Dedekind eta function

1dc520

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

Assumptions:

\tau \in \mathbb{H}

TeX:

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

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("1dc520"),
    Formula(Equal(DedekindEta(tau), Mul(Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 12)), Product(Parentheses(Sub(1, Exp(Mul(Mul(Mul(Mul(2, ConstPi), ConstI), k), tau)))), Tuple(k, 1, Infinity))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

ff587a

\eta\!\left(\tau\right) = {e}^{\pi i \tau / 12} \phi\!\left({e}^{2 \pi i \tau}\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\eta\!\left(\tau\right) = {e}^{\pi i \tau / 12} \phi\!\left({e}^{2 \pi i \tau}\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`EulerQSeries`	$\phi\!\left(q\right)$	Euler's q-series
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("ff587a"),
    Formula(Equal(DedekindEta(tau), Mul(Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 12)), EulerQSeries(Exp(Mul(Mul(Mul(2, ConstPi), ConstI), tau)))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

2e7fdb

\phi\!\left(q\right) = \prod_{k=1}^{\infty} \left(1 - {q}^{k}\right)

Assumptions:

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

TeX:

\phi\!\left(q\right) = \prod_{k=1}^{\infty} \left(1 - {q}^{k}\right)

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`EulerQSeries`	$\phi\!\left(q\right)$	Euler's q-series
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("2e7fdb"),
    Formula(Equal(EulerQSeries(q), Product(Parentheses(Sub(1, Pow(q, k))), Tuple(k, 1, Infinity)))),
    Variables(q),
    Assumptions(And(Element(q, CC), Less(Abs(q), 1))))

8f10b0

\phi\!\left(q\right) = \sum_{k=-\infty}^{\infty} {\left(-1\right)}^{k} {q}^{k \left(3 k - 1\right) / 2}

Assumptions:

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

TeX:

\phi\!\left(q\right) = \sum_{k=-\infty}^{\infty} {\left(-1\right)}^{k} {q}^{k \left(3 k - 1\right) / 2}

q \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \left|q\right| \lt 1

Definitions:

Fungrim symbol	Notation	Short description
`EulerQSeries`	$\phi\!\left(q\right)$	Euler's q-series
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("8f10b0"),
    Formula(Equal(EulerQSeries(q), Sum(Mul(Pow(-1, k), Pow(q, Div(Mul(k, Sub(Mul(3, k), 1)), 2))), Tuple(k, Neg(Infinity), Infinity)))),
    Variables(q),
    Assumptions(And(Element(q, CC), Less(Abs(q), 1))))

9b8c9f

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

TeX:

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

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`ConstI`	$i$	Imaginary unit
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Pow`	${a}^{b}$	Power
`ConstPi`	$\pi$	The constant pi (3.14...)

Source code for this entry:

Entry(ID("9b8c9f"),
    Formula(Equal(DedekindEta(ConstI), Div(GammaFunction(Div(1, 4)), Mul(2, Pow(ConstPi, Div(3, 4)))))))

204acd

\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}

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`Pow`	${a}^{b}$	Power
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function

Source code for this entry:

Entry(ID("204acd"),
    Formula(Equal(DedekindEta(Exp(Div(Mul(Mul(2, ConstPi), ConstI), 3))), Mul(Exp(Neg(Div(Mul(ConstPi, ConstI), 24))), Div(Mul(Pow(3, Div(1, 8)), Pow(GammaFunction(Div(1, 3)), Div(3, 2))), Mul(2, ConstPi))))))

737805

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

Assumptions:

\tau \in \mathbb{H}

TeX:

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

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`JacobiTheta3`	$\theta_3\!\left(z, \tau\right)$	Jacobi theta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("737805"),
    Formula(Equal(DedekindEta(tau), Mul(Exp(Div(Mul(Mul(ConstPi, ConstI), tau), 12)), JacobiTheta3(Div(Add(tau, 1), 2), Mul(3, tau))))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

1bae52

\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("1bae52"),
    Formula(Equal(DedekindEta(Add(tau, 1)), Mul(Exp(Div(Mul(ConstPi, ConstI), 12)), DedekindEta(tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

acee1a

\eta\!\left(\tau + 24\right) = \eta\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\eta\!\left(\tau + 24\right) = \eta\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("acee1a"),
    Formula(Equal(DedekindEta(Add(tau, 24)), DedekindEta(tau))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

3b806f

\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta\!\left(\tau\right)

Assumptions:

\tau \in \mathbb{H}

TeX:

\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta\!\left(\tau\right)

\tau \in \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`Pow`	${a}^{b}$	Power
`ConstI`	$i$	Imaginary unit
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("3b806f"),
    Formula(Equal(DedekindEta(Neg(Div(1, tau))), Mul(Pow(Neg(Mul(ConstI, tau)), Div(1, 2)), DedekindEta(tau)))),
    Variables(tau),
    Assumptions(Element(tau, HH)))

29d9ab

{\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}

Assumptions:

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

TeX:

