# Theta functions identites

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.

• I suggest reading DLMF Chapter 20 on Theta functions. There is an infinite product expansion. Apr 17, 2020 at 11:24
• What's the definition of this particular theta function? Apr 17, 2020 at 12:13
• Yeah sorry i edited the post @AnginaSeng Apr 17, 2020 at 15:27

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}
• @Something Integrate $\theta'/\theta$ over the boundary of a fundamental region. Apr 17, 2020 at 16:11