Difference between revisions of "Dedekind eta"
From specialfunctionswiki
| Line 20: | Line 20: | ||
=References= | =References= | ||
[http://eta.math.georgetown.edu/ A collection of over 6200 identities for the Dedekind Eta Function] | [http://eta.math.georgetown.edu/ A collection of over 6200 identities for the Dedekind Eta Function] | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Revision as of 18:27, 24 May 2016
Let $q=e^{2\pi i \tau}$. We define the Dedekind eta function by the formula $$\eta(\tau) = e^{\frac{\pi i \tau}{12}} \displaystyle\prod_{n=1}^{\infty} (1-q^n).$$
Real part of $\eta$.
Imaginary part of $\eta$.
Properties
Theorem: The following formula holds for $\tau$ with $\mathrm{Im} \hspace{2pt} \tau > 0$: $$\eta \left( -\dfrac{1}{\tau} \right) = (-i\tau)^{\frac{1}{2}}\eta(\tau).$$
Proof: █
References
A collection of over 6200 identities for the Dedekind Eta Function
