# Omega constant

The omega constant is a mathematical constant defined as the unique real number that satisfies the equation

${\displaystyle \Omega e^{\Omega }=1.}$

It is the value of W(1), where W is Lambert's W function. The name is derived[citation needed] from the alternate name for Lambert's W function, the omega function. The numerical value of Ω is given by

Ω = 0.567143290409783872999968662210... (sequence A030178 in the OEIS).
1/Ω = 1.763222834351896710225201776951... (sequence A030797 in the OEIS).

## Properties

### Fixed point representation

The defining identity can be expressed, for example, as

${\displaystyle \ln({\tfrac {1}{\Omega }})=\Omega .}$

or

${\displaystyle -\ln(\Omega )=\Omega }$

or

${\displaystyle e^{-\Omega }=\Omega .}$

### Computation

One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence

${\displaystyle \Omega _{n+1}=e^{-\Omega _{n}}.}$

This sequence will converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function ex.

It is much more efficient to use the iteration

${\displaystyle \Omega _{n+1}={\frac {1+\Omega _{n}}{1+e^{\Omega _{n}}}},}$

because the function

${\displaystyle f(x)={\frac {1+x}{1+e^{x}}},}$

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, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Lambert W function § Numerical evaluation).

${\displaystyle \Omega _{j+1}=\Omega _{j}-{\frac {\Omega _{j}e^{\Omega _{j}}-1}{e^{\Omega _{j}}(\Omega _{j}+1)-{\frac {(\Omega _{j}+2)(\Omega _{j}e^{\Omega _{j}}-1)}{2\Omega _{j}+2}}}}.}$

### Integral representations

An identity due to Victor Adamchik[citation needed] is given by the relationship

${\displaystyle \int _{-\infty }^{\infty }{\frac {dt}{(e^{t}-t)^{2}+\pi ^{2}}}={\frac {1}{1+\Omega }}.}$

Another relation due to Mező is[1]

${\displaystyle \Omega ={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\log \left({\frac {e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}}\right)dt.}$

The latter identity can be extended to other values of the W function (see also Lambert W function § Representations).

## Transcendence

The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, e−Ω is transcendental, but Ω = e−Ω, which is a contradiction. Therefore, it must be transcendental.