Constant omega

In Mathematical, the constant omega , noted Ω , is a constant definite as being a particular value of the Fonction W of Lambert.

It should not be confused with the Oméga of Chaitin, constant mathematics in algorithmic theory of information.

Definition

By definition,

\ Omega

is the value of the Fonction W of Lambert into 1:

\ Omega=W (1)

The name of the constant comes from the other name of this function: the function omega .

Because of definition of the function W:

\ Omega \, e^ {\ Omega} =1

where

e

is the bases natural logarithms.

Properties

Approximate value

The approximate value of

\ Omega

is:

Ω = 0,5671432904… ().

Other definitions

\ Omega

can be perceived like a kind of Golden section applied to the exponential one, since:

\ Omega = \ frac {1} {e^ {\ Omega}}

\ Omega = \ ln \ left (\ frac {1} {\ Omega} \ right)

One can calculate $\ Omega$ in an iterative way , by beginning with an initial value $\ Omega_0$ and by calculating the terms of the continuation

\ Omega_ {n+1} =e^ {- \ Omega_n}

This continuation converges towards $\ Omega$ .

Irrationality and transcendence

\ Omega

is a irrational Nombre. This rises owing to the fact that E is transcendent. Indeed, if

\ Omega

were rational, then there would exist entireties

p

and

q

such as:

\ Omega = \ frac {p} {Q}

and thus:

1 = \ frac {p e^ {\ frac {p} {Q}}} {Q}

that is to say:

e = \ sqrt {\ frac {q^q} {p^q}}

and E would be thus algebraic of degree p . However E being transcendent, $\ Omega$ is irrational.

The fact that $\ Omega$ is a transcendent Nombre is a direct consequence of the Théorème of Lindemann-Weierstrass. If $\ Omega$ were algebraic, $e^ {\ Omega}$ would be transcendent and $\ frac {1} {e^ {\ Omega}}$ also. But that contradicted the assumption according to which it would be algebraic.

See too

Internal bonds

Function W of Lambert

External bonds

'' Constant Omega '' (Eric W. Weisstein, MathWorld )
'' The Omega constant '' (Gerard P. Michon, Numerical Constant )
Value of the first decimals of Ω ( Plouffe' S Inverter )

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