Theta Function - Jacobi Identities

Jacobi Identities

Jacobi's identities describe how theta functions transform under the modular group, which is generated by τ ↦ τ+1 and τ ↦ -1/τ. We already have equations for the first transformation; for the second, let

$\alpha = (-i \tau)^{\frac{1}{2}} \exp\!\left(\frac{\pi}{\tau} i z^2 \right).\,$

Then

$\begin{align} \vartheta_{00}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{00}(z; \tau)\quad& \vartheta_{01}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{10}(z; \tau)\\ \vartheta_{10}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = \alpha\,\vartheta_{01}(z; \tau)\quad& \vartheta_{11}\!\left({\textstyle\frac{z}{\tau}; \frac{-1}{\tau}}\right)& = -i\alpha\,\vartheta_{11}(z; \tau). \end{align}$

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