Definition:Jacobi Theta Function

Definition

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

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

Source of Name

This entry was named for Carl Gustav Jacob Jacobi.

Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $21.11$: The four types of Theta-functions

• Weisstein, Eric W. "Jacobi Theta Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiThetaFunctions.html