# proof of Jacobi’s identity for $\vartheta$ functions

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

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

Applying the Poisson summation formula, we obtain the following:

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

The left hand equals $\vartheta_{3}(z\mid\tau)$. The right hand can be rewritten as follows:

 $\sum_{n=-\infty}^{+\infty}e^{-i{(z+\pi n)^{2}\over\pi\tau}}=e^{-i{z^{2}\over% \pi\tau}}\sum_{n=-\infty}^{+\infty}e^{-i{\pi n^{2}\over\tau}-{2inz\over\tau}}=% e^{-i{z^{2}\over\pi\tau}}\vartheta_{3}(z/\tau\mid-1/\tau)$

Combining the two expressions yields

 $\vartheta_{3}(z\mid\tau)=e^{-i{z^{2}\over\pi\tau}}\vartheta_{3}(z/\tau\mid-1/\tau)$
Title proof of Jacobi’s identity for $\vartheta$ functions ProofOfJacobisIdentityForvarthetaFunctions 2013-03-22 14:47:01 2013-03-22 14:47:01 rspuzio (6075) rspuzio (6075) 19 rspuzio (6075) Proof msc 33E05