Category:Definitions/Jacobi Theta Functions

This category contains definitions related to Jacobi Theta Functions.
Related results can be found in Category:Jacobi Theta Functions.

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

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

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

First Type

$\ds \map {\vartheta_1} {z, q} = 2 \sum_{n \mathop = 0}^\infty \paren {-1}^n q^{\paren {n + \frac 1 2}^2} \sin \paren {2 n + 1} z$

Second Type

$\ds \map {\vartheta_2} {z, q} = 2 \sum_{n \mathop = 0}^\infty q^{\paren {n + \frac 1 2}^2} \map \cos {2 n + 1} z$

Third Type

$\ds \map {\vartheta_3} {z, q} = 1 + 2 \sum_{n \mathop = 1}^\infty q^{n^2} \cos 2 n z$

Fourth Type

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