Wright Omega function

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

: $$omega(z) = W_{ig lceil frac{mathrm{Im}(z) - pi}{2 pi} ig ceil}(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 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:

: $$int w^n , dz = egin{cases} frac{omega^{n+1} -1 }{n+1} + frac{omega^n}{n} & mbox{if } n eq -1, \\ ln(omega) - frac{1}{omega} & mbox{if } n = -1.end{cases}

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

: $$q_n(w) = sum_{k=0}^{n-1} igg langle ! ! igg langle egin{matrix} n+1 \\ kend{matrix} igg angle ! ! igg angle (-1)^k w^{k+1}

in which

: $$igg langle ! ! igg langle egin{matrix} n \\ kend{matrix} igg angle ! ! igg angle

is a second-order Eulerian number.

Values

:$$egin{array}{lll}omega(0) &= W_0(1) &approx 0.56714 \\omega(1) &= 1 & \\omega(-1 pm i pi) &= -1 & \\omega(-frac{1}{3} + ln left ( frac{1}{3} ight ) + i pi ) &= -frac{1}{3} & \\omega(-frac{1}{3} + ln left ( frac{1}{3} ight ) - i pi ) &= W_{-1} left ( -frac{1}{3} e^{-frac{1}{3 ight ) &approx -2.237147028 \\end{array}

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]