Dedekind's eta function

The Dedekind eta function in the complex plane

The eta function (η function) named after the German mathematician Richard Dedekind is a holomorphic function on the upper half-plane . ${\displaystyle \mathbb {H} =\{\tau \in \mathbb {C} \mid \mathrm {Im} \,\tau >0\}}$

It plays an important role in the theory of elliptic functions and theta functions .

definition

The eta function is usually defined as an infinite product as follows:

${\displaystyle \eta (\tau ):=e^{\pi i\tau /12}\prod _{n=1}^{\infty }(1-e^{2\pi in\tau })}$.

From the definition it follows immediately that in has no zeros. ${\displaystyle \eta }$ ${\displaystyle \mathbb {H} }$

The function is closely related to the discriminant it is ${\displaystyle \Delta }$

${\displaystyle \Delta (\tau )\,=\,(2\pi )^{12}\eta ^{24}(\tau )}$.

To calculate the function, the pentagonal set of numbers can be used when looking at the representation

${\displaystyle \eta (\tau )=q^{\frac {1}{24}}\prod _{n=1}^{\infty }(1-q^{n})}$

used with the usual abbreviation . ${\displaystyle q=e^{2\pi i\tau }}$

Transformation behavior

The function gets its importance from its transformation behavior among the substitutions of the producers of the module group

${\displaystyle \Gamma :=\mathrm {SL} _{2}(\mathbb {Z} )=\{{\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}\mid a,b,c,d\in \mathbb {Z} ,ad-bc=1\}}$,

namely:

${\displaystyle \eta (\tau +1)\,=\,e^{\pi i/12}\eta (\tau )}$

and

${\displaystyle \eta \left({\frac {-1}{\tau }}\right)={\sqrt {\frac {\tau }{i}}}\,\eta (\tau )}$.

literature

• Tom M. Apostol: Modular Functions and Dirichlet Series in Number Theory . Springer-Verlag, Berlin / Heidelberg / New York (1990), ISBN 3-540-97127-0
• Eberhard Freitag, Rolf Busam: Function Theory 1 . 4th edition. Springer-Verlag, Berlin (2006), ISBN 3-540-31764-3
• Max Koecher , Aloys Krieg : Elliptical functions and modular forms . 2nd Edition. Springer-Verlag, Berlin (2007), ISBN 978-3-540-49324-2

Web links

• Eric W. Weisstein : Dedekind Eta Function . In: MathWorld (English).