Omega constant
Encyclopedia
The Omega constant is a mathematical constant
defined by
It is the value of W(1) where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the Omega function.
The value of Ω is approximately 0.5671432904097838729999686622 . It has properties that
or equivalently,
One can calculate Ω iteratively
, by starting with an initial guess Ω0, and considering the sequence
This sequence will converge
towards Ω as n→∞.
A beautiful identity due to Victor Adamchik is given by the relationship
from the fact that e
is transcendental
; if Ω were rational, then there would exist integers p and q such that
so that
and e would therefore be algebraic
of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental
as the direct consequence of Lindemann–Weierstrass theorem
. If Ω were algebraic, exp(Ω) would be transcendental and so would be exp−1(Ω). But this contradicts the assumption that it was algebraic.
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
defined by
It is the value of W(1) where W is Lambert's W function. The name is derived from the alternate name for Lambert's W function, the Omega function.
The value of Ω is approximately 0.5671432904097838729999686622 . It has properties that
or equivalently,
One can calculate Ω iteratively
Iterative method
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
, by starting with an initial guess Ω0, and considering the sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
This sequence will converge
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
towards Ω as n→∞.
A beautiful identity due to Victor Adamchik is given by the relationship
Irrationality and transcendence
Ω can be proven irrationalIrrational number
In mathematics, an irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number....
from the fact that e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
is transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
; if Ω were rational, then there would exist integers p and q such that
so that
and e would therefore be algebraic
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
as the direct consequence of Lindemann–Weierstrass theorem
Lindemann–Weierstrass theorem
In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ..., are algebraic numbers which are linearly independent over the rational numbers ', then 1, ..., are algebraically...
. If Ω were algebraic, exp(Ω) would be transcendental and so would be exp−1(Ω). But this contradicts the assumption that it was algebraic.