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 = 0.56714 32904 09783 87299 99686 62210 ... .
1/Ω = 1.76322 28343 51896 71022 52017 76951 ... .

Properties

Fixed point representation

The defining identity can be expressed, for example, as

\ln \left(\tfrac{1}{\Omega} \right)=\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_je^{\Omega_j}-1)}{2\Omega_j+2}}.

Integral representations

An identity due to Victor Adamchik[citation needed] 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[1][2] and Kalugin-Jeffrey-Corless[3] 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.[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.
  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.
  4. ^ Mező, István & Baricz, Árpád (November 2017). "On the Generalization of the Lambert W Function". Transactions of the American Mathematical Society. 369 (11): 7928. doi:10.1090/tran/6911. Retrieved 28 April 2023.