Theta function

Remarks

For $\operatorname{Im} \tau > 0$ the terms decay like $e^{-\pi n^2 \operatorname{Im} \tau}$ times a factor locally bounded in $z$, so the series converges absolutely and locally uniformly. So $\vartheta$ is holomorphic in both variables. It satisfies $\vartheta(z + 1, \tau) = \vartheta(z, \tau)$ and $\vartheta(z + \tau, \tau) = \exp(-\pi i \tau - 2 \pi i z)\, \vartheta(z, \tau)$. Check of the second rule: replace $n$ by $m - 1$ and use $n^2 + 2n = (n + 1)^2 - 1$. The theta constant satisfies $\theta(-1/\tau) = \sqrt{-i\tau}\, \theta(\tau)$, by the Poisson summation formula. Example: $\theta(\tau)^4 = \sum_{n \ge 0} r_4(n) e^{\pi i n \tau}$, where $r_4(n)$ counts the ways to write $n$ as a sum of four squares. Non-example: for real $\tau$ the series diverges, because every term has absolute value $1$ when $z$ is also real; this is why $\tau$ must lie in $\mathbb{H}$. Other authors use the nome $q = e^{\pi i \tau}$ and the four functions $\vartheta_1, \ldots, \vartheta_4$; the function above is then $\vartheta_3$.