Omega constant

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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 (sequence A030178 in OEIS). It has properties that are akin to those of the golden ratio, in that

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

or equivalently,

log(1 / Ω) = Ω.

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→∞.

[edit] 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

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

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

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.