Wright Omega function

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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:

\omega(z) = W_{\big \lceil \frac{\mathrm{Im}(z) - \pi}{2 \pi} \big \rceil}(e^z).

Uses[edit]

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 \neq 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[edit]

The Wright omega function satisfies the relation W_k(z) = \omega(\ln(z) + 2 \pi i k).

It also satisfies the differential equation

 \frac{d\omega}{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 = 
\begin{cases} 
  \frac{\omega^{n+1} -1 }{n+1} + \frac{\omega^n}{n}  & \mbox{if } n \neq -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-1}}\frac{(z-a)^n}{n!}

where

q_n(w) = \sum_{k=0}^{n-1} \bigg \langle \! \! \bigg \langle 
\begin{matrix}
  n+1 \\
  k
\end{matrix} 
\bigg \rangle \! \! \bigg \rangle (-1)^k w^{k+1}

in which

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

is a second-order Eulerian number.

Values[edit]


\begin{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} \right ) + i \pi ) &= -\frac{1}{3} & \\
\omega(-\frac{1}{3} + \ln \left ( \frac{1}{3} \right ) - i \pi ) &= W_{-1} \left ( -\frac{1}{3} e^{-\frac{1}{3}} \right ) &\approx -2.237147028 \\
\end{array}

Plots[edit]

Notes[edit]

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

References[edit]