The omega constant is a

mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...

defined as the unique

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that satisfies the equation :

\Omega e^\Omega = 1.

It is the value of , where is Lambert's function. The name is derived from the alternate name for Lambert's function, the ''omega function''. The numerical value of is given by : . : .

Properties

Fixed point representation

The defining identity can be expressed, for example, as :

\ln \left(\tfrac \right)=\Omega.

or :

-\ln(\Omega)=\Omega

as well as :

e^= \Omega.

Computation

One can calculate iteratively, by starting with an initial guess , and considering the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

\Omega_=e^.

This sequence will

converge Converge may refer to: * Converge (band), American hardcore punk band * Converge (Baptist denomination), American national evangelical Baptist body * Limit (mathematics) In mathematics, a limit is the value that a function (or sequence) app ...

to as approaches infinity. This is because is an attractive fixed point of the function . It is much more efficient to use the iteration :

\Omega_=\frac,

because the function :

f(x)=\frac,

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 In numerical analysis, Halley's method is a root-finding algorithm used for functions of one real variable with a continuous second derivative. Edmond Halley was an English mathematician and astronomer who introduced the method now called by his ...

, can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also ). :

\Omega_=\Omega_j-\frac.

Integral representations

An identity due to Victor Adamchik is given by the relationship :

\int_^\infty\frac = \frac.

Other relations due to Mező and Kalugin-Jeffrey-Corless. are: :

\Omega=\frac\operatorname\int_0^\pi\log\left(\frac\right) dt,

\Omega=\frac\int_0^\pi\log\left(1+\frace^\right)dt.

The latter two identities can be extended to other values of the function (see also ).

Transcendence

The constant is transcendental. This can be seen as a direct consequence of the

Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transc ...

. For a contradiction, suppose that is algebraic. By the theorem, is transcendental, but , which is a contradiction. Therefore, it must be transcendental.

References

External links

* * {{Irrational number

Omega Omega (, ; uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet. In the Greek numerals, Greek numeric system/isopsephy (gematria), it has a value ...

Articles containing proofs Real transcendental numbers