The omega constant is defined as
 |
(1)
|
(OEIS A030178), where 
is the Lambert W-function.
It is available in the Wolfram Language
using the function ProductLog[1].
can be considered a sort of "golden ratio" for exponentials since
![exp[-W(1)]=W(1),](/images/equations/OmegaConstant/NumberedEquation2.svg) |
(2)
|
giving
![ln[1/(W(1))]=W(1).](/images/equations/OmegaConstant/NumberedEquation3.svg) |
(3)
|
The omega constant is also given by the power tower
 |
(4)
|
where 
.
A beautiful integral involving 
given by
is due to V. Adamchik (OEIS A115287; Moll
2006; typo corrected).
See also
Golden Ratio,
Lambert W-Function,
omega2 Constant,
Power
Tower
Explore with Wolfram|Alpha
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. 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 Web Resource. https://mathworld.wolfram.com/OmegaConstant.html
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