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Wright omega function

Mathematical function

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Source updated April 12, 2026

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In mathematics, the Wright omega function or Wright function, 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

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

01Uses

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.

02Properties

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.

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Sources and credits

This article is adapted from the Wikipedia article “Wright omega function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

WrightOmega.png by Sam Derbyshire, CC BY-SA 3.0

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