value of one omega

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→∞. This convergence is due to the fact that 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).

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

«»^[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

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

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