Omega constant

The defining identity can be expressed, for example, as

:$\backslash ln\; \backslash left(\backslash tfrac\{1\}\{\backslash Omega\}\; \backslash right)=\backslash Omega.$

or

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

as well as

:$e^\{-\backslash Omega\}=\; \backslash Omega.$

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

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

This sequence will converge to {{math|Ω}} as {{mvar|n}} approaches infinity. This is because {{math|Ω}} is an attractive fixed point of the function {{math|e^−x}}.

It is much more efficient to use the iteration

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

because the function

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

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, {{math|Ω}} can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also {{section link|Lambert W function|Numerical evaluation}}).

:$\backslash Omega\_\{j+1\}=\backslash Omega\_j-\backslash frac\{\backslash Omega\_j\; e^\{\backslash Omega\_j\}-1\}\{e^\{\backslash Omega\_j\}(\backslash Omega\_j+1)-\backslash frac\{(\backslash Omega\_j+2)(\backslash Omega\_je^\{\backslash Omega\_j\}-1)\}\{2\backslash Omega\_j+2\}\}.$

An identity due to Victor Adamchik{{cn|date=February 2025}} is given by the relationship

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

Other relations due to Mező{{cite web|first=István|last=Mező|title=An integral representation for the principal branch of the Lambert W function|url=https://sites.google.com/site/istvanmezo81/other-things |access-date=24 April 2022}}{{cite arXiv

| last = Mező | first = István

| title = An integral representation for the Lambert W function

| date = 2020| class = math.CA

| eprint = 2012.02480

}}.

and Kalugin-Jeffrey-Corless{{cite arXiv

| first1=German A. | last1=Kalugin | first2=David J. | last2=Jeffrey | first3=Robert M. | last3=Corless

| title = Stieltjes, Poisson and other integral representations for functions of Lambert W

| date = 2011| class = math.CV

| eprint = 1103.5640

}}.

are:

:$\backslash Omega=\backslash frac\{1\}\{\backslash pi\}\backslash operatorname\{Re\}\backslash int\_0^\backslash pi\backslash log\backslash left(\backslash frac\{e^\{e^\{it\}\}-e^\{-it\}\}\{e^\{e^\{it\}\}-e^\{it\}\}\backslash right)\; dt,$

:$\backslash Omega=\backslash frac\{1\}\{\backslash pi\}\backslash int\_0^\backslash pi\backslash log\backslash left(1+\backslash frac\{\backslash sin\; t\}\{t\}e^\{t\backslash cot\; t\}\backslash right)dt.$

The latter two identities can be extended to other values of the {{mvar|W}} function (see also {{section link|Lambert W function|Representations}}).

The constant {{math|Ω}} is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that {{math|Ω}} is algebraic. By the theorem, {{math|e^−Ω}} is transcendental, but {{math|1=Ω = e^−Ω}}, which is a contradiction. Therefore, it must be transcendental.{{cite journal |last1=Mező |first1=István |last2=Baricz |first2=Árpád |title=On the Generalization of the Lambert W Function |journal=Transactions of the American Mathematical Society |date=November 2017 |volume=369 |issue=11 |page=7928 |doi=10.1090/tran/6911 |url=https://www.ams.org/journals/tran/2017-369-11/S0002-9947-2017-06911-7/S0002-9947-2017-06911-7.pdf |access-date=28 April 2023}}

{{Reflist}}

• {{MathWorld|urlname=OmegaConstant|title=Omega Constant}}
• {{citation|title=Omega constant (1,000,000 digits)|url=http://ankokudan.org/d/d.htm?mathlistindex-e.html|work=Darkside communication group (in Japan) |access-date=2017-12-25}}

{{Irrational number}}

Omega

Category:Articles containing proofs

Category:Real transcendental numbers