Wright Omega function

The Wright Omega function along part of the real axis

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

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

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 $W_k(z) = \omega(\ln(z) + 2 \pi i k)$.

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:

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

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
• 

  z = Re(ω(x + i y))

• 

  z = Im(ω(x + i y))

• 

  z = |ω(x + i y)|

Notes

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

References

• "On the Wright ω function", Robert Corless and David Jeffrey