OMEGA CONSTANT

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 . It has properties that are akin to those of the golden ratio, in that
:$e^\{-Omega\}=Omega,,$
or equivalently,
:$log\; (1/Omega)\; =\; Omega.$
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''→∞.

Contents

Irrationality and transcendence

See also

External links

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
:
so that
:
:
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

★ Lambert's W function

External links

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