Omega constant

The Omega constant is a mathematical constant defined by

$\Omega \,\exp(\Omega )=1.\,$

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. It has properties that are akin to those of the golden ratio, in that

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

or equivalently,

$\log(1/\Omega )=\Omega .$

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→∞.

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

$1={\frac {pe^{p/q}}{q}}$

$e={\sqrt[{p}]{\frac {q^{q}}{p^{q}}}}$

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's W function

External links

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