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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!

  • 0
    ...which "Stein and Shakarchi"?2011-10-22
  • 0
    it's Exercise 3 from Chapter 5 of Complex Analysis.2011-10-22

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