It is the value of W(1), where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the omega function. The numerical value of Ω is given by
Ω = 0.567143290409783872999968662210... (sequence A030178 in the OEIS).
1/Ω = 1.763222834351896710225201776951... (sequence A030797 in the OEIS).
Propertiesedit
Fixed point representationedit
The defining identity can be expressed, for example, as
or
as well as
Computationedit
One can calculate Ωiteratively, by starting with an initial guess Ω0, and considering the sequence
This sequence will converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function e−x.
It is much more efficient to use the iteration
because the function
in addition to having the same fixed point, also has a derivative that vanishes there. This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration.
The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, e−Ω is transcendental, but Ω = e−Ω, which is a contradiction. Therefore, it must be transcendental.[4]
Referencesedit
^Mező, István. "An integral representation for the principal branch of the Lambert W function". Retrieved 24 April 2022.
^Mező, István (2020). "An integral representation for the Lambert W function". arXiv:2012.02480 [math.CA]..
^Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2011). "Stieltjes, Poisson and other integral representations for functions of Lambert W". arXiv:1103.5640 [math.CV]..
^Mező, István; Baricz, Árpád (November 2017). "On the Generalization of the Lambert W Function" (PDF). Transactions of the American Mathematical Society. 369 (11): 7928. Retrieved 28 April 2023.