The curve y = xe^x is increasing for x > 0 and crosses y = 1 at exactly x = Ω ≈ 0.56714. This is the definition: Ωe^Ω = 1.
Starting from any positive value, the iteration Ωₙ₊₁ = e⁻Ωₙ converges to Ω ≈ 0.56714. It converges because at Ω the function e⁻ₓ has slope magnitude less than 1, making each step closer than the last.
The Omega constant Ω is the unique positive real number satisfying Ωe^Ω = 1, or equivalently e^(-Ω) = Ω. Its value is approximately 0.56714329040978387299. It is the value of the Lambert W function at 1: the solution to W(x)e^W(x) = x evaluated at x = 1.
Ω is transcendental. The proof is a consequence of the Lindemann-Weierstrass theorem: if Ω were algebraic, then e^Ω = 1/Ω would be transcendental, but since Ω satisfies Ωe^Ω = 1, this leads to a contradiction. Ω is also conjectured to be normal, but this is unproved.
The Lambert W function itself arises in a surprising range of contexts: the time for a radioactive material to decay to a given fraction, the number of comparisons in certain sorting algorithms, the position of the Lagrange points in orbital mechanics, and the solution to iterated exponentials like x = a^(a^(a^⋯)).
The Omega constant Ω = W(1) is one value of the Lambert W function W(xe^x) = x. This function solves equations of the form xe^x = c and appears wherever a variable appears both inside and outside an exponential.
Omega can be computed by Newton's method applied to f(x) = x*e^x - 1, or by the simple iteration Omega(n+1) = e^(-Omega_n), which converges from any positive starting point. Starting from 1.0 gives: 0.3679, 0.6922, 0.5002, 0.6065, 0.5452, ... converging to Omega ≈ 0.56714. About 10 iterations gives 6 correct decimal places.
Omega satisfies the infinite tower: Omega = e^(-e^(-e^(-...))). An infinite stack of negative exponentials converges to Omega. This follows directly from the iteration formula: the fixed point of x maps to e^(-x), which is exactly Omega.
The Omega constant satisfies Omega * e^Omega = 1, so Omega ≈ 0.56714. It is the value of the Lambert W function at 1, and satisfies e^(-Omega) = Omega. The simple iteration Omega_new = e^(-Omega_old) converges from any positive starting value. Omega is transcendental. It satisfies the infinite tower Omega = e^(-e^(-e^(-...))). It appears in the analysis of algorithms and solutions to delay differential equations.