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| {{For|the Dirichlet series|Dirichlet eta function}}
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| [[Image:Dedekind Eta.jpg|right|thumb|500px|Dedekind η-function in the complex plane]]
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| In mathematics, the '''Dedekind eta function''', named after [[Richard Dedekind]], is a function defined on the [[upper half-plane]] of [[complex number]]s, where the imaginary part is positive. For any such complex number τ, let ''q'' = exp(2πiτ), and define the eta function by,
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| :<math>\eta(\tau) = e^{\frac{\pi \rm{i} \tau}{12}} \prod_{n=1}^{\infty} (1-q^{n}) .</math>
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| (The notation <math>q \equiv e^{2\pi \rm{i} \tau}\,</math> is now standard in [[number theory]], though many older books use ''q'' for the [[nome (mathematics)|nome]] <math>e^{\pi \rm{i} \tau}\,</math>.) Note that,
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| :<math>\Delta=(2\pi)^{12}\eta^{24}(\tau)</math>
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| where Δ is the [[modular discriminant]]. The presence of [[24 (number)|24]] can be understood by connection with other occurrences, such as in the 24-dimensional [[Leech lattice]]. | |
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| The eta function is [[holomorphic]] on the upper half-plane but cannot be continued analytically beyond it.
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| [[File:Q-Eulero.jpeg|thumb|right|Modulus of Euler phi on the unit disc, colored so that black=0, red=4]]
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| [[Image:Discriminant real part.jpeg|thumb|right|The real part of the modular discriminant as a function of ''q''.]]
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| The eta function satisfies the [[functional equation]]s<ref>{{cite journal|author=Siegel, C.L.|title=A Simple Proof of <math>\eta(-1/\tau) = \eta(\tau)\sqrt{\tau/{\rm{i}}}\,</math>|journal=Mathematika|year=1954|volume=1|page=4|doi=10.1112/S0025579300000462}}</ref>
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| :<math>\eta(\tau+1) =e^{\frac{\pi {\rm{i}}}{12}}\eta(\tau),\,</math>
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| :<math>\eta(-\tau^{-1}) = \sqrt{-{\rm{i}}\tau} \eta(\tau).\,</math>
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| More generally, suppose ''a'', ''b'', ''c'', ''d'' are integers with ''ad'' − ''bc'' = 1, so that
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| :<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math>
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| is a transformation belonging to the [[modular group]]. We may assume that either ''c'' > 0, or ''c'' = 0 and ''d'' = 1. Then
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| :<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) =
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| \epsilon (a,b,c,d) (c\tau+d)^{\frac{1}{2}} \eta(\tau),</math>
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| where
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| :<math>\epsilon (a,b,c,d)=e^{\frac{b{\rm{i}} \pi}{12}}\quad(c=0,d=1);</math>
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| :<math>\epsilon (a,b,c,d)=e^{{\rm{i}}\pi [\frac{a+d}{12c} - s(d,c)
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| -\frac{1}{4}]}\quad(c>0).</math>
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| Here <math>s(h,k)\,</math> is the [[Dedekind sum]]
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| :<math>s(h,k)=\sum_{n=1}^{k-1} \frac{n}{k}
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| \left( \frac{hn}{k} - \left\lfloor \frac{hn}{k} \right\rfloor -\frac{1}{2} \right).</math>
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| Because of these functional equations the eta function is a [[modular form]] of weight 1/2 and level 1 for a certain character of order 24 of the [[metaplectic group|metaplectic double cover]] of the modular group, and can be used to define other modular forms. In particular the [[modular discriminant]] of [[Weierstrass]] can be defined as
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| :<math>\Delta(\tau) = (2 \pi)^{12} \eta(\tau)^{24}\,</math>
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| and is a modular form of weight 12. (Some authors omit the factor of (2π)<sup>12</sup>, so that the series expansion has integral coefficients).
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| The [[Jacobi triple product]] implies that the eta is (up to a factor) a Jacobi [[theta function]] for special values of the arguments:
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| :<math>\eta(z) = \sum_{n=1}^\infty \chi(n) \exp(\tfrac{1}{12} \pi i n^2 z),</math> | |
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| where <math>\chi(n)</math> is the [[Dirichlet character]] modulo 12 with <math>\chi(\pm1) = 1</math>,
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| <math>\chi(\pm 5)=-1</math>.
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| The [[Euler function]]
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| :<math>\phi(q) = \prod_{n=1}^{\infty} \left(1-q^n\right),</math>
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| related to <math>\eta \,</math> by <math>\phi(q)= q^{-1/24} \eta(\tau)\,</math>, has a power series
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| by the [[Pentagonal number theorem|Euler identity]]:
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| :<math>\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.</math>
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| Because the eta function is easy to compute numerically from either [[power series]], it is often helpful in computation to express other functions in terms of it when possible, and products and quotients of eta functions, called eta quotients, can be used to express a great variety of modular forms.
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| The picture on this page shows the modulus of the Euler function: the additional factor of <math>q^{1/24}</math> between this and eta makes almost no visual difference whatsoever (it only introduces a tiny pinprick at the origin). Thus, this picture can be taken as a picture of eta as a function of ''q''.
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| ==Special values==
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| The above connection with the Euler function together with the special values of the latter, it can be easily deduced that
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| : <math>
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| \eta(i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2 \pi ^{3/4}},
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| </math>
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| : <math>
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| \eta\left(\frac{i}{2}\right)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{7/8} \pi ^{3/4}},
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| </math>
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| : <math>
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| \eta(2i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{{11}/8} \pi ^{3/4}},
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| </math>
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| : <math>
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| \eta(4i)=\frac{\sqrt[4]{\sqrt{2}-1} \Gamma \left(\frac{1}{4}\right)}{2^{{29}/16} \pi ^{3/4}}.
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| </math>
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| ==See also==
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| * [[Chowla–Selberg formula]]
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| * [[q-series]]
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| * [[Weierstrass's elliptic functions]]
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| * [[partition function (number theory)]]
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| * [[Kronecker limit formula]]
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| * [[superstring theory]]
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| ==References==
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| <references/>
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| * Tom M. Apostol, ''Modular functions and Dirichlet Series in Number Theory'' (2 ed), Graduate Texts in Mathematics '''41''' (1990), Springer-Verlag, ISBN 3-540-97127-0 ''See chapter 3.''
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| * Neil Koblitz, ''Introduction to Elliptic Curves and Modular Forms'' (2 ed), Graduate Texts in Mathematics '''97''' (1993), Springer-Verlag, ISBN 3-540-97966-2
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| [[Category:Fractals]]
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| [[Category:Modular forms]]
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| [[Category:Elliptic functions]]
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