TOPICS
Omega Constant
The omega constant is defined as
 | (1) |
(OEISA030178), where
is theLambert W-function. It is available in theWolfram Language using the functionProductLog[1].
can be considered a sort of "golden ratio" for exponentials since
![exp[-W(1)]=W(1),](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fOmegaConstant%2fNumberedEquation2.svg&f=jpg&w=240) | (2) |
giving
![ln[1/(W(1))]=W(1).](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fOmegaConstant%2fNumberedEquation3.svg&f=jpg&w=240) | (3) |
The omega constant is also given by thepower tower
 | (4) |
where
.
A beautiful integral involving
given by
 |  |  | (5) |
 |  |  | (6) |
is due to V. Adamchik (OEISA115287; Moll2006; typo corrected).
See also
Golden Ratio,Lambert W-Function,omega2 Constant,Power TowerExplore with Wolfram|Alpha
More things to try:
References
Michon, G. P. "Final Answers: Numerical Constants."http://home.att.net/~numericana/answer/constants.htm#mertens.Moll, V. H. "Some Questions in the Evaluation of Definite Integrals." MAA Short Course, San Antonio, TX. Jan. 2006.http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf.Sloane, N. J. A. SequencesA030178 andA115287 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Omega ConstantCite this as:
Weisstein, Eric W. "Omega Constant." FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/OmegaConstant.html