{\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}

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`SL2Z`	$\operatorname{SL}_2(\mathbb{Z})$	Modular group

Source code for this entry:

Entry(ID("29d9ab"),
    Formula(Equal(Pow(DedekindEta(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), 24), Mul(Pow(Add(Mul(c, tau), d), 12), Pow(DedekindEta(tau), 24)))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), SL2Z))))

9f19c1

\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)

Assumptions:

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

TeX:

\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)

\tau \in \mathbb{H} \,\mathbin{\operatorname{and}}\, \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`Pow`	${a}^{b}$	Power
`HH`	$\mathbb{H}$	Upper complex half-plane
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)

Source code for this entry:

Entry(ID("9f19c1"),
    Formula(Equal(DedekindEta(Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d))), Mul(Mul(DedekindEtaEpsilon(a, b, c, d), Pow(Add(Mul(c, tau), d), Div(1, 2))), DedekindEta(tau)))),
    Variables(tau, a, b, c, d),
    Assumptions(And(Element(tau, HH), Element(Matrix2x2(a, b, c, d), PSL2Z))))

f04e01

\varepsilon\!\left(1, b, 0, 1\right) = {e}^{\pi i b / 12}

b \in \mathbb{Z}

TeX:

\varepsilon\!\left(1, b, 0, 1\right) = {e}^{\pi i b / 12}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("f04e01"),
    Formula(Equal(DedekindEtaEpsilon(1, b, 0, 1), Exp(Div(Mul(Mul(ConstPi, ConstI), b), 12)))),
    Variables(b),
    Element(b, ZZ))

921ef0

\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)

Assumptions:

a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \,\mathbin{\operatorname{and}}\, c \gt 0

TeX:

\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)

a \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, d \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, a d - b c = 1 \,\mathbin{\operatorname{and}}\, c \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEtaEpsilon`	$\varepsilon\!\left(a, b, c, d\right)$	Root of unity in the functional equation of the Dedekind eta function
`Exp`	${e}^{z}$	Exponential function
`ConstPi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("921ef0"),
    Formula(Equal(DedekindEtaEpsilon(a, b, c, d), Exp(Mul(Mul(ConstPi, ConstI), Sub(Sub(Div(Add(a, d), Mul(12, c)), DedekindSum(d, c)), Div(1, 4)))))),
    Variables(a, b, c, d),
    Assumptions(And(Element(a, ZZ), Element(b, ZZ), Element(c, ZZ), Element(d, ZZ), Equal(Sub(Mul(a, d), Mul(b, c)), 1), Greater(c, 0))))

e06d87

\operatorname{HolomorphicDomain}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \mathbb{H}

TeX:

\operatorname{HolomorphicDomain}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \mathbb{H}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("e06d87"),
    Formula(Equal(HolomorphicDomain(DedekindEta(tau), tau, HH), HH)))

04f4a0

\operatorname{Poles}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

TeX:

\operatorname{Poles}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("04f4a0"),
    Formula(Equal(Poles(DedekindEta(tau), tau, Union(HH, Set(UnsignedInfinity))), Set())))

f2e2c2

\operatorname{BranchPoints}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

TeX:

\operatorname{BranchPoints}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity

Source code for this entry:

Entry(ID("f2e2c2"),
    Formula(Equal(BranchPoints(DedekindEta(tau), tau, Union(HH, Set(UnsignedInfinity))), Set())))

6d7668

\operatorname{BranchCuts}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \left\{\right\}

TeX:

\operatorname{BranchCuts}\!\left(\eta\!\left(\tau\right), \tau, \mathbb{H}\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("6d7668"),
    Formula(Equal(BranchCuts(DedekindEta(tau), tau, HH), Set())))

39fb36

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} \eta\!\left(\tau\right) = \left\{\right\}

TeX:

\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} \eta\!\left(\tau\right) = \left\{\right\}

Definitions:

Fungrim symbol	Notation	Short description
`DedekindEta`	$\eta\!\left(\tau\right)$	Dedekind eta function
`HH`	$\mathbb{H}$	Upper complex half-plane

Source code for this entry:

Entry(ID("39fb36"),
    Formula(Equal(Zeros(DedekindEta(tau), tau, Element(tau, HH)), Set())))

23961e

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)

Assumptions:

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, k \gt 0 \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1

TeX:

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)

n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, k \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, k \gt 0 \,\mathbin{\operatorname{and}}\, \gcd\!\left(n, k\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DedekindSum`	$s\!\left(n, k\right)$	Dedekind sum
`ZZ`	$\mathbb{Z}$	Integers
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("23961e"),
    Formula(Equal(DedekindSum(n, k), Sum(Mul(Div(r, k), Sub(Sub(Div(Mul(n, r), k), Floor(Div(Mul(n, r), k))), Div(1, 2))), Tuple(r, 1, Sub(k, 1))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZ), Element(k, ZZ), Greater(k, 0), Equal(GCD(n, k), 1))))

Fourier series (q-series)

Special values

Connection formulas

Modular transformations

Analytic properties

Dedekind sums

Related topics