Wright omega function

In mathematics, the Wright omega function or Wright function,^{[note 1]} denoted ω, is defined in terms of the Lambert W function as:

${\displaystyle \omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).}$

It is simpler to be defined by its inverse function

${\displaystyle z(\omega )=\ln(\omega )+\omega }$

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^−ω(π i).

y = ω(z) is the unique solution, when ${\displaystyle z\neq x\pm i\pi }$ for x ≤ −1, of the equation y + ln(y) = z. Except for those two values, the Wright omega function is continuous, even analytic.

The Wright omega function satisfies the relation ${\displaystyle W_{k}(z)=\omega (\ln(z)+2\pi ik)}$.

It also satisfies the differential equation

${\displaystyle {\frac {d\omega }{dz}}={\frac {\omega }{1+\omega }}}$

wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation ${\displaystyle \ln(\omega )+\omega =z}$, and as a consequence its integral can be expressed as:

${\displaystyle \int \omega ^{n}\,dz={\begin{cases}{\frac {\omega ^{n+1}-1}{n+1}}+{\frac {\omega ^{n}}{n}}&{\mbox{if }}n\neq -1,\\\ln(\omega )-{\frac {1}{\omega }}&{\mbox{if }}n=-1.\end{cases}}}$

Its Taylor series around the point ${\displaystyle a=\omega _{a}+\ln(\omega _{a})}$ takes the form :

${\displaystyle \omega (z)=\sum _{n=0}^{+\infty }{\frac {q_{n}(\omega _{a})}{(1+\omega _{a})^{2n-1}}}{\frac {(z-a)^{n}}{n!}}}$

where

${\displaystyle q_{n}(w)=\sum _{k=0}^{n-1}{\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n+1\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }(-1)^{k}w^{k+1}}$

in which

${\displaystyle {\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }}$

is a second-order Eulerian number.

${\displaystyle {\begin{array}{lll}\omega (0)&=W_{0}(1)&\approx 0.56714\\\omega (1)&=1&\\\omega (-1\pm i\pi )&=-1&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)+i\pi )&=-{\frac {1}{3}}&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)-i\pi )&=W_{-1}\left(-{\frac {1}{3}}e^{-{\frac {1}{3}}}\right)&\approx -2.237147028\\\end{array}}}$

• Plots of the Wright omega function on the complex plane
• ${\displaystyle z=\Re \{\omega (x+iy)\}}$
• ${\displaystyle z=\Im \{\omega (x+iy)\}}$
• ${\displaystyle z=|\omega (x+iy)|}$

• "The Wright ω function", Robert Corless and David Jeffrey