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おめが is given by

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

1/Ωおめが = 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おめが iteratively, by starting with an initial guess Ωおめが₀, and considering the sequence

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

This sequence will converge to Ωおめが as n approaches infinity. This is because Ωおめが 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おめが 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おめが is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ωおめが is algebraic. By the theorem, e^{−Ωおめが} is transcendental, but Ωおめが = e^{−Ωおめが}, 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