# Definition:Jacobi Theta Function

## Definition

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

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

Then the Jacobi Theta functions are defined for all complex $z$ by:

### First Type

$\displaystyle \vartheta_1 \left({z, q}\right) = 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{\left({n + \frac 1 2}\right)^2} \sin \left({2 n + 1}\right) z$

### Second Type

$\displaystyle \vartheta_2 \left({z, q}\right) = 2 \sum_{n \mathop = 0}^\infty q^{\left({n + \frac 1 2}\right)^2} \cos \left({2 n + 1}\right) z$

### Third Type

$\displaystyle \vartheta_3 \left({z, q}\right) = 1 + 2 \sum_{n \mathop = 0}^\infty q^{n^2} \cos 2 n z$

### Fourth Type

$\displaystyle \vartheta_4 \left({z, q}\right) = 1 + 2 \sum_{n \mathop = 0}^\infty \left({-1}\right)^n q^{n^2} \cos 2 n z$

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.