Wright omega function

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

• "The Wright ω function", Robert Corless and David Jeffrey