Difference between revisions of "Euler phi"
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| − | The Euler phi function is defined for $q \in \mathbb{C}$ with $|q|<1$ by | + | 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:Eulerphiplot.png|Graph of $\phi$ | + | File:Eulerphiplot.png|Graph of $\phi$. |
| − | File: | + | File:Complexeulerphiplot.png|[[Domain coloring]] $\phi$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
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=Properties= | =Properties= | ||
[[Relationship between Euler phi and q-Pochhammer]]<br /> | [[Relationship between Euler phi and q-Pochhammer]]<br /> | ||
| + | |||
| + | =References= | ||
| + | |||
| + | {{: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 $$\phi(q) = \displaystyle\prod_{k=1}^{\infty} 1-q^k.$$
Graph of $\phi$.
Domain coloring $\phi$.
Properties
Relationship between Euler phi and q-Pochhammer
