# Argument transformations for Jacobi theta functions

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

## Reflection symmetry

### Even-odd symmetry

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

### Conjugate symmetry

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

## Periodicity

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

## Quasi-periodicity

### Single shifts

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

### General shifts

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

## Half-period or quarter-period shifts

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

## Theta functions represented in terms of each other

### Theta 1

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

### Theta 2

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

### Theta 3

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

### Theta 4

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

## Double argument

### Theta 1

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

### Theta 2

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

### Theta 3

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

### Theta 4

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

## Relations involving sums and differences of arguments

### Cross-products with two factors and double lattice parameter

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

### Cross-products with four factors

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

### Cross-products of squares

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