In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as:
ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( e z ) . {\displaystyle \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 ( ω ) = ln ( ω ) + ω {\displaystyle 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 ≠ x ± i π {\displaystyle 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 ) = ω ( ln ( z ) + 2 π i k ) {\displaystyle W_{k}(z)=\omega (\ln(z)+2\pi ik)} . It also satisfies the differential equation
d ω d z = ω 1 + ω {\displaystyle {\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 ( ω ) + ω = z {\displaystyle \ln(\omega )+\omega =z} , and as a consequence its integral can be expressed as:
∫ ω n d z = { ω n + 1 − 1 n + 1 + ω n n if n ≠ − 1 , ln ( ω ) − 1 ω if n = − 1. {\displaystyle \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 = ω a + ln ( ω a ) {\displaystyle a=\omega _{a}+\ln(\omega _{a})} takes the form :
ω ( z ) = ∑ n = 0 + ∞ q n ( ω a ) ( 1 + ω a ) 2 n − 1 ( z − a ) n n ! {\displaystyle \omega (z)=\sum _{n=0}^{+\infty }{\frac {q_{n}(\omega _{a})}{(1+\omega _{a})^{2n-1}}}{\frac {(z-a)^{n}}{n!}}}
where
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