Wright Omega function

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

:

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 $z\; eq\; x\; pm\; i\; pi$ 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 $W\_k(z)\; =\; omega(ln(z)\; +\; 2\; pi\; i\; k)$.

It also satisfies the differential equation

: $frac\{domega\}\{dz\}\; =\; frac\{omega\}\{1\; +\; omega\}$

wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation $ln(omega)+omega\; =\; z$), and as a consequence its integral can be expressed as:

:

Its Taylor series around the point $a\; =\; omega\_a\; +\; ln(omega\_a)$ takes the form :

: $omega(z)\; =\; sum\_\{n=0\}^\{+infty\}\; frac\{q\_n(omega\_a)\}\{(1+omega\_a)^\{2n-1frac\{(z-a)^n\}\{n!\}$

where

:

in which

:

is a second-order Eulerian number.

Values

:

Plots

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References

* [http://www.orcca.on.ca/TechReports/TechReports/2000/TR-00-12.pdf "On the Wright ω function", Robert Corless and David Jeffrey]