Kalpana Kalpana (Editor)

Omega constant

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The omega constant is a mathematical constant defined by

Contents

Ω e Ω = 1.

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... (sequence A030178 in the OEIS).

Fixed point representation

The defining identity can be expressed, for example, as

ln ( 1 Ω ) = Ω .

or

ln ( Ω ) = Ω

or

e Ω = Ω .

In Mathematica

Plot[{x, Exp[-x], -Log[x]}, {x, 0, 1}]

Computation

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

Ω n + 1 = e Ω n .

init = 0.5;

FixedPoint[Exp[-#] &, init]

This sequence will converge towards Ω as n→∞. This convergence is because Ω is an attractive fixed point of the function ex.

It is much more efficient to use the iteration

Ω n + 1 = 1 + Ω n 1 + e Ω n ,

because the function

f ( x ) = 1 + x 1 + e x = 1 + x 1 + e x e x

has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration).

Integral representation

A beautiful identity due to Victor Adamchik[1] is given by the relationship

d t ( e t t ) 2 + π 2 = 1 1 + Ω

or

Ω = 1 d t ( e t t ) 2 + π 2 1.

Irrationality and transcendence

Ω can be proven irrational from the fact that e is transcendental; if Ω were rational, then there would exist integers p and q such that

p q = Ω

so that

1 = p e ( p q ) q

and

e = ( q p ) ( q p ) = q q p q p

The number 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, e−Ω would be transcendental; but Ω=exp(-Ω), so these cannot both be true.

References

Omega constant Wikipedia


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