Omega constant

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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]

Fixed point representation[edit]

The defining identity can be expressed, for example, as

or

or

In Mathematica

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

Computation[edit]

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 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).

Integral representation[edit]

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

or

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

See also[edit]

References[edit]

  1. ^ Adamchik, Victor. http://www-bcf.usc.edu/~adamchik/.  Missing or empty |title= (help)

External links[edit]