The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
\Omegae\Omega=1.
It is the value of, where is Lambert's function. The name is derived from the alternate name for Lambert's function, the omega function. The numerical value of is given by
.
.
The defining identity can be expressed, for example, as
ln(\tfrac{1}{\Omega})=\Omega.
-ln(\Omega)=\Omega
e-\Omega=\Omega.
One can calculate iteratively, by starting with an initial guess, and considering the sequence
\Omegan+1
| -\Omegan | |
| =e |
.
This sequence will converge to as approaches infinity. This is because is an attractive fixed point of the function .
It is much more efficient to use the iteration
\Omegan+1=
| 1+\Omegan | ||||
|
,
because the function
| f(x)= | 1+x |
| 1+ex |
,
in addition to having the same fixed point, also has a derivative that vanishes there. This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration.
Using Halley's method, can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also).
\Omegaj+1
| =\Omega | |||||||||||||||||||||||||||||||||||||||
|
.
An identity due to Victor Adamchik is given by the relationship
| ||||
| \int | ||||
| -infty |
=
| 1 | |
| 1+\Omega |
.
Other relations due to Mező[1] [2] and Kalugin-Jeffrey-Corless[3] are:
| \Omega= | 1 |
| \pi |
| ||||||||||||||||
| \operatorname{Re}\int | ||||||||||||||||
| 0 |
\right)dt,
| \Omega= | 1 |
| \pi |
| ||||
| \int | ||||
| 0 |
et\cot\right)dt.
The constant is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that is algebraic. By the theorem, is transcendental, but, which is a contradiction. Therefore, it must be transcendental.[4]