# Series and product representations of Jacobi theta functions

See Jacobi theta functions for an introduction to these functions.

## Fourier series

### Trigonometric Fourier series

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

### Exponential Fourier series

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

### Pure exponential series

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

## Fourier series for derivatives

### Exponential Fourier series

$\theta^{(r)}_{1}\!\left(z , \tau\right) = -i {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$
$\theta^{(r)}_{2}\!\left(z , \tau\right) = {\left(\pi i\right)}^{r} {e}^{\pi i \tau / 4} \sum_{n=-\infty}^{\infty} {\left(2 n + 1\right)}^{r} {q}^{n \left(n + 1\right)} {w}^{2 n + 1}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$
$\theta^{(r)}_{3}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$
$\theta^{(r)}_{4}\!\left(z , \tau\right) = {\left(2 \pi i\right)}^{r} \sum_{n=-\infty}^{\infty} {\left(-1\right)}^{n} {n}^{r} {q}^{{n}^{2}} {w}^{2 n}\; \text{ where } q = {e}^{\pi i \tau},\;w = {e}^{\pi i z}$

## Infinite products

### Infinite q-products with trigonometric factors

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

### Infinite q-products with exponential factors

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

### Jacobi triple product

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

### Trigonometric infinite products

$\theta_{1}\!\left(z , \tau\right) = \frac{\theta'_{1}\!\left(0 , \tau\right)}{\pi} \sin\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(n \tau + z\right)\right) \sin\!\left(\pi \left(n \tau - z\right)\right)}{\sin^{2}\!\left(\pi n \tau\right)}$
$\theta_{2}\!\left(z , \tau\right) = \theta_{2}\!\left(0 , \tau\right) \cos\!\left(\pi z\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(n \tau + z\right)\right) \cos\!\left(\pi \left(n \tau - z\right)\right)}{\cos^{2}\!\left(\pi n \tau\right)}$
$\theta_{3}\!\left(z , \tau\right) = \theta_{3}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \cos\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\cos^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}$
$\theta_{4}\!\left(z , \tau\right) = \theta_{4}\!\left(0 , \tau\right) \prod_{n=1}^{\infty} \frac{\sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau + z\right)\right) \sin\!\left(\pi \left(\left(n - \frac{1}{2}\right) \tau - z\right)\right)}{\sin^{2}\!\left(\pi \left(n - \frac{1}{2}\right) \tau\right)}$

## Series for logarithmic derivatives

### Lambert series with trigonometric factors

$\frac{1}{\pi} \frac{\theta'_{1}\!\left(z , \tau\right)}{\theta_{1}\!\left(z , \tau\right)} = \cot\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\frac{1}{\pi} \frac{\theta'_{2}\!\left(z , \tau\right)}{\theta_{2}\!\left(z , \tau\right)} = -\tan\!\left(\pi z\right) + 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{2 n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\frac{1}{\pi} \frac{\theta'_{3}\!\left(z , \tau\right)}{\theta_{3}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} {\left(-1\right)}^{n} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}$
$\frac{1}{\pi} \frac{\theta'_{4}\!\left(z , \tau\right)}{\theta_{4}\!\left(z , \tau\right)} = 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 - {q}^{2 n}} \sin\!\left(2 \pi n z\right)\; \text{ where } q = {e}^{\pi i \tau}$

### Reciprocal trigonometric series

$\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{1}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\sin^{2}\!\left(\pi \left(z + n \tau\right)\right)}$
$\frac{d^{2}}{{d z}^{2}} \log\!\left(\theta_{2}\!\left(z , \tau\right)\right) = {\pi}^{2} \sum_{n=-\infty}^{\infty} \frac{1}{\cos^{2}\!\left(\pi \left(z + n \tau\right)\right)}$

## Taylor series

$\theta_{j}\!\left(z + x , \tau\right) = \sum_{n=0}^{\infty} \frac{\theta^{(n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}$
$\theta_{j}\!\left(z , \tau + x\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(4 \pi i\right)}^{n}} \frac{\theta^{(2 n)}_{j}\!\left(z , \tau\right)}{n !} {x}^{n}$

## Theta constants

### Fourier series (q-series) with linear exponents

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

### Infinite products for quotients

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

### Lambert series

$\theta_{3}\!\left(0 , \tau\right) = 1 + 2 \sum_{n=1}^{\infty} \frac{\lambda(n) {q}^{n}}{1 - {q}^{n}}\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{3}^{2}\!\left(0, \tau\right) = 1 + 4 \sum_{n=1}^{\infty} \frac{{q}^{n}}{1 + {q}^{2 n}}\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{2}^{4}\!\left(0, \tau\right) = 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{3}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} + 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 - {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{4}^{4}\!\left(0, \tau\right) = 1 + 8 \sum_{n=0}^{\infty} \frac{2 n {q}^{2 n}}{1 + {q}^{2 n}} - 8 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}$
$\theta_{4}^{4}\!\left(0, \tau\right) - \theta_{2}^{4}\!\left(0, \tau\right) = 1 - 24 \sum_{n=0}^{\infty} \frac{\left(2 n + 1\right) {q}^{2 n + 1}}{1 + {q}^{2 n + 1}}\; \text{ where } q = {e}^{\pi i \tau}$