Dedekind eta function

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The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. For any such complex number <math>\tau<math>, we define the nome q = eiπτ, and define the eta function by

<math>\eta(\tau) = q^{1/12} \prod_{n=1}^{\infty} (1-q^{2n})<math>

The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it.

Missing image
Q-euler.jpeg
Modulus of Euler phi on the unit disc, colored so that black=0, red=4
Missing image
Discriminant_real_part.jpeg
The real part of the modular discriminant as a function of the nome q.

The eta function satisfies the functional equations

<math>\eta(\tau+1) = \exp(2 \pi i/24)\eta(\tau)<math>
<math>\eta(-1/\tau) = \sqrt {-i\tau} \eta(\tau)<math>

More generally,

<math>\eta \left( \frac{a\tau+b}{c\tau+d} \right) =

\epsilon (a,b,c,d) \left( -i(c\tau+d) \right)^{1/2} \eta(z)<math> where a,b,c,d are integers, with ad-bc=1, and thus being a transform belonging to the modular group, and

<math>\epsilon (a,b,c,d)=\exp i\pi \left(

\frac{a+d}{12c} + s(-d,c) \right) <math> and 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 -\frac{1}{2} \right)<math>

Because of these functional equations the eta function is a modular form of weight 1/2, 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. Because the eta function is easy to compute, it is often helpful 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

<math>\phi(z) = \prod_{n=1}^{\infty} \left(1-z^n\right)<math>

where <math>z=q^2<math>. Note that the additional factor of <math>z^{1/12}<math> between this and the Dedekind 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.

Note that 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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References

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