Theta function (awaiting review)

The (classical) theta function $\theta(z)$ is a holomorphic modular form of half-integral weight $1/2$ and level 4, defined by the formula \[ \theta(z)=\sum_{n=-\infty}^\infty e^{\pi iz n^2}. \]

It has the following product expansion: \[ \theta(z)=\prod_{n\geq1}(1+q^{n-1/2})^2(1-q^n), \] where $q=e^{2\pi iz}$.

The (classical) theta function is related to the Dedekind eta function via the formula: \[ \theta(z)=\frac{\eta^5(z)}{\eta^2(z/2)\eta^2(2z)}=\frac{\eta^2((z+1)/2)}{\eta(z+1)}. \]