# Dedekind eta function

## Fourier series (q-series)

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

## Special values

$\eta\!\left(i\right) = \frac{\Gamma\!\left(\frac{1}{4}\right)}{2 {\pi}^{3 / 4}}$
$\eta\!\left({e}^{2 \pi i / 3}\right) = {e}^{-\pi i / 24} \frac{{3}^{1 / 8} {\left(\Gamma\!\left(\frac{1}{3}\right)\right)}^{3 / 2}}{2 \pi}$

## Connection formulas

$\eta\!\left(\tau\right) = {e}^{\pi i \tau / 12} \theta_3\!\left(\frac{\tau + 1}{2}, 3 \tau\right)$

## Modular transformations

Related topics: Modular transformations

$\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta\!\left(\tau\right)$
$\eta\!\left(\tau + 24\right) = \eta\!\left(\tau\right)$
$\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta\!\left(\tau\right)$
${\left(\eta\!\left(\frac{a \tau + b}{c \tau + d}\right)\right)}^{24} = {\left(c \tau + d\right)}^{12} {\left(\eta\!\left(\tau\right)\right)}^{24}$
$\eta\!\left(\frac{a \tau + b}{c \tau + d}\right) = \varepsilon\!\left(a, b, c, d\right) {\left(c \tau + d\right)}^{1 / 2} \eta\!\left(\tau\right)$
$\varepsilon\!\left(1, b, 0, 1\right) = {e}^{\pi i b / 12}$
$\varepsilon\!\left(a, b, c, d\right) = \exp\!\left(\pi i \left(\frac{a + d}{12 c} - s\!\left(d, c\right) - \frac{1}{4}\right)\right)$

## Analytic properties

$\operatorname{HolomorphicDomain}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \mathbb{H}$
$\operatorname{Poles}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchPoints}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}$
$\operatorname{BranchCuts}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \left\{\right\}$
$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} \eta\!\left(\tau\right) = \left\{\right\}$

## Dedekind sums

$s\!\left(n, k\right) = \sum_{r=1}^{k - 1} \frac{r}{k} \left(\frac{n r}{k} - \left\lfloor \frac{n r}{k} \right\rfloor - \frac{1}{2}\right)$

Related topics: Partition function

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

2019-06-18 07:49:59.356594 UTC