Editing Dedekind eta function (section)

==Definition==
For any complex number {{mvar|τ}} with {{math|Im(''τ'') > 0}}, let {{math|''q'' {{=}} ''e''<sup>2''πiτ''</sup>}}; then the eta function is defined by,

:<math>\eta(\tau) = e^\frac{\pi i \tau}{12} \prod_{n=1}^\infty \left(1-e^{2 n\pi i \tau}\right) = q^\frac{1}{24} \prod_{n=1}^\infty \left(1 - q^n\right) .</math>

Raising the eta equation to the 24th power and multiplying by {{math|(2''π'')<sup>12</sup>}} gives

:<math>\Delta(\tau)=(2\pi)^{12}\eta^{24}(\tau)</math>

where {{math|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]].

The eta function is [[Holomorphic function|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 {{mvar|q}}.]]

The eta function satisfies the [[functional equation]]s<ref>{{cite journal|last=Siegel|first=C. L.|title=A Simple Proof of ''η''(−1/''τ'') {{=}} ''η''(''τ''){{sqrt|''τ''/''i''}}|journal=[[Mathematika]]|year=1954|volume=1|page=4|doi=10.1112/S0025579300000462}}</ref>

:<math>\begin{align}
\eta(\tau+1) &=e^\frac{\pi i}{12}\eta(\tau),\\
\eta\left(-\frac{1}{\tau}\right) &= \sqrt{-i\tau}\, \eta(\tau).\,
\end{align}</math>

In the second equation the [[Complex square root|branch of the square root]] is chosen such that {{math|{{sqrt|−''iτ''}} {{=}} 1}} when {{math|''τ'' {{=}} ''i''}}.

More generally, suppose {{math|''a'', ''b'', ''c'', ''d''}} are integers with {{math|''ad'' − ''bc'' {{=}} 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 {{math|''c'' > 0}}, or {{math|''c'' {{=}} 0}} and {{math|''d'' {{=}} 1}}.  Then

:<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) = \epsilon (a,b,c,d) \left(c\tau+d\right)^\frac12 \eta(\tau),</math>

where

:<math>\epsilon (a,b,c,d)= \begin{cases}
e^\frac{bi \pi}{12} &c=0,\,d=1, \\
e^{i\pi \left(\frac{a+d}{12c} - s(d,c)-\frac14\right)} &c>0.
\end{cases}</math>

Here {{math|''s''(''h'',''k'')}} 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 -\frac12 \right).</math>

Because of these functional equations the eta function is a [[modular form]] of weight {{sfrac|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 the [[Weierstrass elliptic function]] with

:<math>\omega_2=\tau\omega_1</math>

can be defined as

:<math>\Delta(\tau) = (2 \pi\omega_1)^{12} \eta(\tau)^{24}\,</math>

and is a modular form of weight 12. Some authors omit the factor of {{math|(2''π'')<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:<ref>{{citation|first=Daniel|last= Bump|title=Automorphic Forms and Representations|year=1998|publisher=Cambridge University Press|isbn=0-521-55098-X}}</ref>

:<math>\eta(\tau) = \sum_{n=1}^\infty \chi(n) \exp\left(\frac {\pi i n^2 \tau}{12}\right),</math>

where {{math|''χ''(''n'')}} is "the" [[Dirichlet character]] modulo 12 with {{math|''χ''(±1) {{=}} 1}} and {{math|''χ''(±5) {{=}} −1}}. Explicitly,{{Citation needed|date=September 2016}}

:<math>\eta(\tau) = e^\frac{\pi i \tau}{12}\vartheta\left(\frac{\tau+1}{2}; 3\tau\right).</math>
The [[Euler function]]

:<math>\begin{align}
\phi(q) &= \prod_{n=1}^\infty \left(1-q^n\right) \\
&= q^{-\frac{1}{24}} \eta(\tau),
\end{align}</math>

has a power series by the [[Pentagonal number theorem|Euler identity]]:

:<math>\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^\frac{3n^2-n}{2}.</math>

Note that by using [[Pentagonal number theorem| Euler Pentagonal number theorem]] for <math> \mathfrak{I} (\tau )>0 </math>, the eta function can be expressed as

:<math>\eta(\tau)=\sum_{n=-\infty}^\infty e^{\pi i n}e^{3\pi i \left(n+\frac{1}{6}\right)^2 \tau}.</math>
This can be proved by using <math>x=2\pi i \tau</math> in [[Pentagonal number theorem| Euler Pentagonal number theorem]] with the definition of eta function.

Another way to see the Eta function is through the following limit 

<math>\lim_{z \to 0} \frac{\vartheta_1(z|\tau)}{z}=2\pi \eta^3(\tau)</math>

Which alternatively is: 

<math>
\sum_{n=0}^\infty (-1)^n (2n+1)q^{\frac{(2n+1)^2}8}=\eta^3(\tau)</math>

Where <math> \vartheta_1(z|\tau)</math> is the [[Jacobi Theta function ]] and <math> \vartheta_1(z|\tau)=-\vartheta_{11}(z;\tau)</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''<sup>{{sfrac|1|24}}</sup>}} between this and eta makes almost no visual difference whatsoever.  Thus, this picture can be taken as a picture of eta as a function of {{mvar|q}}.