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Wright Omega function

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

Contents

ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( 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 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 ) .

It also satisfies the differential equation

d ω d z = ω 1 + ω

wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation ln ⁡ ( ω ) + ω = 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.

Its Taylor series around the point a = ω a + ln ⁡ ( ω a ) takes the form :

ω ( z ) = ∑ n = 0 + ∞ q n ( ω a ) ( 1 + ω a ) 2 n − 1 ( z − a ) n n !

where

q n ( w ) = ∑ k = 0 n − 1 ⟨ ⟨ n + 1 k ⟩ ⟩ ( − 1 ) k w k + 1

in which

⟨ ⟨ n k ⟩ ⟩

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

Plots

  • Plots of the Wright Omega function on the complex plane

    • References

    Wright Omega function Wikipedia
    (Text) CC BY-SA