Wright omega function

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:

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

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

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:

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

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
• 
  $z=\mathrm{\Re }\{\omega (x+iy)\}$
• 
  $z=\mathrm{\Im }\{\omega (x+iy)\}$
• 
  $z=|\omega (x+iy)|$

References

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