So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5:
Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function $$\theta(z | r) = \sum_{n=-\infty}^{\infty}{ e^{\pi i n^{2} \tau} e^{2\pi i n z} }$$ is of (growth) order $2$ as a function of z
They give this hint: $-n^{2}t + 2n|z| \leq - n^{2}t /2$ when $t > 0$ and $n \geq 4 |z|/t$, but I don't understand how to use it.
Any help is to be well received //or any reference of course.
Thanks!