Definition:Jacobi Theta Function

Definition

The Jacobi theta functions are defined as follows:

First Type

The Jacobi Theta function of the first type is defined as:

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

where:

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

$\tau \in \C$ such that $\map \Im \tau > 0$

Second Type

The Jacobi Theta function of the second type is defined as:

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

where:

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

$\tau \in \C$ such that $\map \Im \tau > 0$

Third Type

The Jacobi Theta function of the third type is defined for all complex $z$ by:

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

where:

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

$\tau \in \C$ such that $\map \Im \tau > 0$

Fourth Type

The Jacobi Theta function of the fourth type is defined as:

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

where:

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

$\tau \in \C$ such that $\map \Im \tau > 0$

Also see

• Results about the Jacobi theta functions can be found here.

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