Difference between revisions of "Euler phi"

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The Euler phi function is defined as
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The Euler phi function (not to be confused with the [[Euler totient]]) is defined for $q \in \mathbb{C}$ with $|q|<1$ by
 
$$\phi(q) = \displaystyle\prod_{k=1}^{\infty} 1-q^k.$$
 
$$\phi(q) = \displaystyle\prod_{k=1}^{\infty} 1-q^k.$$
  
 
<div align="center">
 
<div align="center">
 
<gallery>
 
<gallery>
File:Qpochhammer(q,q)infty.png|Plot of $(q,q)_{\infty}$ for $q \in [-1,1]$.
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File:Eulerphiplot.png|Graph of $\phi$.
File:Complex qpochhammer (q,q) infty.png|[[Domain coloring]] of [[analytic continuation]] of $(q,q)_{\infty}$ to the unit disk.
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File:Complexeulerphiplot.png|[[Domain coloring]] $\phi$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>
  
 
=Properties=
 
=Properties=
{{:Relationship between Euler phi and q-Pochhammer}}
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[[Relationship between Euler phi and q-Pochhammer]]<br />
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=References=
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{{:Number theory functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 06:33, 22 June 2016

The Euler phi function (not to be confused with the Euler totient) is defined for $q \in \mathbb{C}$ with $|q|<1$ by $$\phi(q) = \displaystyle\prod_{k=1}^{\infty} 1-q^k.$$

Properties

Relationship between Euler phi and q-Pochhammer

References

Number theory functions