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Dedekind Eta Function

\begin{figure}\begin{center}\BoxedEPSF{DedekindEtaReIm.epsf scaled 620}\end{center}\end{figure}

Let

\begin{displaymath}
q=e^{2\pi iz},
\end{displaymath} (1)

then the Dedekind eta function is defined over the upper half-plane $H=\{\tau:\Im[\tau]>0\}$ by
\begin{displaymath}
\eta(z)\equiv q^{1/24}\prod_{n=1}^\infty (1-q^{n}),
\end{displaymath} (2)

which can be written as
\begin{displaymath}
\eta(z)=q^{1/24}\left\{{1+\sum_{n=1}^\infty (-1)^n [q^{n(3n-1)/2}+q^{n(3n+1)/2}]}\right\}
\end{displaymath} (3)

(Weber 1902, pp. 85 and 112; Atkin and Morain 1993). $\eta$ is a Modular Form. Letting $\zeta_{24}=e^{2\pi
i/24}$ be a Root of Unity, $\eta(z)$ satisfies
$\displaystyle \eta(z+1)$ $\textstyle =$ $\displaystyle \zeta_{24}\eta(z)$ (4)
$\displaystyle \eta\left({-{1\over z}}\right)$ $\textstyle =$ $\displaystyle \sqrt{-iz}\,\eta(z)$ (5)

(Weber 1902, p. 113; Atkin and Morain 1993).

See also Dirichlet Eta Function, Theta Function, Weber Functions


References

Atkin, A. O. L. and Morain, F. ``Elliptic Curves and Primality Proving.'' Math. Comput. 61, 29-68, 1993.

Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1902.




© 1996-9 Eric W. Weisstein
1999-05-24