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 0.5671432904097838729999686622 (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%2B1}=e^{-\Omega_n}.\,

This sequence will converge towards Ω as n→∞.

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%2B\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

External links