“Omega constant”的意思、由来-开放百科全书

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Omega constant

释义

Properties
Fixed point representation Computation Integral representations
Transcendence
See also
References
External links

{{about|the Ω constant from analysis|the Ω constant from information theory|Chaitin's constant}}

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's {{mvar|W}} function. The name is derived{{cn|date=September 2018}} 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

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=September 2018}} is given by the relationship

Another relation due to Mező is^[1]

The latter identity 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.

References

1. ^{{cite web|last1=István|first1=Mező|title=An integral representation for the principal branch of Lambert the W function|url=https://sites.google.com/site/istvanmezo81/others|accessdate=7 November 2017}}

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) |accessdate=2017-12-25}}

4 : Transcendental numbers|Mathematical constants|Articles containing proofs|Real transcendental numbers

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