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GeezerVison is showing mathworld.wolfram.com — Omega Constant -- from Wolfram MathWorld

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Omega Constant

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The omega constant is defined as

W(1)=0.5671432904...
(1)

(OEIS A030178), where W(x) is the Lambert W-function. It is available in the Wolfram Language using the function ProductLog[1]. W(1) can be considered a sort of "golden ratio" for exponentials since

exp[-W(1)]=W(1),
(2)

giving

ln[1/(W(1))]=W(1).
(3)

The omega constant is also given by the power tower

W(1)=u^(u^(·^(·^·))),
(4)

where u=1/e.

An integral involving W(1) is given by

int_(-infty)^infty(dx)/((e^x-x)^2+pi^2)=1/(1+W(1))
(5)
=0.638103743....
(6)

This evaluation is attributed to V. Adamchik (OEIS A115287; Moll 2006, with a sign typo corrected).

See also

omega2 Constant, Golden Ratio, Lambert W-Function, Power Tower

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References

Michon, G. P. "Final Answers: Numerical Constants." https://www.numericana.com/answer/constants.htm#mertens.Moll, V. H. "Some Questions in the Evaluation of Definite Integrals." MAA Short Course, San Antonio, TX. Jan. 2006. https://www.experimentalmath.info/maa-course/Moll-MAA.pdf.Sloane, N. J. A. Sequences A030178 and A115287 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Omega Constant

Cite this as:

Weisstein, Eric W. "Omega Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OmegaConstant.html

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