TITLE: How to prove the transformation formula for Jacobi classic theta function
QUESTION [6 upvotes]: How to prove the following transformation formula:
$$
\theta(x)=\frac{1}{\sqrt{x}} \theta\left(\frac{1}{x}\right),
$$
where $\theta$ is the Jacobi theta function $\theta(x)=\sum_{n\in \mathbb{Z}} e^{-\pi n^2 x}$?

REPLY [7 votes]: One can use the Poisson summation formula:
$$ \sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\hat{f}(k),$$
where $\hat{f}(\nu)$ denotes the Fourier transform of $f(t)$,
$$ \hat{f}(\nu)=\int_{-\infty}^{\infty}f(t)e^{-2\pi i \nu t}dt.$$
Namely, setting $f(t)=e^{-\pi x t^2}$ in the above, we obtain
$$\theta(x)=\sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\underbrace{\int_{-\infty}^{\infty}e^{-\pi x t^2-2\pi i k t}dt}_{\hat{f}(k)}=
\sum_{k\in\mathbb{Z}}\frac{e^{-\pi k^2/x}}{\sqrt{x}}=\frac{\theta\left(x^{-1}\right)}{\sqrt{x}}.$$