Omega constant

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

Ω e Ω = 1. {\displaystyle \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 ⁡ ( 1 Ω ) = Ω . {\displaystyle \ln({\tfrac {1}{\Omega }})=\Omega .}

or

− ln ⁡ ( Ω ) = Ω {\displaystyle -\ln(\Omega )=\Omega }

as well as

e − Ω = Ω . {\displaystyle e^{-\Omega }=\Omega .}

Computation

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

Ω n + 1 = e − Ω n . {\displaystyle \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

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

because the function

f ( x ) = 1 + x 1 + e x , {\displaystyle 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).

Ω j + 1 = Ω j − Ω j e Ω j − 1 e Ω j ( Ω j + 1 ) − ( Ω j + 2 ) ( Ω j e Ω j − 1 ) 2 Ω j + 2 . {\displaystyle \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 [citation needed]Victor Adamchik[citation needed] is given by the relationship

∫ − ∞ ∞ d t ( e t − t ) 2 + π 2 = 1 1 + Ω . {\displaystyle \int _{-\infty }^{\infty }{\frac {dt}{(e^{t}-t)^{2}+\pi ^{2}}}={\frac {1}{1+\Omega }}.}

Other relations due to Mező[1][2] and Kalugin-Jeffrey-Corless[3] are:

Ω = 1 π Re ⁡ ∫ 0 π log ⁡ ( e e i t − e − i t e e i t − e i t ) d t , {\displaystyle \Omega ={\frac {1}{\pi }}\operatorname {Re} \int _{0}^{\pi }\log \left({\frac {e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}}\right)dt,}
Ω = 1 π ∫ 0 π log ⁡ ( 1 + sin ⁡ t t e t cot ⁡ t ) d t . {\displaystyle \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.[4]

References

  1. ^ Mező, István. "An integral representation for the principal branch of the Lambert W function". Retrieved 24 April 2022.
  2. ^ Mező, István (2020). "An integral representation for the Lambert W function". arXiv:2012.02480 [math.CA]..
  3. ^ Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2011). "Stieltjes, Poisson and other integral representations for functions of Lambert W". arXiv:1103.5640 [math.CV]..
  4. ^ Mező, István; Baricz, Árpád (November 2017). "On the Generalization of the Lambert W Function" (PDF). Transactions of the American Mathematical Society. 369 (11): 7928. Retrieved 28 April 2023.

External links

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