# Dedekind eta function

## Definitions

Symbol: DedekindEta $\eta(\tau)$ Dedekind eta function
Symbol: EulerQSeries $\phi(q)$ Euler's q-series

## Fourier series (q-series)

$\eta(\tau) = {e}^{\pi i \tau / 12} \prod_{k=1}^{\infty} \left(1 - {e}^{2 \pi i k \tau}\right)$
$\eta(\tau) = {e}^{\pi i \tau / 12} \phi\!\left({e}^{2 \pi i \tau}\right)$
$\phi(q) = \prod_{k=1}^{\infty} \left(1 - {q}^{k}\right)$
$\phi(q) = \sum_{k=-\infty}^{\infty} {\left(-1\right)}^{k} {q}^{k \left(3 k - 1\right) / 2}$

## Specific values

### Limiting values

$\eta\!\left(i \infty\right) = \lim_{\tau \to i \infty} \eta(\tau) = 0$
$\lim_{\varepsilon \to {0}^{+}} \eta\!\left(i \varepsilon\right) = 0$

$\eta(i) = \frac{\Gamma\!\left(\frac{1}{4}\right)}{2 {\pi}^{3 / 4}}$
$\eta'(i) = -\frac{i}{4} \eta(i)$
$\eta\!\left(2 i\right) = \frac{\eta(i)}{{2}^{3 / 8}}$
$\eta\!\left(3 i\right) = \frac{\eta(i)}{{3}^{3 / 8} {\left(2 + \sqrt{3}\right)}^{1 / 12}}$
$\eta\!\left(4 i\right) = \frac{\eta(i)}{{2}^{13 / 16} {\left(1 + \sqrt{2}\right)}^{1 / 4}}$
$\eta\!\left(5 i\right) = \frac{\eta(i)}{\sqrt{5 \varphi}}$
$\eta\!\left(6 i\right) = \frac{1}{{6}^{3 / 8}} {\left(\frac{5 - \sqrt{3}}{2} - \frac{{3}^{3 / 4}}{\sqrt{2}}\right)}^{1 / 6} \eta(i)$
$\eta\!\left(7 i\right) = \frac{1}{\sqrt{7}} {\left(-\frac{7}{2} + \sqrt{7} + \frac{1}{2} \sqrt{-7 + 4 \sqrt{7}}\right)}^{1 / 4} \eta(i)$
$\eta\!\left(8 i\right) = \frac{1}{{2}^{41 / 32}} \frac{{\left({2}^{1 / 4} - 1\right)}^{1 / 2}}{{\left(1 + \sqrt{2}\right)}^{1 / 8}} \eta(i)$
$\eta\!\left(16 i\right) = \frac{1}{{2}^{113 / 64}} \frac{{\left({2}^{1 / 4} - 1\right)}^{1 / 4}}{{\left(1 + \sqrt{2}\right)}^{1 / 16}} {\left(-{2}^{5 / 8} + \sqrt{1 + \sqrt{2}}\right)}^{1 / 2} \eta(i)$
$\eta\!\left(\sqrt{3} i\right) = \frac{{3}^{1 / 8}}{{2}^{4 / 3}} \frac{{\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3 / 2}}{\pi}$

### Third roots of unity

$\eta\!\left({e}^{2 \pi i / 3}\right) = {e}^{-\pi i / 24} \frac{{3}^{1 / 8} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3 / 2}}{2 \pi}$
$\eta'({e}^{2 \pi i / 3}) = \frac{i \sqrt{3}}{6} \eta\!\left({e}^{2 \pi i / 3}\right)$

## Connection formulas

$\eta(\tau) = {e}^{\pi i \tau / 12} \theta_{3}\!\left(\frac{\tau + 1}{2} , 3 \tau\right)$

## Modular transformations

Related topics: Modular transformations

Symbol: DedekindEtaEpsilon $\varepsilon\!\left(a, b, c, d\right)$ Root of unity in the functional equation of the Dedekind eta function
$\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta(\tau)$
$\eta\!\left(\tau + 24\right) = \eta(\tau)$
$\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta(\tau)$
$\eta^{24}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{12} \eta^{24}\!\left(\tau\right)$
$\eta\!\left(\frac{a \tau + b}{c \tau + d}\right) = \varepsilon\!\left(a, b, c, d\right) {\left(c \tau + d\right)}^{1 / 2} \eta(\tau)$
$\varepsilon\!\left(1, b, 0, 1\right) = {e}^{\pi i b / 12}$
$\varepsilon\!\left(a, b, c, d\right) = \exp\!\left(\pi i \left(\frac{a + d}{12 c} - s\!\left(d, c\right) - \frac{1}{4}\right)\right)$
$\eta\!\left(\tau + \frac{1}{2}\right) = {e}^{\pi i / 24} \frac{\eta^{3}\!\left(2 \tau\right)}{\eta(\tau) \eta\!\left(4 \tau\right)}$

## Derivatives

$\eta'(\tau) = \frac{i \pi}{12} \eta(\tau) E_{2}\!\left(\tau\right)$
$\eta'(\tau) = \frac{i}{2 \pi} \eta(\tau) \zeta\!\left(\frac{1}{2}, \tau\right)$
$36 {\left(y'(\tau)\right)}^{2} - 24 y''(\tau) y(\tau) + y'''(\tau) = 0\; \text{ where } y(\tau) = \frac{\eta'(\tau)}{\eta(\tau)}$
$\eta^{2}\!\left(\tau\right) \left(33 {\left(\eta''(\tau)\right)}^{2} + \eta(\tau) {\eta}^{(4)}(\tau)\right) - 18 {\left(\eta'(\tau)\right)}^{4} + 12 \eta(\tau) \eta''(\tau) {\left(\eta'(\tau)\right)}^{2} - 28 \eta^{2}\!\left(\tau\right) \eta'''(\tau) \eta'(\tau) = 0$

## Analytic properties

$\eta(\tau) \text{ is holomorphic on } \tau \in \mathbb{H}$
$\mathop{\operatorname{poles}\,}\limits_{\tau \in \mathbb{H} \cup \left\{{\tilde \infty}\right\}} \eta(\tau) = \left\{\right\}$
$\operatorname{BranchPoints}\!\left(\eta(\tau), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(\eta(\tau), \tau, \mathbb{H}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} \eta(\tau) = \left\{\right\}$

## Dedekind sums

Symbol: DedekindSum $s\!\left(n, k\right)$ Dedekind sum
$s\!\left(n, k\right) = \sum_{r=1}^{k - 1} \frac{r}{k} \left(\frac{n r}{k} - \left\lfloor \frac{n r}{k} \right\rfloor - \frac{1}{2}\right)$
$s\!\left(n, k\right) = \sum_{r=1}^{k - 1} Q\!\left(\frac{r}{k}\right) Q\!\left(\frac{n r}{k}\right)\; \text{ where } Q(x) = \begin{cases} x - \left\lfloor x \right\rfloor - \frac{1}{2}, & x \notin \mathbb{Z}\\0, & x \in \mathbb{Z}\\ \end{cases}$

Related topics: Partition function

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

2020-04-08 16:14:44.404316 UTC