INFINITE PRODUCTS AND JACOBI THETA FUNCTION

This entry has some of the infinite product related to Jacobi Theta function proved.

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\begin{align*} Q_0\left(e^{-\pi}\right)= \prod_{n=1}^\infty(1-e^{-2\pi n})=\frac{e^{\pi/12}\Gamma\left(\frac{1}{4} \right)}{2\pi^{3/4}} \end{align*}

The Jacobi triple product proved here, plays a central role in their proof as well some special values of the Jacobi theta function, see here



\begin{align*} \sum_{n=-\infty}^{\infty} q^{n^{2}} z^{n}&=\prod_{n=1}^{\infty}\left(1+z q^{2 n-1}\right)\left(1+z^{-1} q^{2 n-1}\right)\left(1-q^{2 n}\right) \end{align*}

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