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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:

Contents

UsesEdit

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.

PropertiesEdit

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.

ValuesEdit

 

PlotsEdit

NotesEdit

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

ReferencesEdit