\lambda)^2}-\frac{1}{\lambda^2}\right) \\
& =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z+\lambda)^2}-\frac{1}{\lambda^2}\right) \\
& =\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus\{0\}}\left(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2}\right)=\wp(z).
\end{align}</math>
:The second last equality holds because <math>\{-\lambda:\lambda \in \Lambda\}=\Lambda</math>. Since the sum converges absolutely this rearrangement does not change the limit.
* The derivative of <math>\wp</math> is given by:<ref name=":1">{{citation|surname1=Apostol, Tom M.|title=Modular functions and Dirichlet series in number theory|publisher=Springer-Verlag|publication-place=New York|at=p.&nbsp;11|isbn=0-387-90185-X|date=1976| language=German}}</ref> <math display="block">\wp'(z)=-2\sum_{\lambda \in \Lambda}\frac1{(z-\lambda)^3}.</math>
* <math>\wp</math> and <math>\wp'</math> are [[Doubly periodic function|doubly periodic]] with the periods <math>\omega_1 </math> and <math>\omega_2</math>.<ref name=":1" /> This means: <math display="block">\begin{aligned}
\wp(z+\omega_1) &= \wp(z) = \wp(z+\omega_2),\ \textrm{and} \\[3mu]
\wp'(z+\omega_1) &= \wp'(z) = \wp'(z+\omega_2).
\end{aligned}</math> It follows that <math>\wp(z+\lambda)=\wp(z)</math> and <math>\wp'(z+\lambda)=\wp'(z)</math> for all <math>\lambda \in \Lambda</math>.