Omega constant

Fixed point representation

The defining identity can be expressed, for example, as

or

or

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

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 e^−x.

It is much more efficient to use the iteration

because the function

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

or

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

so that

and

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.