Omega constant

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This article is about the Ω constant from analysis. For the Ω constant from information theory, see Chaitin's constant.

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

(sequence A030178 in the OEIS).

Properties[edit]

The defining identity can be expressed, for example, as

or

or

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

or

Computation[edit]

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

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

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).

Irrationality and transcendence[edit]

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

See also[edit]

References[edit]

External links[edit]