Wright omega function explained

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

\omega(z) = W _{\lceil (Im (z) - \pi)/(2 \pi) \rceil} (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 ≠ x \pm i \pi for x ≤ -1, of the equation y + ln(y) = z. Except for those two values, 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

(d\omega)/(dz) = (\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 \omega ^{n} dz = \begin{cases} (\omega ^{n+1} -1)/(n +1) + (\omega ^{n})/(n) & if n ≠ -1 , \\ ln (\omega) - (1)/(\omega) & if n = -1 . \end{cases}

Its Taylor series around the point

a = \omega _{a} + ln (\omega _{a)} takes the form :\omega(z) = \sum ^{+inf ty} _{n =0} (q _{n(\omega} _{a)})/(( 1+ \omega ^{2 n -1} _{a)}) ((z - a) ^{n})/(n!)

where

q _{n(w)} = \sum ^{n -1} _{k =0} \langle \langle \begin{matrix} n +1 \\ k \end{matrix} \rangle \rangle ( -1 ) ^{k} w ^{k+1}

in which

\langle \langle \begin{matrix} n \\ k \end{matrix} \rangle \rangle

is a second-order Eulerian number.

Values

\begin{array}{lll} \omega( 0 ) & = W _{0(1)} & ≈ 0 . 56714 \\ \omega( 1 ) & = 1 & \\ \omega( -1 \pm i \pi) & = -1 & \\ \omega( - (1)/(3) + ln \left ( (1)/(3) \right ) + i \pi ) & =- (1)/(3) & \\ \omega( - (1)/(3) + ln \left ( (1)/(3) \right ) - i \pi ) & = W _{-1} \left ( - (1)/(3) e ^{- (1)/(3)} \right ) & ≈ -2 . 237147028 \\ \end{array}

References

• "On the Wright ω function", Robert Corless and David Jeffrey

Notes and References

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