Fungrim home page

# Jacobi theta functions

## Definitions

Symbol: JacobiTheta $\theta_{j}\!\left(z , \tau\right)$ Jacobi theta function

## Illustrations

Main topic: Illustrations of Jacobi theta functions

### Variable argument

Image: X-ray of $\theta_{3}\!\left(z , i\right)$ on $z \in \left[-2, 2\right] + \left[-2, 2\right] i$ ### Variable lattice parameter

Image: X-ray of $\theta_{4}\!\left(0 , \tau\right)$ on $\tau \in \left[-\frac{5}{2}, \frac{5}{2}\right] + \left[0, 2\right] i$ Image: X-ray of $\theta_{1}\!\left(\frac{1}{3} + \frac{3}{4} i , \tau\right)$ on $\tau \in \left[-\frac{5}{2}, \frac{5}{2}\right] + \left[0, 2\right] i$ ## Series and product representations

### Trigonometric Fourier series

$\theta_{1}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {\left(-1\right)}^{n} {q}^{n \left(n + 1\right)} \sin\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{2}\!\left(z , \tau\right) = 2 {e}^{\pi i \tau / 4} \sum_{n=0}^{\infty} {q}^{n \left(n + 1\right)} \cos\!\left(\left(2 n + 1\right) \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{3}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{4}\!\left(z , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} {q}^{{n}^{2}} \cos\!\left(2 n \pi z\right)\; \text{ where } q = {e}^{\pi i \tau}$

### Jacobi triple product

$\theta_{3}\!\left(z , \tau\right) = \sum_{n=-\infty}^{\infty} {q}^{{n}^{2}} {w}^{2 n} = \prod_{n=1}^{\infty} \left(1 - {q}^{2 n}\right) \left(1 + {q}^{2 n - 1} {w}^{2}\right) \left(1 + {q}^{2 n - 1} {w}^{-2}\right)\; \text{ where } q = {e}^{\pi i \tau},\,w = {e}^{\pi i z}$

## Zeros

$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{1}\!\left(z , \tau\right) = \left\{ m + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{2}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + n \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{3}\!\left(z , \tau\right) = \left\{ \left(m + \frac{1}{2}\right) + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$
$\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \theta_{4}\!\left(z , \tau\right) = \left\{ m + \left(n + \frac{1}{2}\right) \tau : m \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \right\}$

## Argument transformations

### 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

$\theta_{1}\!\left(z + 2 n , \tau\right) = \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z + 2 n , \tau\right) = \theta_{2}\!\left(z , \tau\right)$
$\theta_{3}\!\left(z + 2 n , \tau\right) = \theta_{3}\!\left(z , \tau\right)$
$\theta_{4}\!\left(z + 2 n , \tau\right) = \theta_{4}\!\left(z , \tau\right)$

### Quasi-periodicity

$\theta_{1}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m + n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{m} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{2}\!\left(z , \tau\right)$
$\theta_{3}\!\left(z + m + n \tau , \tau\right) = {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{3}\!\left(z , \tau\right)$
$\theta_{4}\!\left(z + m + n \tau , \tau\right) = {\left(-1\right)}^{n} {e}^{-\pi i \left(\tau {n}^{2} + 2 n z\right)} \theta_{4}\!\left(z , \tau\right)$

## Lattice transformations

### Modular transformations

$\theta_{3}\!\left(z , \tau + 1\right) = \theta_{4}\!\left(z , \tau\right)$
$\theta_{3}\!\left(z , \frac{-1}{\tau}\right) = \sqrt{\frac{\tau}{i}} {e}^{\pi i \tau {z}^{2}} \theta_{3}\!\left(\tau z , \tau\right)$
$\theta_{j}\!\left(z , \frac{a \tau + b}{c \tau + d}\right) = \varepsilon_{j}\!\left(a, b, c, d\right) \sqrt{\frac{v}{i}} {e}^{\pi i c v {z}^{2}} \theta_{S_{j}\!\left(a, b, c, d\right)}\!\left(v z , \tau\right)\; \text{ where } v = c \tau + d$

### Multiplication of the lattice parameter

$\theta_{3}\!\left(z , \frac{\tau}{2}\right) = \frac{\theta_{2}^{2}\!\left(z, \tau\right) + \theta_{3}^{2}\!\left(z, \tau\right)}{\theta_{3}\!\left(0 , \frac{\tau}{2}\right)}$
$\theta_{3}\!\left(2 z , 2 \tau\right) = \frac{\theta_{1}^{2}\!\left(z, \tau\right) + \theta_{2}^{2}\!\left(z, \tau\right)}{2 \theta_{2}\!\left(0 , 2 \tau\right)}$
$\theta_{3}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) + \theta_{4}\!\left(z , \tau\right)}{2}$

## Differential equations

$\frac{d^{r}}{{d z}^{r}} \theta_{j}\!\left(z , \tau\right) = \theta^{(r)}_{j}\!\left(z , \tau\right)$
$\frac{d^{r}}{{d \tau}^{r}} \theta^{(s)}_{j}\!\left(z , \tau\right) = \frac{1}{{\left(4 \pi i\right)}^{r}} \theta^{(2 r + s)}_{j}\!\left(z , \tau\right)$

### Heat equation

$\theta''_{j}\!\left(z , \tau\right) - 4 \pi i \frac{d}{d \tau}\, \theta_{j}\!\left(z , \tau\right) = 0$

### Jacobi's differential equation

${\left(30 {D}_{1}^{3} - 15 {D}_{0} {D}_{1} {D}_{2} + {D}_{0}^{2} {D}_{3}\right)}^{2} + 32 {\left({D}_{0} {D}_{2} - 3 {D}_{1}^{2}\right)}^{3} + {\pi}^{2} {\left({D}_{0} {D}_{2} - 3 {D}_{1}^{2}\right)}^{2} {D}_{0}^{10} = 0\; \text{ where } {D}_{r} = \frac{d^{r}}{{d \tau}^{r}} \theta_{j}\!\left(0 , \tau\right)$

### Derivatives of ratios

$\frac{d}{d z}\, \frac{\theta_{1}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = \pi \theta_{2}^{2}\!\left(0, \tau\right) \frac{\theta_{3}\!\left(z , \tau\right) \theta_{4}\!\left(z , \tau\right)}{\theta_{2}^{2}\!\left(z, \tau\right)}$

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

2019-08-21 11:44:15.926409 UTC