= Theta function with characteristics
{title2=$\theta[\alpha,\beta](z,\tau)=\sum_{n\in\mathbb Z}e^{\pi i(n+\alpha)^2\tau+2\pi i(n+\alpha)(z+\beta)}$}
For real characteristics $\alpha,\beta$ and $\operatorname{Im}\tau>0$, Gaussian decay gives normal convergence, an entire function of $z$ and holomorphic dependence on $\tau$. Integer shifts of alpha leave the function unchanged; a unit shift of beta multiplies it by $e^{2\pi i\alpha}$. Its spatial quasi-periods are $\theta(z+1)=e^{2\pi i\alpha}\theta(z)$ and $\theta(z+\tau)=e^{-\pi i\tau-2\pi i(z+\beta)}\theta(z)$. Binary characteristic labels often mean alpha and beta divided by two, so the convention must be stated.
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