In mathematics , the Wright omega function or Wright function ,[ note 1] ω , is defined in terms of the Lambert W function as:
The Wright omega function along part of the real axis
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 }
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 }
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
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}
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})}
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
q
n
(
w
)
=
∑
k
=
0
n
−
1
⟨
⟨
n
+
1
k
⟩
⟩
(
−
1
)
k
w
k
+
1
{\displaystyle 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
⟨
⟨
n
k
⟩
⟩
{\displaystyle {\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }}
is a second-order Eulerian number .
ω
(
0
)
=
W
0
(
1
)
≈
0.56714
ω
(
1
)
=
1
ω
(
−
1
±
i
π
)
=
−
1
ω
(
−
1
3
+
ln
(
1
3
)
+
i
π
)
=
−
1
3
ω
(
−
1
3
+
ln
(
1
3
)
−
i
π
)
=
W
−
1
(
−
1
3
e
−
1
3
)
≈
−
2.237147028
{\displaystyle {\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 of the Wright omega function on the complex plane
z
=
ℜ
{
ω
(
x
+
i
y
)
}
{\displaystyle z=\Re \{\omega (x+iy)\}}
z
=
ℑ
{
ω
(
x
+
i
y
)
}
{\displaystyle z=\Im \{\omega (x+iy)\}}
z
=
|
ω
(
x
+
i
y
)
|
{\displaystyle z=|\omega (x+iy)|}
