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

:\Omega e^\Omega = 1.

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

:\ln \left(\tfrac{1}{\Omega} \right)=\Omega.

or

:-\ln(\Omega)=\Omega

as well as

:e^{-\Omega}= \Omega.

= Computation =

One can calculate {{math|Ω}} iteratively, by starting with an initial guess {{math|Ω0}}, and considering the sequence

:\Omega_{n+1}=e^{-\Omega_n}.

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|ex}}.

It is much more efficient to use the iteration

:\Omega_{n+1}=\frac{1+\Omega_n}{1+e^{\Omega_n}},

because the function

:f(x)=\frac{1+x}{1+e^x},

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}}).

:\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}}.

= Integral representations =

An identity due to Victor Adamchik{{cn|date=February 2025}} is given by the relationship

:\int_{-\infty}^\infty\frac{dt}{(e^t-t)^2+\pi^2} = \frac{1}{1+\Omega}.

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:

:\Omega=\frac{1}{\pi}\operatorname{Re}\int_0^\pi\log\left(\frac{e^{e^{it}}-e^{-it}}{e^{e^{it}}-e^{it}}\right) dt,

:\Omega=\frac{1}{\pi}\int_0^\pi\log\left(1+\frac{\sin t}{t}e^{t\cot t}\right)dt.

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

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