# Showing that an elliptic function has no poles

Let $$\Lambda = \{m \omega_1+n\omega_2; m,n \in \mathbb{Z}\}$$ with $$\omega_i \in \mathbb{C}$$ with $$\omega_2/\omega_1 \notin \mathbb{R}$$ be a lattice. Define the Weierstrass $$\mathscr{P}$$ function on the torus $$\mathbb{C}/\Lambda$$ as $$\mathscr{P}(z) = \frac{1}{z^2}+\sum\limits_{\omega \in \Lambda} (\frac{1}{(z-\omega)^2}-\frac{1}{\omega^2})$$

I am asked to show that if we choose $$a \in \mathbb{C}/\Lambda$$ with $$a \notin \frac{1}{2}\Lambda$$ then the elliptic function $$h(z) = (\mathscr{P}(z-a) − \mathscr{P}(z+a))(\mathscr{P}(z)-\mathscr{P}(a))^2-\mathscr{P}'(z)\mathscr{P}'(a)$$

has no poles and is hence constant. To do this I tried to show that $$h$$ is analytic. I attempted to do this by taking Laurent expansions around $$0,a,-a$$ and showing that $$h$$ was analytic in a disc around all those points (as those are the only possible points where $$h$$ can have a pole) but was not able to get this result. I don't have any other ideas for what to do.

How should I go about doing this problem? Any hints are appreciated!

This function (with the sum over $$\omega\in\Lambda\color{red}{\setminus\{0\}}$$) is commonly denoted $$\wp(z)$$.
You have $$\wp(-z)=\wp(z)$$, $$\wp(z)=z^{-2}+O(z^2),\quad\wp'(z)=-2z^{-3}+O(z)\qquad(z\to 0)$$ and $$\wp'''(z)=12\wp(z)\wp'(z)$$ (follows from $$\wp''(z)=6\wp^2(z)-g_2/2$$ which in turn follows from well-known $$\wp'^2(z)=4\wp^3(z)-g_2\wp(z)-g_3$$, or, the more elementary way, can just be deduced from Laurent expansions of $$\wp$$, $$\wp'$$ and $$\wp'''$$ — all the negative powers of $$z$$ except $$-24z^{-5}$$ vanish).
This is sufficient. With $$z\to 0$$ you have \begin{align}h(z)&=\big(-2\wp'(a)z-\wp'''(a)z^3/3+O(z^5)\big)\big(z^{-2}-\wp(a)+O(z^2)\big)^2+2\wp'(a)z^{-3}+O(z)\\ &=-2\wp'(a)z^{-3}\big(1+2\wp(a)z^2+O(z^4)\big)\big(1-2\wp(a)z^2+O(z^4)\big)+2\wp'(a)z^{-3}+O(z)\\ &=8\wp^2(a)\wp'(a)z+O(z)=O(z),\end{align} and even easier things at $$z\to\pm a$$ when the double pole of $$\wp(z\mp a)$$ is compensated by (the) double zero of $$\big(\wp(z)-\wp(a)\big)^2$$.