PlanetMath: proof of Jacobi's identity for $\vartheta$ functions

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proof of Jacobi's identity for $\vartheta$ functions

(Proof)

We start with the Fourier transform of $f(x) = e^{i \pi \tau x^2 + 2 i x z}$ :

$\displaystyle \int_{-\infty}^{+\infty} e^{i \pi \tau x^2 + 2 i x z} e^{2 \pi ixy} \, dx = (- i \tau)^{-1/2} e^{-i {(z + \pi y)^2 \over \pi \tau}}$

Applying the Poisson summation formula, we obtain the following:

$\displaystyle \sum_{n=-\infty}^{+\infty} e^{i \pi \tau n^2 + 2 i n z} = (- i \tau)^{-1/2} \sum_{n=-\infty}^{+\infty} e^{-i {(z + \pi n)^2 \over \pi \tau}}$

The left hand side equals $\vartheta_3 (z \mid \tau)$ . The right hand side can be rewritten as follows:

$\displaystyle \sum_{n=-\infty}^{+\infty} e^{-i {(z + \pi n)^2 \over \pi \tau}} ... ...over \tau}} = e^{-i {z^2 \over \pi \tau}} \vartheta_3 (z / \tau \mid -1 / \tau)$