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The Wright omega function along part of the real axis

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

\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

z(\omega )=\ln(\omega )+\omega

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^−ω(π i).

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

Properties

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

It also satisfies the differential equation

{\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 $\ln(\omega )+\omega =z$ , and as a consequence its integral can be expressed as:

\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 $a=\omega _{a}+\ln(\omega _{a})$ takes the form :

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

where

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

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

is a second-order Eulerian number.

Values

{\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

Plots of the Wright omega function on the complex plane
$\Re \{\omega (z)\}$
$\Im \{\omega (z)\}$
$|\omega (z)|$

Notes

^ Not to be confused with the Fox–Wright function, also known as Wright function.

References

Corless, R.M.; Jeffrey, D.J. (June 2002). “The Wright ω function” (PDF). International Conference on Artificial Intelligence and Symbolic Computation. Lecture Notes in Computer Science. Vol. 2385. Springer. pp. 76–89. doi:10.1007/3-540-45470-5_10. ISBN 3-540-45470-5.
Mezo, Istvan (2022). “3. Unwinding number and branch differences §3.4 The Wright ω function”. The Lambert W function: its generalizations and applications. Chapman and Hall/CRC. pp. 82–87. ISBN 978-1-003-16810-2.

[rdp-we-cite_note-1] Not to be confused with the Fox–Wright function, also known as Wright function.

[note 1]