Omega constant

This article is about the Ω constant from analysis. For the Ω constant from information theory, see Chaitin's 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 the OEIS).

Properties

The defining identity can be expressed, for example, as

\ln({\tfrac {1}{\Omega }})=\Omega .

-\ln(\Omega )=\Omega

e^{-\Omega }=\Omega .

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

\displaystyle \int \limits _{-\infty }^{\infty }{\frac {\,dt}{(e^{t}-t)^{2}+\pi ^{2}}}={\dfrac {1}{1+\Omega }}

\Omega ={\frac {1}{\displaystyle \int \limits _{-\infty }^{\infty }{\tfrac {\,dt}{(e^{t}-t)^{2}+\pi ^{2}}}}}-1.

Computation

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→∞. This convergence is because Ω is an attractive fixed point of the function e^−x.

It is much more efficient to use the iteration

\Omega _{n+1}={\frac {1+\Omega _{n}}{1+e^{\Omega _{n}}}},

because the function

f(x)={\frac {1+x}{1+e^{x}}}={\frac {1+x}{1+e^{-x}}}\cdot e^{-x}

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).

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 {pe^{\left({\frac {p}{q}}\right)}}{q}}

and

e=\left({\frac {q}{p}}\right)^{\left({\frac {q}{p}}\right)}={\sqrt[{p}]{\frac {q^{q}}{p^{q}}}}

The number 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, e^−Ω would be transcendental; but Ω=exp(-Ω), so these cannot both be true.

References

External links

Weisstein, Eric W. "Omega Constant". MathWorld.

This article is issued from Wikipedia - version of the 10/9/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.