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).
Fixed point representation
The defining identity can be expressed, for example, as
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.
An identity due to Victor Adamchik is given by the relationship
Another relation due to Mező is
The latter identity 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.
- István, Mező. "An integral representation for the principal branch of Lambert the W function". Retrieved 7 November 2017.