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Dedekind eta function

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

Fourier series (q-series)

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η ⁣(τ)=eπiτ/12k=1(1e2πikτ)\eta\!\left(\tau\right) = {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\!\left(\tau\right) = {e}^{\pi i \tau / 12} \phi\!\left({e}^{2 \pi i \tau}\right)
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ϕ ⁣(q)=k=1(1qk)\phi\!\left(q\right) = \prod_{k=1}^{\infty} \left(1 - {q}^{k}\right)
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ϕ ⁣(q)=k=(1)kqk(3k1)/2\phi\!\left(q\right) = \sum_{k=-\infty}^{\infty} {\left(-1\right)}^{k} {q}^{k \left(3 k - 1\right) / 2}

Special values

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η ⁣(i)=Γ ⁣(14)2π3/4\eta\!\left(i\right) = \frac{\Gamma\!\left(\frac{1}{4}\right)}{2 {\pi}^{3 / 4}}
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η ⁣(e2πi/3)=eπi/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}

Connection formulas

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η ⁣(τ)=eπiτ/12θ3 ⁣(τ+12,3τ)\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

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η ⁣(τ+1)=eπi/12η ⁣(τ)\eta\!\left(\tau + 1\right) = {e}^{\pi i / 12} \eta\!\left(\tau\right)
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η ⁣(τ+24)=η ⁣(τ)\eta\!\left(\tau + 24\right) = \eta\!\left(\tau\right)
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η ⁣(1τ)=(iτ)1/2η ⁣(τ)\eta\!\left(-\frac{1}{\tau}\right) = {\left(-i \tau\right)}^{1 / 2} \eta\!\left(\tau\right)
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(η ⁣(aτ+bcτ+d))24=(cτ+d)12(η ⁣(τ))24{\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}
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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\!\left(\tau\right)
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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+d12cs ⁣(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)

Analytic properties

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

Dedekind sums

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s ⁣(n,k)=r=1k1rk(nrknrk12)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.

2019-05-23 08:00:13.607731 UTC