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Dedekind eta function

Table of contents: Definitions - Fourier series (q-series) - Specific values - Connection formulas - Modular transformations - Derivatives - Analytic properties - Dedekind sums - Related topics

Definitions

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Symbol: DedekindEta — η(τ)\eta(\tau) — Dedekind eta function
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Symbol: EulerQSeries — ϕ(q)\phi(q) — Euler's q-series

Fourier series (q-series)

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η(τ)=eπiτ/12∏k=1∞(1−e2πikτ)\eta(\tau) = {e}^{\pi i \tau / 12} \prod_{k=1}^{\infty} \left(1 - {e}^{2 \pi i k \tau}\right)
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η(τ)=eπiτ/12ϕ ⁣(e2πiτ)\eta(\tau) = {e}^{\pi i \tau / 12} \phi\!\left({e}^{2 \pi i \tau}\right)
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ϕ(q)=∏k=1∞(1−qk)\phi(q) = \prod_{k=1}^{\infty} \left(1 - {q}^{k}\right)
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ϕ(q)=∑k=−∞∞(−1)kqk(3k−1)/2\phi(q) = \sum_{k=-\infty}^{\infty} {\left(-1\right)}^{k} {q}^{k \left(3 k - 1\right) / 2}

Specific values

Limiting values

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η ⁣(i∞)=lim⁡τ→i∞η(τ)=0\eta\!\left(i \infty\right) = \lim_{\tau \to i \infty} \eta(\tau) = 0
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lim⁡ε→0+η ⁣(iε)=0\lim_{\varepsilon \to {0}^{+}} \eta\!\left(i \varepsilon\right) = 0

Imaginary quadratic points

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η(i)=Γ ⁣(14)2π3/4\eta(i) = \frac{\Gamma\!\left(\frac{1}{4}\right)}{2 {\pi}^{3 / 4}}
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η′(i)=−i4η(i)\eta'(i) = -\frac{i}{4} \eta(i)
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η ⁣(2i)=η(i)23/8\eta\!\left(2 i\right) = \frac{\eta(i)}{{2}^{3 / 8}}
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η ⁣(3i)=η(i)33/8(2+3)1/12\eta\!\left(3 i\right) = \frac{\eta(i)}{{3}^{3 / 8} {\left(2 + \sqrt{3}\right)}^{1 / 12}}
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η ⁣(4i)=η(i)213/16(1+2)1/4\eta\!\left(4 i\right) = \frac{\eta(i)}{{2}^{13 / 16} {\left(1 + \sqrt{2}\right)}^{1 / 4}}
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η ⁣(5i)=η(i)5φ\eta\!\left(5 i\right) = \frac{\eta(i)}{\sqrt{5 \varphi}}
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η ⁣(6i)=163/8(5−32−33/42)1/6η(i)\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)
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η ⁣(7i)=17(−72+7+12−7+47)1/4η(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)
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η ⁣(8i)=1241/32(21/4−1)1/2(1+2)1/8η(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)
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η ⁣(16i)=12113/64(21/4−1)1/4(1+2)1/16(−25/8+1+2)1/2η(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)
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η ⁣(3i)=31/824/3(Γ ⁣(13))3/2π\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

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η ⁣(e2πi/3)=e−πi/2431/8(Γ ⁣(13))3/22π\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}
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η′(e2πi/3)=i36η ⁣(e2πi/3)\eta'({e}^{2 \pi i / 3}) = \frac{i \sqrt{3}}{6} \eta\!\left({e}^{2 \pi i / 3}\right)

Connection formulas

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η(τ)=eπiτ/12θ3 ⁣(τ+12,3τ)\eta(\tau) = {e}^{\pi i \tau / 12} \theta_{3}\!\left(\frac{\tau + 1}{2} , 3 \tau\right)

Modular transformations

Related topics: Modular transformations

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Symbol: DedekindEtaEpsilon — ε ⁣(a,b,c,d)\varepsilon\!\left(a, b, c, d\right) — Root of unity in the functional equation of the Dedekind eta function
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η ⁣(τ+1)=eπi/12η(τ)\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta(\tau)
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η ⁣(τ+24)=η(τ)\eta\!\left(\tau + 24\right) = \eta(\tau)
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η ⁣(−1τ)=(−iτ)1/2η(τ)\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta(\tau)
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η24 ⁣(aτ+bcτ+d)=(cτ+d)12η24 ⁣(τ)\eta^{24}\!\left(\frac{a \tau + b}{c \tau + d}\right) = {\left(c \tau + d\right)}^{12} \eta^{24}\!\left(\tau\right)
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η ⁣(aτ+bcτ+d)=ε ⁣(a,b,c,d)(cτ+d)1/2η(τ)\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)
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ε ⁣(1,b,0,1)=eπib/12\varepsilon\!\left(1, b, 0, 1\right) = {e}^{\pi i b / 12}
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ε ⁣(a,b,c,d)=exp⁡ ⁣(πi(a+d12c−s ⁣(d,c)−14))\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)
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η ⁣(τ+12)=eπi/24η3 ⁣(2τ)η(τ)η ⁣(4τ)\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

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η′(τ)=iπ12η(τ)E2 ⁣(τ)\eta'(\tau) = \frac{i \pi}{12} \eta(\tau) E_{2}\!\left(\tau\right)
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η′(τ)=i2πη(τ)ζ ⁣(12,τ)\eta'(\tau) = \frac{i}{2 \pi} \eta(\tau) \zeta\!\left(\frac{1}{2}, \tau\right)
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36(y′(τ))2−24y′′(τ)y(τ)+y′′′(τ)=0   where y(τ)=η′(τ)η(τ)36 {\left(y'(\tau)\right)}^{2} - 24 y''(\tau) y(\tau) + y'''(\tau) = 0\; \text{ where } y(\tau) = \frac{\eta'(\tau)}{\eta(\tau)}
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η2 ⁣(τ)(33(η′′(τ))2+η(τ)η(4)(τ))−18(η′(τ))4+12η(τ)η′′(τ)(η′(τ))2−28η2 ⁣(τ)η′′′(τ)η′(τ)=0\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

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η(τ) is holomorphic on τ∈H\eta(\tau) \text{ is holomorphic on } \tau \in \mathbb{H}
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poles⁡ τ∈H∪{∞~}η(τ)={}\mathop{\operatorname{poles}\,}\limits_{\tau \in \mathbb{H} \cup \left\{{\tilde \infty}\right\}} \eta(\tau) = \left\{\right\}
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BranchPoints⁡ ⁣(η(τ),τ,H∪{∞~})={}\operatorname{BranchPoints}\!\left(\eta(\tau), \tau, \mathbb{H} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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BranchCuts⁡ ⁣(η(τ),τ,H)={}\operatorname{BranchCuts}\!\left(\eta(\tau), \tau, \mathbb{H}\right) = \left\{\right\}
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zeros⁡ τ∈Hη(τ)={}\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathbb{H}} \eta(\tau) = \left\{\right\}

Dedekind sums

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Symbol: DedekindSum — s ⁣(n,k)s\!\left(n, k\right) — Dedekind sum
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s ⁣(n,k)=∑r=1k−1rk(nrk−⌊nrk⌋−12)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)
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s ⁣(n,k)=∑r=1k−1Q ⁣(rk)Q ⁣(nrk)   where Q(x)={x−⌊x⌋−12,x∉Z0,x∈Zs\!\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.

2021-03-15 19:12:00.328586 UTC