TITLE: Function transformation of exponentials
QUESTION [5 upvotes]: I came across the following function transformation:
$$
\sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j \in \mathbb{Z}$ (i.e. integers).
Can anyone help me to understand why this relation is true? Thanks!

REPLY [5 votes]: This is the functional equation for the theta function. A nice proof (using Poisson summation) can be found here.