Difference between revisions of "Dedekind eta"

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