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
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]
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