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).
Fixed point representation
The defining identity can be expressed, for example, as
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 Victor Adamchik is given by the relationship
Other relations due to Mező and Kalugin-Jeffrey-Corless are:
The latter two identities can be extended to other values of the W function (see also Lambert W function § Representations).
- ^ Mező, István. "An integral representation for the principal branch of the Lambert W function". Retrieved 24 April 2022.
- ^ Mező, István (2020). "An integral representation for the Lambert W function". arXiv:2012.02480 [math.CA]..
- ^ Kalugin, German A.; Jeffrey, David J.; Corless, Robert M. (2011). "Stieltjes, Poisson and other integral representations for functions of Lambert W". arXiv:1103.5640 [math.CV]..
- Weisstein, Eric W. "Omega Constant". MathWorld.
- "Omega constant (1,000,000 digits)", Darkside communication group (in Japan), retrieved 2017-12-25