# Wright omega function

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{{Short description|Mathematical function}}
{{Use American English|date = March 2019}}
thumb|right|250px|The Wright omega function along part of the real axis

In [mathematics](/source/mathematics), the '''Wright omega function''' or '''Wright function''',<ref group="note">Not to be confused with the [Fox–Wright function](/source/Fox%E2%80%93Wright_function), also known as Wright function.</ref> denoted ''ω'', is defined in terms of the [Lambert W function](/source/Lambert_W_function) as:

: <math>\omega(z) = W_{\big \lceil \frac{\mathrm{Im}(z) - \pi}{2 \pi} \big \rceil}(e^z).</math>

It is simpler to be defined by its inverse function

: <math> z (\omega) = \ln(\omega)+\omega </math>

==Uses==
One of the main applications of this function is in the resolution of the equation ''z''&nbsp;=&nbsp;ln(''z''), as the only solution is given by ''z''&nbsp;=&nbsp;''e''<sup>&minus;ω(''&pi;''&nbsp;''i'')</sup>.

''y'' = ω(''z'') is the unique solution, when <math>z \neq x \pm i \pi</math> for ''x''&nbsp;&le;&nbsp;&minus;1, of the equation ''y''&nbsp;+&nbsp;ln(''y'')&nbsp;=&nbsp;''z''. Except for those two values, the Wright omega function is [continuous](/source/continuous_function), even [analytic](/source/analytic_function).

==Properties==
The Wright omega function satisfies the relation <math>W_k(z) = \omega(\ln(z) + 2 \pi i k)</math>.

It also satisfies the [differential equation](/source/differential_equation)

: <math> \frac{d\omega}{dz} = \frac{\omega}{1 + \omega}</math>

wherever ω is analytic (as can be seen by performing [separation of variables](/source/separation_of_variables) and recovering the equation <math>\ln(\omega)+\omega = z</math>, and as a consequence its [integral](/source/integral) can be expressed as:

: <math>
\int \omega^n \, dz = 
\begin{cases} 
  \frac{\omega^{n+1} -1 }{n+1} + \frac{\omega^n}{n}  & \mbox{if } n \neq -1, \\
  \ln(\omega) - \frac{1}{\omega} & \mbox{if } n = -1.
\end{cases}
</math>

Its [Taylor series](/source/Taylor_series) around the point <math> a = \omega_a + \ln(\omega_a) </math> takes the form :

: <math>\omega(z) = \sum_{n=0}^{+\infty} \frac{q_n(\omega_a)}{(1+\omega_a)^{2n-1}}\frac{(z-a)^n}{n!}</math>

where

: <math>q_n(w) = \sum_{k=0}^{n-1} \bigg \langle \! \! \bigg \langle 
\begin{matrix}
  n+1 \\
  k
\end{matrix} 
\bigg \rangle \! \! \bigg \rangle (-1)^k w^{k+1}</math>

in which

: <math>\bigg \langle \! \! \bigg \langle 
\begin{matrix}
  n \\
  k
\end{matrix} 
\bigg \rangle \! \! \bigg \rangle</math>

is a [second-order Eulerian number](/source/Eulerian_number).

==Values==

:<math>
\begin{array}{lll}
\omega(0) &= W_0(1) &\approx 0.56714 \\
\omega(1) &= 1 & \\
\omega(-1 \pm i \pi) &= -1 & \\
\omega(-\frac{1}{3} + \ln \left ( \frac{1}{3} \right ) + i \pi ) &= -\frac{1}{3} & \\
\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 \\
\end{array}
</math>

==Plots==
<gallery caption="Plots of the Wright omega function on the complex plane">
Image:Wright omega function - real.png|<math>\Re\{\omega(z)\}</math>
Image:Wright omega - imaginary.png|<math>\Im\{\omega(z)\}</math>
Image:Wright omega - magnitude.png|<math>|\omega(z)|</math>
</gallery>

==Notes==
{{reflist|group=note}}

==References==
{{refbegin}}
*{{cite conference |last=Corless |first=R.M. |last2=Jeffrey |first2=D.J. |title=The Wright ω function |book-title=International Conference on Artificial Intelligence and Symbolic Computation |publisher=Springer |date=June 2002 |isbn=3-540-45470-5 |pages=76–89 |url=https://www.uwo.ca/apmaths/faculty/jeffrey/pdfs/wrightomega.pdf |doi=10.1007/3-540-45470-5_10 |series=Lecture Notes in Computer Science |volume=2385 }}
*{{cite book |first=Istvan |last=Mezo |chapter=3. Unwinding number and branch differences §3.4 The Wright ω function |title=The Lambert W function: its generalizations and applications |publisher=Chapman and Hall/CRC |date=2022 |isbn=978-1-003-16810-2 |pages=82–87 |url={{GBurl|iJwIEQAAQBAJ|pg=PR9}} }}
{{refend}}

Category:Special functions

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Adapted from the Wikipedia article [Wright omega function](https://en.wikipedia.org/wiki/Wright_omega_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Wright_omega_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.