Omega constant

This article is about a specific value of Lambert's W function. For other omega constants, see omega (disambiguation) § Mathematics.

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

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

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 numerical value of Ω is given by

Ω = 0.567143290409783872999968662210... (sequence A030178 in the OEIS).

1/Ω = 1.763222834351896710225201776951... (sequence A030797 in the OEIS).

Properties

Fixed point representation

The defining identity can be expressed, for example, as

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

or

$-\ln(\Omega )=\Omega$

as well as

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

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 to Ω as n 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

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

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 Lambert W function § Numerical evaluation).

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

Integral representations

An identity due to Victor Adamchik is given by the relationship

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

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

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

$\Omega ={\frac {1}{\pi }}\int _{0}^{\pi }\log \left(1+{\frac {\sin t}{t}}e^{t\cot t}\right)dt.$

The latter two identities can be extended to other values of the W function (see also Lambert W function § Representations).

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.

External links

• Weisstein, Eric W. "Omega Constant". MathWorld.
• "Omega constant (1,000,000 digits)", Darkside communication group (in Japan), retrieved 2017-12-25