Wright omega function

Mathematical function

The Wright omega function along part of the real axis

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

ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( e z ) . {\displaystyle \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 ≠ x ± i π {\displaystyle 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 ) = ω ( ln ⁡ ( z ) + 2 π i k ) {\displaystyle W_{k}(z)=\omega (\ln(z)+2\pi ik)} .

It also satisfies the differential equation

d ω d z = ω 1 + ω {\displaystyle {\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 ⁡ ( ω ) + ω = z {\displaystyle \ln(\omega )+\omega =z} ), and as a consequence its integral can be expressed as:

∫ w n d z = { ω n + 1 − 1 n + 1 + ω n n if  n ≠ − 1 , ln ⁡ ( ω ) − 1 ω if  n = − 1. {\displaystyle \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 = ω a + ln ⁡ ( ω a ) {\displaystyle a=\omega _{a}+\ln(\omega _{a})} takes the form :

ω ( z ) = ∑ n = 0 + ∞ q n ( ω a ) ( 1 + ω a ) 2 n − 1 ( z − a ) n n ! {\displaystyle \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 ) = ∑ k = 0 n − 1 ⟨ ⟨ n + 1 k ⟩ ⟩ ( − 1 ) k w k + 1 {\displaystyle 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

⟨ ⟨ n k ⟩ ⟩ {\displaystyle {\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }}

is a second-order Eulerian number.

Values

ω ( 0 ) = W 0 ( 1 ) ≈ 0.56714 ω ( 1 ) = 1 ω ( − 1 ± i π ) = − 1 ω ( − 1 3 + ln ⁡ ( 1 3 ) + i π ) = − 1 3 ω ( − 1 3 + ln ⁡ ( 1 3 ) − i π ) = W − 1 ( − 1 3 e − 1 3 ) ≈ − 2.237147028 {\displaystyle {\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 = Re(ω(x + i y))
  • z = Im(ω(x + i y))
    z = Im(ω(x + i y))
  • ω(x + i y)
    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