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 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).

1/Ω = 1.763222834351896710225201776951... (sequence A030797 in the OEIS).

The defining identity can be expressed, for example, as

or

as well as

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 e^−x.

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).

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

Other relations due to Mező^[1][2] and Kalugin-Jeffrey-Corless^[3] are:

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

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.^[4]

• Weisstein, Eric W. "Omega Constant". MathWorld.
• "Omega constant (1,000,000 digits)", Darkside communication group (in Japan), retrieved 2017-12-25