# Lattice transformations for Jacobi theta functions

This topic lists identities for how Jacobi theta functions $\theta_{j}\!\left(z , \tau\right)$ transform when the lattice parameter $\tau$ is transformed. See Argument transformations for Jacobi theta functions for identities involving the argument $z$ when $\tau$ is fixed. See Jacobi theta functions for other properties of these functions.

## Reflection symmetry

$\theta_{j}\!\left(z , -\overline{\tau}\right) = \overline{\theta_{j}\!\left(\overline{z} , \tau\right)}$

## Basic modular transformations

### Single shift

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

### Single inversion

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

## General shifts

$\theta_{1}\!\left(z , \tau + n\right) = {e}^{\pi i n / 4} \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z , \tau + n\right) = {e}^{\pi i n / 4} \theta_{2}\!\left(z , \tau\right)$
$\theta_{3}\!\left(z , \tau + n\right) = \begin{cases} \theta_{3}\!\left(z , \tau\right), & n \text{ even}\\\theta_{4}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}$
$\theta_{4}\!\left(z , \tau + n\right) = \begin{cases} \theta_{4}\!\left(z , \tau\right), & n \text{ even}\\\theta_{3}\!\left(z , \tau\right), & n \text{ odd}\\ \end{cases}$
$\theta_{1}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z , \tau + 2 n\right) = {i}^{n} \theta_{2}\!\left(z , \tau\right)$
$\theta_{3}\!\left(z , \tau + 2 n\right) = \theta_{3}\!\left(z , \tau\right)$
$\theta_{4}\!\left(z , \tau + 2 n\right) = \theta_{4}\!\left(z , \tau\right)$
$\theta_{1}\!\left(z , \tau + 4 n\right) = {\left(-1\right)}^{n} \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z , \tau + 4 n\right) = {\left(-1\right)}^{n} \theta_{2}\!\left(z , \tau\right)$
$\theta_{1}\!\left(z , \tau + 8 n\right) = \theta_{1}\!\left(z , \tau\right)$
$\theta_{2}\!\left(z , \tau + 8 n\right) = \theta_{2}\!\left(z , \tau\right)$

## Helper functions for general modular transformations

### Index permutations

Symbol: JacobiThetaPermutation $S_{j}\!\left(a, b, c, d\right)$ Index permutation in modular transformation of Jacobi theta functions
$S_{j}\!\left(a, b, c, d\right) = \begin{cases} 1, & j = 1\\T\!\left(c, d\right), & j = 2\\T\!\left(a + c, b + d\right), & j = 3\\T\!\left(a, b\right), & j = 4\\ \end{cases}\; \text{ where } T\!\left(m, n\right) = \begin{cases} 1, & \left(m, n\right) \equiv \left(0, 0\right) \pmod {2}\\2, & \left(m, n\right) \equiv \left(0, 1\right) \pmod {2}\\4, & \left(m, n\right) \equiv \left(1, 0\right) \pmod {2}\\3, & \left(m, n\right) \equiv \left(1, 1\right) \pmod {2}\\ \end{cases}$

