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| [[Image:WrightOmega.png|thumb|right|250px|The Wright Omega function along part of the real axis]]
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| In [[mathematics]], the '''Wright omega function''' or '''Wright function''',<ref>Not to be confused with the [[Fox–Wright function]], also known as Wright function.</ref> denoted ω, is defined in terms of the [[Lambert W function]] as:
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| : <math>\omega(z) = W_{\big \lceil \frac{\mathrm{Im}(z) - \pi}{2 \pi} \big \rceil}(e^z).</math>
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| ==Uses== | |
| One of the main applications of this function is in the resolution of the equation ''z'' = ln(''z''), as the only solution is given by ''z'' = ''e''<sup>−ω(''π'' ''i'')</sup>.
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| ''y'' = ω(''z'') is the unique solution, when <math>z \neq x \pm i \pi</math> for ''x'' ≤ −1, of the equation ''y'' + ln(''y'') = ''z''. Except on those two rays, the Wright omega function is [[continuous function|continuous]], even [[analytic function|analytic]].
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| ==Properties==
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| The Wright omega function satisfies the relation <math>W_k(z) = \omega(\ln(z) + 2 \pi i k)</math>.
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| It also satisfies the [[differential equation]]
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| : <math> \frac{d\omega}{dz} = \frac{\omega}{1 + \omega}</math>
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| wherever ω is analytic (as can be seen by performing [[separation of variables]] and recovering the equation <math>\ln(\omega)+\omega = z</math>), and as a consequence its [[integral]] can be expressed as:
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| : <math> | |
| \int w^n \, dz =
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| \begin{cases}
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| \frac{\omega^{n+1} -1 }{n+1} + \frac{\omega^n}{n} & \mbox{if } n \neq -1, \\
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| \ln(\omega) - \frac{1}{\omega} & \mbox{if } n = -1.
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| \end{cases}
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| </math>
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| Its [[Taylor series]] around the point <math> a = \omega_a + \ln(\omega_a) </math> takes the form :
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| : <math>\omega(z) = \sum_{n=0}^{+\infty} \frac{q_n(\omega_a)}{(1+\omega_a)^{2n-1}}\frac{(z-a)^n}{n!}</math>
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| where
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| : <math>q_n(w) = \sum_{k=0}^{n-1} \bigg \langle \! \! \bigg \langle
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| \begin{matrix}
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| n+1 \\
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| k
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| \end{matrix}
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| \bigg \rangle \! \! \bigg \rangle (-1)^k w^{k+1}</math>
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| in which
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| : <math>\bigg \langle \! \! \bigg \langle
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| \begin{matrix}
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| n \\
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| k
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| \end{matrix}
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| \bigg \rangle \! \! \bigg \rangle</math>
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| is a second-order [[Eulerian number]].
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| ==Values==
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| :<math>
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| \begin{array}{lll}
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| \omega(0) &= W_0(1) &\approx 0.56714 \\
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| \omega(1) &= 1 & \\
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| \omega(-1 \pm i \pi) &= -1 & \\
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| \omega(-\frac{1}{3} + \ln \left ( \frac{1}{3} \right ) + i \pi ) &= -\frac{1}{3} & \\
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| \omega(-\frac{1}{3} + \ln \left ( \frac{1}{3} \right ) - i \pi ) &= W_{-1} \left ( -\frac{1}{3} e^{-\frac{1}{3}} \right ) &\approx -2.237147028 \\
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| \end{array}
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| </math>
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| ==Plots==
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| <gallery caption="Plots of the Wright Omega function on the complex plane"> | |
| Image:WrightOmegaRe.png| ''z'' = Re(ω(''x'' + ''i'' ''y''))
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| Image:WrightOmegaIm.png| ''z'' = Im(ω(''x'' + ''i'' ''y''))
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| Image:WrightOmegaAbs.png| ''z'' = |ω(''x'' + ''i'' ''y'')|
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| </gallery> | |
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf "On the Wright ω function", Robert Corless and David Jeffrey]
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| [[Category:Special functions]]
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