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 OEIS).

Properties

The defining identity can be expressed, for example, as

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

or

-\ln(\Omega)=\Omega

or

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}

or

\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 ex.

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}

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{p e^{\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.

See also

References

    External links

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