Theta functions identites

Question

I have been reading Rick Miranda's Book on Riemann surfaces and to indroduce meromorphic functions on the complex torues $\mathbb{C}$\ $L$ he talks about theta functions. I was able to see that $\theta(z+1)=\theta(z)$ but I cant quite seem to see why $\theta(\gamma+z)=e^{-\pi i(\gamma+2z)} \theta(z)$, maybe theres some identity about series that i am not seeing, also I cant seem to show that the only zeros of $\theta$ are at the points $(1/2)+(\gamma/2) +m+n\gamma$. The definition of this $\theta$-function is $\theta(z)=\sum_{n=-\infty}^{\infty}e^{\pi i[n^2\gamma+2nz]}$.

Any help is aprecciated, thanks in advance.

Answer 1

From $$\theta(z)=\sum_{n=-\infty}^{\infty}\exp(\pi i[n^2\gamma+2nz])$$ we get \begin{align} \theta(z+\gamma)&=\sum_{n=-\infty}^{\infty}\exp(\pi i[n^2\gamma+2nz+2n\gamma])\\ &=\exp(-\pi i\gamma)\sum_{n=-\infty}^{\infty}\exp(\pi i[(n+1)^2\gamma+2nz])\\ &=\exp(-\pi i\gamma-2\pi inz) \sum_{n=-\infty}^{\infty}\exp(\pi i[(n+1)^2\gamma+2(n+1)z])\\ &=\exp(-\pi i\gamma-2\pi inz)\theta(z). \end{align}