# Definition:Jacobi Theta Function/Fourth 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 fourth type is defined for all complex $z$ by:

$\ds \map {\vartheta_4} {z, q} = 1 + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n q^{n^2} \cos 2 n z$