Omega constant

This article is about the Ω constant from analysis. For the Ω constant from information theory, see Chaitin's constant.

The omega constant, also known as the Adamchik 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

$0.5671432904097838729999686622...$ (sequence A030178 in OEIS).

Properties

The defining identity can be expressed, for example, as

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

or

$-\ln(\Omega)=\Omega$

or

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

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

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

or

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

Computation

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 because Ω is an attractive fixed point of the function e^−x.

It 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).

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

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

and

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

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.

See also

• Lambert W function

External links

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