In mathematics, the Wright omega function or Wright function, denoted ÃÂ, is defined in terms of the Lambert W function as:
It is simpler to be defined by its inverse function
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<sup>−ÃÂ(π i)</sup>.
y = ÃÂ(z) is the unique solution, when for x ≤ −1, of the equation y + ln(y) = z. Except for those two values, the Wright omega function is continuous, even analytic.
The Wright omega function satisfies the relation .
It also satisfies the differential equation
wherever ÃÂ is analytic (as can be seen by performing separation of variables and recovering the equation , and as a consequence its integral can be expressed as:
Its Taylor series around the point takes the form :
where
in which
is a second-order Eulerian number.