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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]]
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 &tau;, let ''q'' = exp(2&pi;i&tau;), and define the eta function by,


:<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,
 
:<math>\Delta=(2\pi)^{12}\eta^{24}(\tau)</math>
 
where &Delta; 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]].
 
The eta function is [[holomorphic]] on the upper half-plane but cannot be continued analytically beyond it.
 
[[File:Q-Eulero.jpeg|thumb|right|Modulus of Euler phi on the unit disc, colored so that black=0, red=4]]
[[Image:Discriminant real part.jpeg|thumb|right|The real part of the modular discriminant as a function of ''q''.]]
 
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>
 
:<math>\eta(\tau+1) =e^{\frac{\pi {\rm{i}}}{12}}\eta(\tau),\,</math>
 
:<math>\eta(-\tau^{-1}) = \sqrt{-{\rm{i}}\tau} \eta(\tau).\,</math>
 
More generally, suppose ''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;''d'' are integers with ''ad''&nbsp;&minus;&nbsp;''bc''&nbsp;=&nbsp;1, so that
 
:<math>\tau\mapsto\frac{a\tau+b}{c\tau+d}</math>
 
is a transformation belonging to the [[modular group]]. We may assume that either ''c''&nbsp;&gt;&nbsp;0, or ''c''&nbsp;=&nbsp;0 and ''d''&nbsp;=&nbsp;1.  Then
 
:<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) =
\epsilon (a,b,c,d) (c\tau+d)^{\frac{1}{2}} \eta(\tau),</math>
 
where
 
:<math>\epsilon (a,b,c,d)=e^{\frac{b{\rm{i}} \pi}{12}}\quad(c=0,d=1);</math>
:<math>\epsilon (a,b,c,d)=e^{{\rm{i}}\pi [\frac{a+d}{12c} - s(d,c)
-\frac{1}{4}]}\quad(c>0).</math>
 
Here <math>s(h,k)\,</math> is the [[Dedekind sum]]
 
:<math>s(h,k)=\sum_{n=1}^{k-1} \frac{n}{k}
\left( \frac{hn}{k} - \left\lfloor \frac{hn}{k} \right\rfloor -\frac{1}{2} \right).</math>
 
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
 
:<math>\Delta(\tau) = (2 \pi)^{12} \eta(\tau)^{24}\,</math>
 
and is a modular form of weight 12. (Some authors omit the factor of (2&pi;)<sup>12</sup>, so that the series expansion has integral coefficients).
 
The [[Jacobi triple product]] implies that the eta is (up to a factor) a Jacobi [[theta function]] for special values of the arguments:
 
:<math>\eta(z) = \sum_{n=1}^\infty \chi(n) \exp(\tfrac{1}{12} \pi i n^2 z),</math>
 
where <math>\chi(n)</math> is the [[Dirichlet character]] modulo 12 with <math>\chi(\pm1) = 1</math>,
<math>\chi(\pm 5)=-1</math>.
 
The [[Euler function]]
 
:<math>\phi(q) = \prod_{n=1}^{\infty} \left(1-q^n\right),</math>
 
related to <math>\eta \,</math> by <math>\phi(q)= q^{-1/24} \eta(\tau)\,</math>, has a power series
by the [[Pentagonal number theorem|Euler identity]]:
 
:<math>\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.</math>
 
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.
 
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''.
 
==Special values==
 
The above connection with the Euler function together with the special values of the latter, it can be easily deduced that
 
: <math>
\eta(i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2 \pi ^{3/4}},
</math>
 
: <math>
\eta\left(\frac{i}{2}\right)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{7/8} \pi ^{3/4}},
</math>
 
: <math>
\eta(2i)=\frac{\Gamma \left(\frac{1}{4}\right)}{2^{{11}/8} \pi ^{3/4}},
</math>
 
: <math>
\eta(4i)=\frac{\sqrt[4]{\sqrt{2}-1} \Gamma \left(\frac{1}{4}\right)}{2^{{29}/16} \pi ^{3/4}}.
</math>
 
==See also==
 
* [[Chowla–Selberg formula]]
* [[q-series]]
* [[Weierstrass's elliptic functions]]
* [[partition function (number theory)]]
* [[Kronecker limit formula]]
* [[superstring theory]]
 
==References==
<references/>
* 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.''
* Neil Koblitz, ''Introduction to Elliptic Curves and Modular Forms'' (2 ed), Graduate Texts in Mathematics '''97''' (1993), Springer-Verlag, ISBN 3-540-97966-2
 
[[Category:Fractals]]
[[Category:Modular forms]]
[[Category:Elliptic functions]]

Latest revision as of 11:06, 12 December 2014

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