Difference between revisions of "Dedekind eta-function"
From Encyclopedia of Mathematics
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The function defined by | The function defined by | ||
$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$ | $$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$ | ||
| − | for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the [[ | + | for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the [[infinite product]] converges absolutely, uniformly for $z$ in compact sets (cf. [[Uniform convergence]]), the function $\eta$ is holomorphic (cf. [[Analytic function]]). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^{24}$ is a [[modular form]] of weight $12$ (cf. also [[Modular group]]). |
| − | R. Dedekind [[#References|[a1]]] comments on computations of B. Riemann in connection with | + | R. Dedekind [[#References|[a1]]] comments on computations of B. Riemann in connection with [[theta-function]]s. He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called [[Dedekind sum]]s. See [[#References|[a2]]], Chapt. IV, for a further discussion. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint (1953)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , ''B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass'' , Dover, reprint (1953) {{ZBL|0053.19405}}</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972) {{ZBL|0251.10020}}</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | |||
| + | [[Category:Special functions]] | ||
Latest revision as of 19:54, 12 April 2017
2020 Mathematics Subject Classification: Primary: 11F20 [MSN][ZBL]
The function defined by
$$\eta(z)=e^{\pi iz/12}\prod_{n=1}^\infty(1-e^{2\pi inz})$$
for $z\in\mathbf C$, $\operatorname{Im}z>0$. As the infinite product converges absolutely, uniformly for $z$ in compact sets (cf. Uniform convergence), the function $\eta$ is holomorphic (cf. Analytic function). Moreover, it satisfies $\eta(z+1)=e^{\pi i/12}\eta(z)$ and $\eta(-1/z)=\sqrt{-iz}\eta(z)$. So, $\eta^{24}$ is a modular form of weight $12$ (cf. also Modular group).
R. Dedekind [a1] comments on computations of B. Riemann in connection with theta-functions. He shows that it is basic to understand the transformation behaviour of the logarithm of the function now carrying his name. This study leads him to quantities now called Dedekind sums. See [a2], Chapt. IV, for a further discussion.
References
| [a1] | R. Dedekind, "Erläuterungen zu den fragmenten XXVIII" H. Weber (ed.) , B. Riemann: Gesammelte mathematische Werke und wissenschaftlicher Nachlass , Dover, reprint (1953) Zbl 0053.19405 |
| [a2] | H. Rademacher, E. Grosswald, "Dedekind sums" , Math. Assoc. America (1972) Zbl 0251.10020 |
How to Cite This Entry:
Dedekind eta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_eta-function&oldid=32933
Dedekind eta-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dedekind_eta-function&oldid=32933
This article was adapted from an original article by R.W. Bruggeman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article