Wright omega function

In mathematics, the Wright omega function or Wright function, 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). It is simpler to be defined by its inverse function : z (\omega) = \ln(\omega)+\omega

In mathematics, the Wright omega function or Wright function, 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). It is simpler to be defined by its inverse function z (\omega) = \ln(\omega)+\omega ==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 \neq 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 \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 \omega^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== \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== Image:Wright omega function - real.png| \Re\{\omega(z)\} Image:Wright omega - imaginary.png| \Im\{\omega(z)\} Image:Wright omega - magnitude.png| |\omega(z)| ==Notes== ==References==

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