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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!

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1 Answer 1

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.

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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

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