Difference between revisions of "Dedekind eta"

From specialfunctionswiki
Jump to: navigation, search
 
(9 intermediate revisions by the same user not shown)
Line 1: Line 1:
Let $q=e^{2\pi i \tau}$. We define the Dedekind eta function by the formula
+
Let $q=e^{2\pi i z}$, where $z$ is in the upper half plane. We define the Dedekind eta function by the formula
$$\eta(\tau) = e^{\frac{\pi i \tau}{12}} \displaystyle\prod_{n=1}^{\infty} (1-q^n).$$
+
$$\eta(z) = e^{\frac{\pi i z}{12}} \displaystyle\prod_{n=1}^{\infty} (1-q^n).$$
 +
 
 +
<div align="center">
 +
<gallery>
 +
File:DedekindetaRe.png|Real part of $\eta$.
 +
File:DedekindetaIm.png|Imaginary part of $\eta$.
 +
</gallery>
 +
</div>
 +
 
 +
=Properties=
 +
[[eta(z+1)=e^(i pi/12)eta(z)]]<br />
 +
[[eta(-1/z)=sqrt(-iz)eta(z)]]<br />
 +
 
 +
=References=
 +
[http://eta.math.georgetown.edu/ A collection of over 6200 identities for the Dedekind Eta Function]
 +
 
 +
[[Category:SpecialFunction]]

Latest revision as of 05:20, 12 February 2018

Let $q=e^{2\pi i z}$, where $z$ is in the upper half plane. We define the Dedekind eta function by the formula $$\eta(z) = e^{\frac{\pi i z}{12}} \displaystyle\prod_{n=1}^{\infty} (1-q^n).$$

Properties

eta(z+1)=e^(i pi/12)eta(z)
eta(-1/z)=sqrt(-iz)eta(z)

References

A collection of over 6200 identities for the Dedekind Eta Function