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 = 0.56714 32904 09783 87299 99686 62210 ... . 1/Ω = 1.76322 28343 51896 71022 52017 76951 ... .

Properties

Fixed point representation

The defining identity can be expressed, for example, as or as well as

Computation

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 ).

Integral representations

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 ).

Transcendence

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.