Omega constant

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The omega constant is a mathematical constant defined as the unique real number that satisfies the equation

It is the value of W(1), where W is Lambert's W function. The name is derived[citation needed] from the alternate name for Lambert's W function, the omega function. The numerical value of Ω is given by

Ω = 0.567143290409783872999968662210... (sequence A030178 in the OEIS).

Properties[edit]

Fixed point representation[edit]

The defining identity can be expressed, for example, as

or

or

Computation[edit]

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

This sequence will converge to Ω as n approaches infinity. This is because Ω is an attractive fixed point of the function ex.

It is much more efficient to use the iteration

because the function

in addition to having the same fixed point, also has a derivative that vanishes there. This guarantees quadratic convergence; that is, the number of correct digits is roughly doubled with each iteration.

Using Halley's method, Ω can be approximated with cubic convergence (the number of correct digits is roughly tripled with each iteration): (see also Lambert W function § Numerical evaluation).

Integral representations[edit]

An identity due to Victor Adamchik[citation needed] is given by the relationship

Another relation due to Mező is[1]

The latter identity can be extended to other values of the W function (see also Lambert W function § Representations).

Transcendence[edit]

The constant Ω is transcendental. This can be seen as a direct consequence of the Lindemann–Weierstrass theorem. For a contradiction, suppose that Ω is algebraic. By the theorem, e−Ω is transcendental, but Ω = e−Ω, which is a contradiction. Therefore, it must be transcendental.

See also[edit]

References[edit]

  1. ^ István, Mező. "An integral representation for the principal branch of Lambert the W function". Retrieved 7 November 2017.

External links[edit]