Wright omega function

Mathematical function From Wikipedia, the free encyclopedia

Wright omega function

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

Thumb
The Wright omega function along part of the real axis

It is simpler to be defined by its inverse function

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 = eω(π i).

y = ω(z) is the unique solution, when for x  1, of the equation y + ln(y) = z. Except for those two values, the Wright omega function is continuous, even analytic.

Properties

Summarize
Perspective

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

Plots

Notes

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

References

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