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The Wright Omega function along part of the real axis

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

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

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 $z\ne x\pm i\pi$ 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 $W_k(z)=\omega (\ln (z)+2\pi ik)$.

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:

$\int w^{n}dz=\{\begin{array}{cc}\frac{{\omega }^{n+1}-1}{n+1}+\frac{{\omega }^n}{n}& \text{if }n\ne -1,\\ \ln (\omega )-\frac{1}{\omega }& \text{if }n=-1.\end{array}$

Its Taylor series around the point $a={\omega }_a+\ln ({\omega }_a)$ takes the form :

$\omega (z)=\sum _{n=0}^{+\mathrm{\infty }}\frac{q_n({\omega }_a)}{(1+{\omega }_a{)}^{2n-1}}\frac{(z-a{)}^n}{n!}$

where

$q_n(w)=\sum _{k=0}^{n-1}⟨⟨\begin{array}{c}n+1\\ k\end{array}⟩⟩(-1{)}^k{w}^{k+1}$

in which

$⟨⟨\begin{array}{c}n\\ k\end{array}⟩⟩$

is a second-order Eulerian number.

Values

$\begin{array}{ccc}\omega (0)& =W_0(1)& \approx 0.56714\\ \omega (1)& =1& \\ \omega (-1\pm i\pi )& =-1& \\ \omega (-\frac{1}{3}+\ln \left(\frac{1}{3}\right)+i\pi )& =-\frac{1}{3}& \\ \omega (-\frac{1}{3}+\ln \left(\frac{1}{3}\right)-i\pi )& =W_{-1}(-\frac{1}{3}{e}^{-\frac{1}{3}})& \approx -2.237147028\end{array}$

Plots

• Plots of the Wright Omega function on the complex plane
• 
  z = Re(ω(x + i y))
• 
  z = Im(ω(x + i y))
• 

Notes

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References

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