Wright omega function
Mathematical function
title: "Wright omega function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["special-functions"] description: "Mathematical function" topic_path: "general/special-functions" source: "https://en.wikipedia.org/wiki/Wright_omega_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Mathematical function ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/63/WrightOmega.png" caption="The Wright omega function along part of the real axis"] ::
In mathematics, the Wright omega function or Wright function,Not to be confused with the Fox–Wright function, also known as 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
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 i k).
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
Image:Wright omega function - real.png|\Re{\omega(z)} Image:Wright omega - imaginary.png|\Im{\omega(z)} Image:Wright omega - magnitude.png||\omega(z)|
Notes
References
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