Wright Omega function

In mathematics, the Wright omega 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).

Contents

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 on those two rays, the Wright omega function is continuous, even analytic.

Properties

The Wright omega function satisfies the relation W_k(z) = \omega(\ln(z) %2B 2 \pi i k).

It also satisfies the differential equation

\frac{d\omega}{dz} = \frac{\omega}{1 %2B \omega}

wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation \ln(\omega)%2B\omega = z), and as a consequence its integral can be expressed as:

\int w^n \, dz = \begin{cases} \frac{\omega^{n%2B1} -1 }{n%2B1} %2B \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 %2B \ln(\omega_a) takes the form :

\omega(z) = \sum_{n=0}^{%2B\infty} \frac{q_n(\omega_a)}{(1%2B\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%2B1 \\ k \end{matrix} \bigg \rangle \! \! \bigg \rangle (-1)^k w^{k%2B1}

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} %2B \ln \left ( \frac{1}{3} \right ) %2B i \pi ) &= -\frac{1}{3} & \\ \omega(-\frac{1}{3} %2B \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

References