# Why is Every Elliptic Function of Order $2$ the Möbius Tranform of a $\wp$-function?

I'm trying to prove that every elliptic function of order $2$ has the form

$$f(z)=\frac{a\wp(z-z_0)+b}{c\wp(z-z_0)+d}$$

I've got the following so far. Let $f$ be an elliptic function of order 2. Then $f$ has $2$ poles and $2$ zeroes inside the fundamental parallelogram $P$ counting with multiplicity. Choose $z_0$ s.t. $f(z_0)$ is a pole. Now $f(z)=\wp(z-z_0)$ at $z=z_0$. But now I don't really know what to do, because the multiplicity of my pole of $\wp$ doesn't necessarily match that of $f$ does it? Could someone clarify this for me, and give me a hint on how to proceed?

Many thanks!

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Don't worry - have worked it out now! – Edward Hughes May 31 '12 at 12:56
Could you outline your solution please, so that whoever in the future gets here doesn't have to ask again? Thanks! – Gregor Bruns May 31 '12 at 15:40