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!



Function transformation of exponentials
This is the functional equation for the theta function. A nice proof (using Poisson summation) can be found [here.](http://math.mit.edu/~brubaker/Math784/thetafunctions.pdf)


