\begin{definition}[Definition:Jacobi Theta Function/First Type]
Let $\tau$ be a complex constant with a positive imaginary part. 
Let $q = e^{i \pi \tau}$. 
The '''Jacobi Theta function of the first type''' is defined for all complex $z$ by:
:$\ds \map {\vartheta_1} {z, q} = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n q^{\paren {n + \frac 1 2}^2} \sin \paren {2 n + 1} z$
\end{definition}