Wright Omega function

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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:

Uses[edit]

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 for x ≤ −1, of the equation y + ln(y) = z. Except on those two rays, the Wright omega function is continuous, even analytic.

Properties[edit]

The Wright omega function satisfies the relation .

It also satisfies the differential equation

wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation ), and as a consequence its integral can be expressed as:

Its Taylor series around the point takes the form :

where

in which

is a second-order Eulerian number.

Values[edit]

Plots[edit]

Notes[edit]

  1. ^ Not to be confused with the Fox–Wright function, also known as Wright function.

References[edit]