Starting from x=0.5, repeatedly applying e^(−x) converges to Ω ≈ 0.5671. The fixed point satisfies Ω = e^(−Ω), equivalently Ω·e^Ω = 1.
| Iteration | x | e^(−x) | |x − Ω| |
|---|---|---|---|
| 1 | 0.5 | 0.60653 | 0.067 |
| 2 | 0.60653 | 0.54545 | 0.022 |
| 3 | 0.54545 | 0.57970 | 0.008 |
| 4 | 0.57970 | 0.56007 | 0.003 |
| 5 | 0.56007 | 0.57121 | 0.001 |
| … | … | … | → 0 |
| ∞ | Ω | Ω | 0 |
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.