Omega constant
{{Short description|1=Solution to x * e^x = 1}}
{{about|a specific value of Lambert's W function|other omega constants|omega (disambiguation)#Mathematics}}
The omega constant is a mathematical constant defined as the unique real number that satisfies the equation
:
It is the value of {{math|W(1)}}, where {{mvar|W}} is Lambert W function. The name is derived from the alternate name for Lambert's {{mvar|W}} function, the omega function. The numerical value of {{math|Ω}} is given by
:{{math|1=Ω = {{gaps|0.56714|32904|09783|87299|99686|62210|...}}}} {{OEIS|id=A030178}}.
:{{math|1=1/Ω = {{gaps|1.76322|28343|51896|71022|52017|76951|...}}}} {{OEIS|id=A030797}}.
Properties
= Fixed point representation =
The defining identity can be expressed, for example, as
:
or
:
as well as
:
= Computation =
One can calculate {{math|Ω}} iteratively, by starting with an initial guess {{math|Ω0}}, and considering the sequence
:
This sequence will converge to {{math|Ω}} as {{mvar|n}} approaches infinity. This is because {{math|Ω}} is an attractive fixed point of the function {{math|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.
Using Halley's method, {{math|Ω}} can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also {{section link|Lambert W function|Numerical evaluation}}).
:
= Integral representations =
An identity due to Victor Adamchik{{cn|date=February 2025}} is given by the relationship
:
Other relations due to Mező{{cite web|first=István|last=Mező|title=An integral representation for the principal branch of the Lambert W function|url=https://sites.google.com/site/istvanmezo81/other-things |access-date=24 April 2022}}{{cite arXiv
| last = Mező | first = István
| title = An integral representation for the Lambert W function
| date = 2020| class = math.CA
| eprint = 2012.02480
}}.
and Kalugin-Jeffrey-Corless{{cite arXiv
| first1=German A. | last1=Kalugin | first2=David J. | last2=Jeffrey | first3=Robert M. | last3=Corless
| title = Stieltjes, Poisson and other integral representations for functions of Lambert W
| date = 2011| class = math.CV
| eprint = 1103.5640
}}.
are:
:
:
The latter two identities can be extended to other values of the {{mvar|W}} function (see also {{section link|Lambert W function|Representations}}).
=Transcendence=
The constant {{math|Ω}} is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that {{math|Ω}} is algebraic. By the theorem, {{math|e−Ω}} is transcendental, but {{math|1=Ω = e−Ω}}, which is a contradiction. Therefore, it must be transcendental.{{cite journal |last1=Mező |first1=István |last2=Baricz |first2=Árpád |title=On the Generalization of the Lambert W Function |journal=Transactions of the American Mathematical Society |date=November 2017 |volume=369 |issue=11 |page=7928 |doi=10.1090/tran/6911 |url=https://www.ams.org/journals/tran/2017-369-11/S0002-9947-2017-06911-7/S0002-9947-2017-06911-7.pdf |access-date=28 April 2023}}
References
{{Reflist}}
External links
- {{MathWorld|urlname=OmegaConstant|title=Omega Constant}}
- {{citation|title=Omega constant (1,000,000 digits)|url=http://ankokudan.org/d/d.htm?mathlistindex-e.html|work=Darkside communication group (in Japan) |access-date=2017-12-25}}
{{Irrational number}}