How to prove the transformation formula for Jacobi classic theta function

Question

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 $\theta(x)=\sum_{n\in \mathbb{Z}} e^{-\pi n^2 x}$?

Answer 1

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 $$\theta(x)=\sum_{n\in\mathbb{Z}}f(n)=\sum_{k\in\mathbb{Z}}\underbrace{\int_{-\infty}^{\infty}e^{-\pi x t^2-2\pi i k t}dt}_{\hat{f}(k)}= \sum_{k\in\mathbb{Z}}\frac{e^{-\pi k^2/x}}{\sqrt{x}}=\frac{\theta\left(x^{-1}\right)}{\sqrt{x}}.$$