Definition:Jacobi Theta Function/Fourth 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 fourth type is defined for all complex $z$ by:
 * $\displaystyle \vartheta_4 \left({z, q}\right) = 1 + 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{n^2} \cos 2 n z$