Definition:Jacobi Theta Function

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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.


Sources