# Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent.

Definition 1
$\wp(z)=cf(z)+d$ where $f$ is the elliptic function w.r.t. $\Lambda$ with a pole of order $2$ at $0$ and a zero of order $2$ at $-\frac{1}{2}-\frac{\lambda}{2}$, and $c$, $d$ constants s.t. $c_{-2}=1$, $c_0=0$ in the Laurent expansion of $\wp$ about $0$. In terms of the $\theta$ function $f(z)=e^{2\pi i z}\frac{\theta(z)^2}{\theta(z-\frac{1}{2}-\frac{\lambda}{2})^2}$.

Definition 2
$\wp(z)=\frac{1}{z^2}+\sum_{\lambda\in\Lambda\setminus0}\left(\frac{1}{(z-\lambda)^2}-\frac{1}{\lambda^2}\right)$

Has anyone got a good reference where these are proved to be equivalent, or a nice idea for a quick proof? Many thanks!

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Your probably need to use Liouville's theorem that any bounded entire function on $\mathbb{C}$ is constant. –  Zhen Lin May 30 '12 at 19:23
Unimportant comment: this is usually denoted by $\wp$ (\wp in $\LaTeX$), at least in the number theory literature. –  Dylan Moreland May 30 '12 at 19:24
If you know that both functions are $\Lambda$-periodic, show that their difference is holomorphic. As Zhen says, it follows that the difference is a bounded holomorphic functions on $\mathbb{C}$. –  Qiaochu Yuan May 30 '12 at 19:28
@QiaochuYuan: And the difference is bounded as it's periodic without poles, right? –  Edward Hughes May 30 '12 at 19:40
Thanks Dylan, I'll use that next time! –  Edward Hughes May 30 '12 at 19:45