Definition:Jacobi Theta Function/First Type

Definition
Let $\tau$ be a complex constant with a positive imaginary part.

Let $q = e^{i \pi \tau}$.

Then the Jacobi Theta function of the first type is defined for all complex $z$ by:
 * $\displaystyle \vartheta_1 \left({z, q}\right) = 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{\left({n + \frac 1 2}\right)^2} \sin \left({2 n + 1}\right) z$