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).
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.
omega, constant, this, article, about, specific, value, lambert, function, other, omega, constants, omega, disambiguation, mathematics, omega, constant, mathematical, constant, defined, unique, real, number, that, satisfies, equation, ΩeΩ, displaystyle, omega,. This article is about a specific value of Lambert s W function For other omega constants see omega disambiguation Mathematics The omega constant is a mathematical constant defined as the unique real number that satisfies the equation WeW 1 displaystyle Omega e Omega 1 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 W is given by W 0 5671432904 09783 87299 99686 62210 sequence A030178 in the OEIS 1 W 1 7632228343 51896 71022 52017 76951 sequence A030797 in the OEIS Contents 1 Properties 1 1 Fixed point representation 1 2 Computation 1 3 Integral representations 1 4 Transcendence 2 References 3 External linksProperties editFixed point representation edit The defining identity can be expressed for example as ln 1W W displaystyle ln tfrac 1 Omega Omega nbsp or ln W W displaystyle ln Omega Omega nbsp as well as e W W displaystyle e Omega Omega nbsp Computation edit One can calculate W iteratively by starting with an initial guess W0 and considering the sequence Wn 1 e Wn displaystyle Omega n 1 e Omega n nbsp This sequence will converge to W as n approaches infinity This is because W is an attractive fixed point of the function e x It is much more efficient to use the iteration Wn 1 1 Wn1 eWn displaystyle Omega n 1 frac 1 Omega n 1 e Omega n nbsp because the function f x 1 x1 ex displaystyle f x frac 1 x 1 e x nbsp 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 Using Halley s method W can be approximated with cubic convergence the number of correct digits is roughly tripled with each iteration see also Lambert W function Numerical evaluation Wj 1 Wj WjeWj 1eWj Wj 1 Wj 2 WjeWj 1 2Wj 2 displaystyle Omega j 1 Omega j frac Omega j e Omega j 1 e Omega j Omega j 1 frac Omega j 2 Omega j e Omega j 1 2 Omega j 2 nbsp Integral representations edit An identity due to citation needed Victor Adamchik citation needed is given by the relationship dt et t 2 p2 11 W displaystyle int infty infty frac dt e t t 2 pi 2 frac 1 1 Omega nbsp Other relations due to Mezo 1 2 and Kalugin Jeffrey Corless 3 are W 1pRe 0plog eeit e iteeit eit dt displaystyle Omega frac 1 pi operatorname Re int 0 pi log left frac e e it e it e e it e it right dt nbsp W 1p 0plog 1 sin ttetcot t dt displaystyle Omega frac 1 pi int 0 pi log left 1 frac sin t t e t cot t right dt nbsp The latter two identities can be extended to other values of the W function see also Lambert W function Representations Transcendence edit The constant W is transcendental This can be seen as a direct consequence of the Lindemann Weierstrass theorem For a contradiction suppose that W is algebraic By the theorem e W is transcendental but W e W which is a contradiction Therefore it must be transcendental 4 References edit Mezo Istvan An integral representation for the principal branch of the Lambert W function Retrieved 24 April 2022 Mezo Istvan 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 Mezo Istvan Baricz Arpad November 2017 On the Generalization of the Lambert W Function PDF Transactions of the American Mathematical Society 369 11 7928 Retrieved 28 April 2023 External links editWeisstein Eric W Omega Constant MathWorld Omega constant 1 000 000 digits Darkside communication group in Japan retrieved 2017 12 25 Retrieved from https en wikipedia org w index php title Omega constant amp oldid 1207848789, wikipedia, wiki, book, books, library,