{{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, the '''Wright omega function''' or '''Wright function''',<ref group="note">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:
: <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'' = ln(''z''), as the only solution is given by ''z'' = ''e''<sup>−ω(''π'' ''i'')</sup>.
''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 for those two values, the Wright omega function is continuous, even analytic.
==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
: <math> \frac{d\omega}{dz} = \frac{\omega}{1 + \omega}</math>
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:
: <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 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.
==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