Nick Bostrom

2013-11-24T21:29:19Z

78.146.112.194: /* Simulation argument */ deleted two word to make the article make more sense

The '''omega constant''' is a [[mathematical constant]] defined by

:<math>\Omega\,e^{\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... {{OEIS|id=A030178}}. It has properties that

:<math> e^{-\Omega}=\Omega,\,</math>

or equivalently,

:<math> \ln \Omega = - \Omega.\,</math>

One can calculate Ω [[iterative method|iteratively]], by starting with an initial guess Ω0, and considering the [[sequence]]

:<math> \Omega_{n+1}=e^{-\Omega_n}.\,</math>

This sequence will [[limit of a sequence|converge]] towards Ω as ''n''→∞. This convergence is due to the fact that Ω is an [[Fixed point (mathematics)|attractive fixed point]] of the function ''e''−''x''.

It is much more efficient to use the iteration

:<math>\Omega_{n+1} = \frac{1+\Omega_n}{1+e^{\Omega_n}},</math>

because the function

:<math> f(x) = \frac{1+x}{1+e^x},</math>

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).

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

:<math> \Omega=\frac{1}{\displaystyle \int_{-\infty}^{+\infty}\frac{\,dt}{(e^t-t)^2+\pi^2}}-1 .</math>

==Irrationality and transcendence==

Ω can be proven [[irrational number|irrational]] from the fact that [[e (mathematical constant)|e]] is [[transcendental number|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^{\left( \frac{p}{q} \right)}}{q}</math>



: <math> e = \left( \frac{q}{p} \right)^{\left( \frac{q}{p} \right)} = \sqrt[p]{\frac{q^q}{p^q}} </math>

and ''e'' would therefore be [[Algebraic number|algebraic]] of degree ''p''. However ''e'' is transcendental, so Ω must be irrational.

Ω is in fact [[transcendental number|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 W function]]

==External links==
* {{MathWorld|urlname=OmegaConstant|title=Omega Constant}}

[[Category:Transcendental numbers]]
[[Category:Mathematical constants|Omega]]
[[Category:Articles containing proofs]]

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Nick Bostrom