### Roots of unity

Symbol: JacobiThetaEpsilon $\varepsilon_{j}\!\left(a, b, c, d\right)$ Root of unity in modular transformation of Jacobi theta functions
$\varepsilon_{1}\!\left(a, b, c, d\right) = \begin{cases} \left( \frac{c}{d} \right) \exp\!\left(\frac{\pi i}{4} \left[d \left(b - c - 1\right) + 2\right]\right), & c \text{ even}\\\left( \frac{d}{c} \right) \exp\!\left(\frac{\pi i}{4} \left[c \left(a + d + 1\right) - 3\right]\right), & c \text{ odd}\\ \end{cases}$
$\varepsilon_{j}\!\left(a, b, c, d\right) = \frac{1}{\varepsilon_{1}\!\left(-d, b, c, -a\right)} \begin{cases} \exp\!\left(\frac{\pi i}{4} \left[\left(c - 2\right) d - 2 + 2 \left(1 - c\right) \delta_{d + 1}\right]\right), & j = 2\\\exp\!\left(\frac{\pi i}{4} \left[\left(a + c - 2\right) \left(b + d\right) - 3 + 2 \left(1 - a - c\right) \delta_{b + d + 1}\right]\right), & j = 3\\\exp\!\left(\frac{\pi i}{4} \left[\left(a - 2\right) b - 4 + 2 \left(1 - a\right) \delta_{b + 1}\right]\right), & j = 4\\ \end{cases}\; \text{ where } \delta_{n} = n \bmod 2$
${\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{4} = {\left(-1\right)}^{n}\; \text{ where } n = \begin{cases} a \left(b + d\right) + c d, & j = 1\\a \left(b + d\right), & j = 2\\a d, & j = 3\\d \left(a + c\right), & j = 4\\ \end{cases}$
${\left(\varepsilon_{j}\!\left(a, b, c, d\right)\right)}^{8} = 1$

## General modular transformations

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

## Half parameter

### Theta constants

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

### General arguments

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

## Double parameter

### Theta constants

$2 \theta_{2}\!\left(0 , 2 \tau\right) \theta_{3}\!\left(0 , 2 \tau\right) = \theta_{2}^{2}\!\left(0, \tau\right)$
$2 \theta_{2}^{2}\!\left(0, 2 \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) - \theta_{4}^{2}\!\left(0, \tau\right)$
$2 \theta_{3}^{2}\!\left(0, 2 \tau\right) = \theta_{3}^{2}\!\left(0, \tau\right) + \theta_{4}^{2}\!\left(0, \tau\right)$
$\theta_{4}^{2}\!\left(0, 2 \tau\right) = \theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right)$
$2 \theta'_{1}\!\left(0 , 2 \tau\right) \theta_{4}\!\left(0 , 2 \tau\right) = \theta'_{1}\!\left(0 , \tau\right) \theta_{2}\!\left(0 , \tau\right)$

### General arguments

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

$\theta_{2}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) - \theta_{4}\!\left(z , \tau\right)}{2}$
$\theta_{3}\!\left(2 z , 4 \tau\right) = \frac{\theta_{3}\!\left(z , \tau\right) + \theta_{4}\!\left(z , \tau\right)}{2}$
$\theta_{1}\!\left(4 z , 4 \tau\right) = \frac{\theta_{1}\!\left(z , \tau\right) \theta_{1}\!\left(\frac{1}{4} - z , \tau\right) \theta_{1}\!\left(\frac{1}{4} + z , \tau\right) \theta_{2}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}$
$\theta_{2}\!\left(4 z , 4 \tau\right) = \frac{\theta_{2}\!\left(\frac{1}{8} - z , \tau\right) \theta_{2}\!\left(\frac{1}{8} + z , \tau\right) \theta_{2}\!\left(\frac{3}{8} - z , \tau\right) \theta_{2}\!\left(\frac{3}{8} + z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}$
$\theta_{3}\!\left(4 z , 4 \tau\right) = \frac{\theta_{3}\!\left(\frac{1}{8} - z , \tau\right) \theta_{3}\!\left(\frac{1}{8} + z , \tau\right) \theta_{3}\!\left(\frac{3}{8} - z , \tau\right) \theta_{3}\!\left(\frac{3}{8} + z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}$
$\theta_{4}\!\left(4 z , 4 \tau\right) = \frac{\theta_{4}\!\left(z , \tau\right) \theta_{4}\!\left(\frac{1}{4} - z , \tau\right) \theta_{4}\!\left(\frac{1}{4} + z , \tau\right) \theta_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(0 , \tau\right) \theta_{4}\!\left(0 , \tau\right) \theta_{3}\!\left(\frac{1}{4} , \tau\right)}$