The omega constant is a mathematical constant defined by
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.
It is much more efficient to use the iteration
because the function
has the same fixed point but features a zero derivative at this fixed point, therefore the convergence is quadratic (the number of correct digits is roughly doubled with each iteration).
A beautiful identity due to Victor Adamchik is given by the relationship
Irrationality and transcendence
and e would therefore be algebraic of degree p. However e is transcendental, so Ω must be irrational.
Ω is in fact transcendental as the direct consequence of Lindemann–Weierstrass theorem. If Ω were algebraic, exp(Ω) would be transcendental and so would be exp−1(Ω). But this contradicts the assumption that it was algebraic.