The **omega constant** is a mathematical constant defined by

Ω
e
Ω
=
1.
It is the value of *W*(1) where *W* is Lambert's W function. The name is derived from the alternate name for Lambert's *W* function, the *omega function*. The value of Ω is approximately

0.5671432904097838729999686622...
(sequence

A030178 in the OEIS).

The defining identity can be expressed, for example, as

ln
(
1
Ω
)
=
Ω
.
or

−
ln
(
Ω
)
=
Ω
or

e
−
Ω
=
Ω
.
In Mathematica

`Plot[{x, Exp[-x], -Log[x]}, {x, 0, 1}]`

One can calculate Ω iteratively, by starting with an initial guess Ω_{0}, and considering the sequence

Ω
n
+
1
=
e
−
Ω
n
.
`init = 0.5;`

`FixedPoint[Exp[-#] &, init]`

This sequence will converge towards Ω as *n*→∞. This convergence is because Ω is an attractive fixed point of the function *e*^{−x}.

It is much more efficient to use the iteration

Ω
n
+
1
=
1
+
Ω
n
1
+
e
Ω
n
,
because the function

f
(
x
)
=
1
+
x
1
+
e
x
=
1
+
x
1
+
e
−
x
⋅
e
−
x

has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration).

A beautiful identity due to Victor Adamchik[1] is given by the relationship

∫
−
∞
∞
d
t
(
e
t
−
t
)
2
+
π
2
=
1
1
+
Ω
or

Ω
=
1
∫
−
∞
∞
d
t
(
e
t
−
t
)
2
+
π
2
−
1.
Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers *p* and *q* such that

p
q
=
Ω
so that

1
=
p
e
(
p
q
)
q
and

e
=
(
q
p
)
(
q
p
)
=
q
q
p
q
p
The number *e* would therefore be algebraic of degree *p*. However *e* is transcendental, so Ω must be irrational.

Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, e^{−Ω} would be transcendental; but Ω=exp(-Ω), so these cannot both be true.