Editing 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

:<math>\Omega e^\Omega = 1.</math>

It is the value of {{math|''W''(1)}}, where {{mvar|W}} is [[Lambert W function|Lambert's {{mvar|W}} function]].  The name is derived from the alternate name for Lambert's {{mvar|W}} function, the ''omega function''. The numerical value of {{math|&Omega;}} is given by

:{{math|1=&Omega; = {{gaps|0.56714|32904|09783|87299|99686|62210|...}}}} {{OEIS|id=A030178}}.
:{{math|1=1/&Omega; = {{gaps|1.76322|28343|51896|71022|52017|76951|...}}}} {{OEIS|id=A030797}}.

== Properties ==
=== Fixed point representation ===
The defining identity can be expressed, for example, as
:<math>\ln \left(\tfrac{1}{\Omega} \right)=\Omega.</math>
or
:<math>-\ln(\Omega)=\Omega</math>
as well as
:<math>e^{-\Omega}= \Omega.</math>

=== Computation ===
One can calculate {{math|&Omega;}} [[iterative method|iteratively]], by starting with an initial guess {{math|&Omega;<sub>0</sub>}}, and considering the [[sequence]]

:<math>\Omega_{n+1}=e^{-\Omega_n}.</math>

This sequence will [[limit of a sequence|converge]] to {{math|&Omega;}} as {{mvar|n}} approaches infinity. This is because {{math|&Omega;}} is an [[Fixed point (mathematics)|attractive fixed point]] of the function {{math|''e''<sup>−''x''</sup>}}.

It is much more efficient to use the iteration

:<math>\Omega_{n+1}=\frac{1+\Omega_n}{1+e^{\Omega_n}},</math>

because the function

:<math>f(x)=\frac{1+x}{1+e^x},</math>

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|&Omega;}} 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}}).

:<math>\Omega_{j+1}=\Omega_j-\frac{\Omega_j e^{\Omega_j}-1}{e^{\Omega_j}(\Omega_j+1)-\frac{(\Omega_j+2)(\Omega_je^{\Omega_j}-1)}{2\Omega_j+2}}.</math>

=== Integral representations ===
An identity due to Victor Adamchik{{cn|date=February 2025}} is given by the relationship
:<math>\int_{-\infty}^\infty\frac{dt}{(e^t-t)^2+\pi^2} = \frac{1}{1+\Omega}.</math>

Other relations due to Mező<ref>{{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}}</ref><ref>{{cite arXiv
 | last = Mező | first = István
 | title = An integral representation for the Lambert W function
 | date = 2020| class = math.CA
 | eprint = 2012.02480
 }}.</ref>
and Kalugin-Jeffrey-Corless<ref>{{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
 }}.</ref>
are:
:<math>\Omega=\frac{1}{\pi}\operatorname{Re}\int_0^\pi\log\left(\frac{e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}\right) dt,</math>
:<math>\Omega=\frac{1}{\pi}\int_0^\pi\log\left(1+\frac{\sin t}{t}e^{t\cot t}\right)dt.</math>
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|&Omega;}} is [[transcendental number|transcendental]]. This can be seen as a direct consequence of the [[Lindemann–Weierstrass theorem]]. For a contradiction, suppose that {{math|&Omega;}} is algebraic. By the theorem, {{math|''e''<sup>−&Omega;</sup>}} is transcendental, but {{math|1=&Omega; = ''e''<sup>−&Omega;</sup>}}, which is a contradiction. Therefore, it must be transcendental.<ref name="Mezo">{{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}}</ref>

==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}}

[[Category:Mathematical constants|Omega]]
[[Category:Articles containing proofs]]
[[Category:Real transcendental numbers]]