Omega Constant

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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

(2)

giving

(3)

The omega constant is also given by the power tower

(4)

where .

A beautiful integral involving given by

			(5)
			(6)

is due to V. Adamchik (OEIS A115287; Moll 2006; typo corrected).

See also

Golden Ratio, Lambert W-Function, omega2 Constant, 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."

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Cite this as:

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

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