Jacobi theta function ($\theta (\tau )$)

The Jacobi theta functionsatisfies $\theta (\tau +2)=\theta (\tau )$ and $\theta (-1/\tau )=(-i\tau {)}^{1/2}\theta (\tau )$.

For $|q|<1$ and $z\ne 0$, the Jacobi triple product is

The theta group is ${\Gamma }_{\theta }=\Gamma (2)\cup \Gamma (2)S$. It is generated by $T^2$ and $S$ and has index three in the modular group.

The Jacobi triple product givesEvery factor is nonzero for $|q|<1$, and the product converges to a nonzero limit, so $\theta$ has no zero in the upper half-plane.