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

ω ω ( z ) = W ⌈ I m ( z ) − − π π 2 π π ⌉ ( e z ) . {\displaystyle \omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).}

It is simpler to be defined by its inverse function

z ( ω ω ) = ln ⁡ ⁡ ( ω ω ) + ω ω {\displaystyle z(\omega )=\ln(\omega )+\omega }

• Uses

• Properties

• Values

• Plots

• Notes

• References

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 π {\displaystyle 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.

The Wright omega function satisfies the relation W k ( z ) = ω ω ( ln ⁡ ⁡ ( z ) + 2 π π i k ) {\displaystyle W_{k}(z)=\omega (\ln(z)+2\pi ik)} .

It also satisfies the differential equation

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

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

ω ω ( z ) = ∑ ∑ n = 0 + ∞ ∞ q n ( ω ω a ) ( 1 + ω 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.

ω ω ( 0 ) = W 0 ( 1 ) ≈ ≈ 0.56714 ω ω ( 1 ) = 1 ω ω ( − − 1 ± ± i π π ) = − − 1 ω ω ( − − 1 3 + ln ⁡ ⁡ ( 1 3 ) + i π π ) = − − 1 3 ω ω ( − − 1 3 + ln ⁡ ⁡ ( 1 3 ) − − i π 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}}}

cite-note-1note 1. ↑ Not to be confused with the Fox–Wright function, also known as Wright function.

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