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 $W_{k}(z)=\omega (\ln(z)+2\pi ik)$ .

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{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))

Notes

↑ 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

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