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| [[File:Q-Eulero.jpeg|thumb|right|Modulus of phi on the complex plane, colored so that black=0, red=4]]
| | The writer is recognized by the name of Numbers Wunder. He is truly fond of doing ceramics but he is having difficulties to discover time for it. California is our birth place. In her professional life she is a payroll clerk but she's always wanted her own company.<br><br>Here is my web blog ... [http://Www.gaysphere.net/blog/262619 std testing at home] |
| :''For other meanings, see [[List of topics named after Leonhard Euler]]''.
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| In [[mathematics]], the '''Euler function''' is given by
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| :<math>\phi(q)=\prod_{k=1}^\infty (1-q^k).</math>
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| Named after [[Leonhard Euler]], it is a prototypical example of a [[q-series]], a [[modular form]], and provides the prototypical example of a relation between [[combinatorics]] and [[complex analysis]].
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| ==Properties==
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| The [[coefficient]] <math>p(k)</math> in the [[formal power series]] expansion for <math>1/\phi(q)</math> gives the number of all [[Partition of an integer|partitions]] of k. That is,
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| :<math>\frac{1}{\phi(q)}=\sum_{k=0}^\infty p(k) q^k</math>
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| where <math>p(k)</math> is the [[Partition_function_(number_theory)|partition function]] of k.
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| The '''Euler identity''', also known as the [[Pentagonal number theorem]] is
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| :<math>\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}. </math>
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| Note that <math>(3n^2-n)/2</math> is a [[pentagonal number]].
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| The Euler function is related to the [[Dedekind eta function]] through a [[Ramanujan identity]] as
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| :<math>\phi(q)= q^{-\frac{1}{24}} \eta(\tau)</math>
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| where <math>q=e^{2\pi i\tau}</math> is the square of the [[nome (mathematics)|nome]].
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| Note that both functions have the symmetry of the [[modular group]].
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| The Euler function may be expressed as a [[Q-Pochhammer symbol]]:
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| :<math>\phi(q)=(q;q)_\infty</math>
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| The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q=0, yielding:
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| :<math>\ln(\phi(q))=-\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{1-q^n}</math>
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| which is a [[Lambert series]] with coefficients ''-1/n''. The logarithm of the Euler function may therefore be expressed as:
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| :<math>\ln(\phi(q))=\sum_{n=1}^\infty b_n q^n</math>
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| where
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| :<math>b_n=-\sum_{d|n}\frac{1}{d}=</math> -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see [[OEIS]] [http://oeis.org/A000203/table A000203])
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| On account of the following identity,
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| :<math>\sum_{d|n} d = \sum_{d|n} \frac n d</math>
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| this may also be written as
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| :<math>\ln(\phi(q))=-\sum_{n=1}^\infty \frac{q^n}{n} \sum_{d|n} d</math>
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| ==Special values==
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| The next identities come from [[Ramanujan's lost notebook]], Part V, p. 326.
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| : <math>
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| \phi(e^{-\pi})=\frac{e^{\pi/24}\Gamma\left(\frac14\right)}{2^{7/8}\pi^{3/4}}
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| </math>
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| : <math>
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| \phi(e^{-2\pi})=\frac{e^{\pi/12}\Gamma\left(\frac14\right)}{2\pi^{3/4}}
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| </math>
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| : <math>
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| \phi(e^{-4\pi})=\frac{e^{\pi/6}\Gamma\left(\frac14\right)}{2^{{11}/8}\pi^{3/4}}
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| </math>
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| : <math>
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| \phi(e^{-8\pi})=\frac{e^{\pi/3}\Gamma\left(\frac14\right)}{2^{29/16}\pi^{3/4}}(\sqrt{2}-1)^{1/4}
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| </math>
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| ==References==
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| * {{Apostol IANT}}
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| [[Category:Number theory]]
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| [[Category:Q-analogs]]
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| [[km:អនុគមន៍អឺលែរ]]
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The writer is recognized by the name of Numbers Wunder. He is truly fond of doing ceramics but he is having difficulties to discover time for it. California is our birth place. In her professional life she is a payroll clerk but she's always wanted her own company.
Here is my web blog ... std testing at home