\begin{definition}[Definition:Jacobi Theta Function/Third Type]
Let $\tau$ be a complex constant with a positive imaginary part. 
Let $q = e^{i \pi \tau}$. 
The '''Jacobi Theta function of the third type''' is defined for all complex $z$ by:
:$\ds \map {\vartheta_3} {z, q} = 1 + 2 \sum_{n \mathop = 1}^\infty q^{n^2} \cos 2 n z$
\end{definition}