Omega constant
From Free net encyclopedia
The Omega constant is a mathematical constant defined by
- <math>\Omega\,\exp(\Omega)=1</math>.
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
- <math> e^{-\Omega}=\Omega,</math>
or equivalently,
- <math> \ln (1/\Omega) = \Omega.</math>
One can calculate Ω iteratively, by starting with an initial guess Ω0, and considering the sequence
- <math> \Omega_{n+1}=e^{-\Omega_n}.</math>
This sequence will converge towards Ω as n→∞.
Irrationality
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
- <math> \frac{p}{q} = \Omega </math>
so that
- <math> 1 = \frac{p e^{\frac{p}{q}}}{q} </math>
- <math> e = \sqrt[p]{\frac{q^q}{p^q}} </math>
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.