Definition:Jacobi Theta Function/Second Type

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

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

The Jacobi Theta function of the second type is defined for all complex $z$ by:
 * $\ds \map {\vartheta_2} {z, q} = 2 \sum_{n \mathop = 0}^\infty q^{\paren {n + \frac 1 2}^2} \map \cos {2 n + 1} z$