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Encyclopedia > Dedekind eta function

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 τ, we define the nome q = eiπτ, and define the eta function by In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ... In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

The eta function is holomorphic on the upper half plane but cannot be continued analytically beyond it. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

Enlarge
Modulus of Euler phi on the unit disc, colored so that black=0, red=4
The real part of the modular discriminant as a function of the nome q.

The eta function satisfies the functional equations Eulers Q-series on the complex plane (600x600 pixels) Detailed description This picture shows the modulus on the complex plane, inside the unit circle , where is Eulers q-series for the Partition of an integer. ... Eulers Q-series on the complex plane (600x600 pixels) Detailed description This picture shows the modulus on the complex plane, inside the unit circle , where is Eulers q-series for the Partition of an integer. ... Modular discrimnant, real part, as function of nome. ... Modular discrimnant, real part, as function of nome. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...

η(τ + 1) = exp(2πi / 24)η(τ)

More generally,

where a,b,c,d are integers, with ad-bc=1, and thus being a transform belonging to the modular group, and In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

and s(h,k) is the Dedekind sum In mathematics, Dedekind sums are certain sums of products of a sawtooth function s, and are given by a function D of three integer variables. ...

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 A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion of the constant coefficient a0. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ...

Δ(τ) = (2π)12η(τ)24

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 In mathematics, a q-series is defined as usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ...

where z = q2. Note that the additional factor of z1 / 12 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. In mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi theta function, normally written as a series, as a product. ... In mathematics, theta functions are special functions of several complex variables. ...


Related topics

In mathematics, a q-series is defined as usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. ...

References

  • Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. ISBN 0-387-97127-0 See chapter 3.

  Results from FactBites:
 
ONLINE THESES IN NUMBER THEORY (3211 words)
Evaluation of the Dedekind Eta Function, PhD thesis, William Hart, Macquarie University 2004
The pair correlation of the zeros of the Riemann zeta function, Senior Thesis, Lillian Pierce, Princeton University, 2002
On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD thesis, Christopher Hughes, University of Bristol 2001
PlanetMath: modular discriminant (104 words)
The Dedekind eta function is defined to be
The Dedekind eta function should not be confused with the Weierstrass eta function,
-invariant, as a function of lattices, is defined to be:
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