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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

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

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  • $\begingroup$ This IS the Poisson summation:-) $\endgroup$
    – Alexandre Eremenko
    Commented Jul 22, 2013 at 16:32
  • $\begingroup$ Yes, of course. $\endgroup$
    – Igor Rivin
    Commented Jul 22, 2013 at 16:45

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