Wright Omega function

In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as: 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 ± i π {displaystyle z eq xpm ipi } for x ≤ −1, of the equation y + ln(y) = z. Except on those two rays, the Wright omega function is continuous, even analytic. The Wright omega function satisfies the relation W k ( z ) = ω ( ln ⁡ ( z ) + 2 π i k ) {displaystyle W_{k}(z)=omega (ln(z)+2pi ik)} . It also satisfies the differential equation wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation ln ⁡ ( ω ) + ω = z {displaystyle ln(omega )+omega =z} ), and as a consequence its integral can be expressed as: Its Taylor series around the point a = ω a + ln ⁡ ( ω a ) {displaystyle a=omega _{a}+ln(omega _{a})} takes the form :