Talk:Omega constant

This article is rated Start-class on Wikipedia's content assessment scale. It is of interest to the following WikiProjects:

Mathematics Low‑priority This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics Low This article has been rated as Low-priority on the project's priority scale.

letter The fact that it is called the omega function makes me think someone mistook the letter ω, a vowel, for the letter w, because of visual similarity. Is that what happened? Michael Hardy 20:42, 12 Feb 2004 (UTC)

"It has properties that are akin to those of the golden ratio, in that

${\displaystyle e^{-\Omega }=\Omega ,\,}$

or equivalently,

${\displaystyle \ln(1/\Omega )=\Omega .}$"

I'm not sure I can see any similarity to the golden ration here. Should this be obvious? :/ Woscafrench (talk) 01:00, 26 November 2007 (UTC)Reply

One way to express the similarity in exact terms is as follows: if we replace f(x)=e^x with its linear approximation f(x)=x-1 then the solution to the equation f(x)=1/x would be golden ratio.  Grue  15:36, 26 November 2007 (UTC)Reply

I also doubt that sentence. No one can explain this better; it is just a point of view. I already removed it. --Octra Bond (talk) 02:29, 3 November 2009 (UTC)Reply

I used Windows calculator to evaluate the Omega constant. The value of Omega is:

0.56714329040978387299

99686622103555497485

40291138701110223886

36460700559427176005

70163043896301071636

09769866542711127501

81294450722431215290

864337423...

Garygoh884 (talk) 06:27, 15 July 2011 (UTC)Reply

Reference for part "due to Victor Adamchik"? — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 22:52, 13 December 2012 (UTC)Reply

Let k = 1/Ω, that is, the reciprocal of the Omega constant. Then it holds that:

${\displaystyle {\frac {d}{dx}}k^{x}=k^{x-1}}$

That is, it imitates the power rule of derivatives (even though it's not a polynomial, but an exponential function with base k) in reducing the exponent by 1. The reciprocal Omega constant is the unique exponential base for which this is true.

Proof:

${\displaystyle {\frac {d}{dx}}k^{x}={\frac {d}{dx}}e^{x\ln k}=(\ln k)(e^{x\ln k})=(\ln k)k^{x}}$

But since ${\displaystyle e^{-\Omega }=\Omega }$, it follows that ${\displaystyle e^{\Omega }=1/\Omega =k}$, and therefore ${\displaystyle \ln k=\ln e^{\Omega }=\Omega =1/k}$. Substituting into the above equation, we have:

${\displaystyle {\frac {d}{dx}}k^{x}=(\ln k)k^{x}=(1/k)k^{x}=k^{x-1}}$

as claimed.—69.172.154.75 (talk) 04:53, 27 September 2015 (UTC)Reply

I was asking myself if Ω is the the absolute value of the solution to

          x+ex = 0

I've found it with the approximative form thanks to the comparison between y=x and y=-e^x. If I'm not wrong, we can add it to the principal page. — Preceding unsigned comment added by 87.0.114.210 (talk) 14:35, 15 January 2017 (UTC)Reply

It’s possible to prove this algebraically as well.

${\displaystyle x+e^{x}=0}$

${\displaystyle x=-e^{x}}$

${\displaystyle 1=-{\frac {e^{x}}{x}}}$

${\displaystyle {\begin{aligned}1=-{\frac {e^{-u}}{-u}}&&(u=-x)\end{aligned}}}$

${\displaystyle 1={\frac {1}{ue^{u}}}}$

${\displaystyle 1=ue^{u}}$

${\displaystyle W_{0}(1)=u}$

${\displaystyle {\begin{aligned}\Omega =u&&(\Omega =W_{0}(1))\end{aligned}}}$

${\displaystyle -\Omega =x}$ 107.9.36.50 (talk) 21:11, 15 July 2025 (UTC)Reply

(1/omega)^(1/omega)=e — Preceding unsigned comment added by 191.111.6.32 (talk) 22:07, 31 May 2022 (UTC)Reply

Yes, but not surprising (ie pretty simple to deduce) and nor more snappy. Unless this is especially useful for something (best shown with a souce), I don't think it needs to be mentioned in the article.

I do think the continued fraction of omega would be more interesting. Xario (talk) 12:34, 2 December 2023 (UTC)Reply

OEIS Sequences A370490 and A370491 are the denominators and the numerators of an infinite series that converges to the Omega constant. Dacicus Geometricus (talk) 17:10, 6 April 2024 (UTC)Reply

I want to mention that I am the author of the 2 OEIS sequences. Dacicus Geometricus (talk) 08:17, 7 August 2024 (UTC)Reply