Omega constant

The omega constant is a mathematical constant defined as the unique real number that satisfies the equation

$\mathrm{\Omega }{e}^{\mathrm{\Omega }}=1.$

It is the value of W(1), where Template:Mvar is [[Lambert W function|Lambert's Template:Mvar function]]. The name is derived from the alternate name for Lambert's Template:Mvar function, the omega function. The numerical value of Ω is given by

Ω = Template:Gaps (sequence A030178 in the OEIS).

1/Ω = Template:Gaps (sequence A030797 in the OEIS).

Properties

Fixed point representation

The defining identity can be expressed, for example, as

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

or

$-\ln (\mathrm{\Omega })=\mathrm{\Omega }$

as well as

$e^{-\mathrm{\Omega }}=\mathrm{\Omega }.$

Computation

One can calculate Ω iteratively, by starting with an initial guess Ω₀, and considering the sequence

${\mathrm{\Omega }}_{n+1}=e^{-{\mathrm{\Omega }}_n}.$

This sequence will converge to Ω as Template:Mvar approaches infinity. This is because Ω is an attractive fixed point of the function e^−x.

It is much more efficient to use the iteration

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

because the function

$f(x)=\frac{1+x}{1+e^x},$

in addition to having the same fixed point, also has a derivative that vanishes there. This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration.

Using Halley's method, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Template:Section link).

${\mathrm{\Omega }}_{j+1}={\mathrm{\Omega }}_j-\frac{{\mathrm{\Omega }}_j{e}^{{\mathrm{\Omega }}_j}-1}{e^{{\mathrm{\Omega }}_j}({\mathrm{\Omega }}_j+1)-\frac{({\mathrm{\Omega }}_j+2)({\mathrm{\Omega }}_j{e}^{{\mathrm{\Omega }}_j}-1)}{2{\mathrm{\Omega }}_j+2}}.$

Integral representations

An identity due to Victor Adamchik is given by the relationship

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{dt}{(e^t-t{)}^2+{\pi }^2}=\frac{1}{1+\mathrm{\Omega }}.$

Other relations due to Mező and Kalugin-Jeffrey-Corless are:

$\mathrm{\Omega }=\frac{1}{\pi }\mathrm{Re}\underset{0}{\overset{\pi }{\int }}\log \left(\frac{e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}\right)dt,$

$\mathrm{\Omega }=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\log (1+\frac{\sin t}{t}{e}^{t\cot t})dt.$

The latter two identities can be extended to other values of the Template:Mvar function (see also Template:Section link).

Transcendence

The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, e^−Ω is transcendental, but Ω = e^−Ω, which is a contradiction. Therefore, it must be transcendental.