MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 5 down vote accepted

This is the functional equation for the theta function. A nice proof (using Poisson summation) can be found here.

share|cite|improve this answer
This IS the Poisson summation:-) – Alexandre Eremenko Jul 22 '13 at 16:32
Yes, of course. – Igor Rivin Jul 22 '13 at 16:45
Thank you! That file is very helpful! – Steven Jul 22 '13 at 17:07

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.