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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.2012-05-30
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    Unimportant comment: this is usually denoted by $\wp$ (`\wp` in $\LaTeX$), at least in the number theory literature.2012-05-30
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    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 $\mathbb{C}$.2012-05-30
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    @QiaochuYuan: And the difference is bounded as it's periodic without poles, right?2012-05-30
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    Thanks Dylan, I'll use that next time!2012-05-30
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    Also - does this mean that $\Lambda$-periodic functions are unique up to poles?2012-05-30
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    Not clear what you mean by "unique up to poles". If $f$ is doubly-periodic, so is $17f-23$, and with the same poles.2012-05-31
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    Yes, that's fine. I see my mistake now!2012-05-31
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    See markusevich book on theory of functions of a complex variable PART III chapters 5 and 62013-12-20

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