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This is a question from Stein & Shakarchi's complex analysis book.

Show that if $\rho$ is fixed with $Im(\rho)>0$, then the Jacobi theta function $$\theta(z|\rho)=\sum_{n=-\infty}^\infty e^{\pi in^2\rho}e^{2\pi inz}$$ is of order 2 as a function of z. And here's the hint from the book:

$-n^2t + 2n|z| \le -(n^2t)/2$ when $t \gt 0$ and $n \ge 4|z|/t$

I've been teaching myself complex analysis with Stein's book but it's pretty difficult to do so. I can't quite figure out the steps to do a problem. Any help?

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