I am reading Elliptic Curves by Moll and McKean and it defines Jacobi's theta function on the lattice $\Gamma = \{n+m\omega \mid m,n \in \mathbb{Z}\}$ for a $\omega \in \mathbb{H}$ as below:
$$\vartheta (x) = \sum_{n\in\mathbb{Z}}{(-1)^n \exp{[2\pi inx +\pi in(n+1)\omega]}} = \sum_{n\in\mathbb{Z}}{(-1)^n e^{2\pi inx}e^{\pi i n(n+1)\omega}}$$
An exercise in page 105 wants us to prove the following identity:
$$\wp (x)=-[\frac{\vartheta'(0)}{\vartheta(x_0)}]^2 \times \frac{\vartheta(x-x_0)\vartheta(x+x_0)}{\vartheta^2(x)}\times e^{-2\pi ix_0}$$ where $x_0$ is a zero of $\wp(x)$ in the fundamental cell.
I tried to calculate the right-hand side of the above equation using the definitions of $\vartheta$ and $\wp$ but it does not work and everything gets messy!
Any help is appreciated!
You need to say that $\vartheta(0)=0$, to find how $\vartheta$ transforms under a translation by $\omega$, integrating $\vartheta'/\vartheta$ on $[-1/2,-1/2] +[-1/2,1/2]\omega$ you'll find that its only zeros are simple at the lattice points, from which you'll get that $ \frac{\vartheta(x-x_0)\vartheta(x+x_0)}{\vartheta^2(x)}$ is a constant times $\wp(x)$.
The constant is found by looking at the Laurent coefficient of the double pole at $0$.
(This is not very different to proving the Jacobi triple product, up to a constant)