# Jacobi theta functions

## Exponential Fourier series

$\theta_1\!\left(z, \tau\right) = -i \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}$
$\theta_2\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({\left(n + 1 / 2\right)}^{2} \tau + \left(2 n + 1\right) z\right)}$
$\theta_3\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}$
$\theta_4\!\left(z, \tau\right) = \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i \left({n}^{2} \tau + 2 n z\right)}$

## Trigonometric Fourier series

$\theta_1\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \sin\!\left(\left(2 n + 1\right) \pi z\right)$
$\theta_2\!\left(z, \tau\right) = 2 \sum_{n=0}^{\infty} {e}^{\pi i {\left(n + 1 / 2\right)}^{2} \tau} \cos\!\left(\left(2 n + 1\right) \pi z\right)$
$\theta_3\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)$
$\theta_4\!\left(z, \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {e}^{\pi i {n}^{2} \tau} \cos\!\left(2 n \pi z\right)$

## Zeros

$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_1\!\left(z, \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_2\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_3\!\left(z, \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_4\!\left(z, \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$

## Symmetry

$\theta_1\!\left(-z, \tau\right) = -\theta_1\!\left(z, \tau\right)$
$\theta_2\!\left(-z, \tau\right) = \theta_2\!\left(z, \tau\right)$
$\theta_3\!\left(-z, \tau\right) = \theta_3\!\left(z, \tau\right)$
$\theta_4\!\left(-z, \tau\right) = \theta_4\!\left(z, \tau\right)$

## Periodicity

$\theta_1\!\left(z + 2 n, \tau\right) = \theta_1\!\left(z, \tau\right)$
$\theta_2\!\left(z + 2 n, \tau\right) = \theta_2\!\left(z, \tau\right)$
$\theta_3\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)$
$\theta_4\!\left(z + 2 n, \tau\right) = \theta_3\!\left(z, \tau\right)$

## Quasi-periodicity

$\theta_1\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_1\!\left(z, \tau\right)$
$\theta_2\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_2\!\left(z, \tau\right)$
$\theta_3\!\left(z + m + n \tau, \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_3\!\left(z, \tau\right)$
$\theta_4\!\left(z + m + n \tau, \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_4\!\left(z, \tau\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-06-18 07:49:59.356594 UTC