Omega constant

The Omega constant is a mathematical constant defined by

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

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 0.56714329040978387299996866221035554975381578718651250813513107911868218606033163864180370634902545380773757807516825805267767102176872019047786560057916365319936454603076753022511993394673457440338486 (sequence A030178 in OEIS). It has properties that

$e^{-\Omega}=\Omega,\,$

or equivalently,

$\ln \Omega = - \Omega.\,$

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

$\Omega_{n+1}=e^{-\Omega_n}.\,$

This sequence will converge towards Ω as n→∞. This convergence is due to the fact that $\Omega$ is an attractive fixed points of the function $e^{-x}$.

Anyway, is much more efficient to use the iteration

$\Omega_{n+1} = \frac{1+\Omega_n}{1+e^{\Omega_n}},$

because the function

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

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 is given by the relationship

$\Omega=\frac{1}{\displaystyle \int_{-\infty}^{\infty}\frac{{\rm{d}}x}{(e^x-x)^2+\pi^2}}-1 .\,$

Irrationality and transcendence

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

$\frac{p}{q} = \Omega$

so that

$1 = \frac{p e^{\left( \frac{p}{q} \right)}}{q}$

$e = \left( \frac{q}{p} \right)^{\left( \frac{q}{p} \right)} = \sqrt[p]{\frac{q^q}{p^q}}$

and 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, exp(Ω) would be transcendental and so would be exp⁻¹(Ω). But this contradicts the assumption that it was algebraic.

See also

• Lambert W function

External links

• Weisstein, Eric W., "Omega Constant" from MathWorld.