The omega constant

Exactly one real number satisfies $x e^x = 1$:

$$\omega\,e^{\omega} = 1, \qquad \omega = e^{-\omega} = \ln\frac{1}{\omega} = 0.567143\ldots$$

It is the value at $1$ of Lambert’s $W$, the inverse of $x\mapsto xe^x$, and it behaves like a fixed point because it is one: iterate $x \mapsto e^{-x}$ and you fall into it. On $[0.49,\,0.62]$ the map sends the interval into itself and $|{-e^{-x}}| \le e^{-0.49} < 0.62$, so the contraction argument applies verbatim. There is a small joke in the rate: the asymptotic error ratio is $|{-e^{-\omega}}| = \omega$ itself. The omega constant governs its own convergence.1

1

Lambert introduced the transcendent equation (1758); Euler’s 1783 paper on it is where the series for $W$ effectively appears.

Newton on $f(x) = xe^x - 1$ doubles digits per step and is how libraries actually evaluate $W$ near the point; the iteration above is the one you can run in your head.

Where the trees are

The same function organizes combinatorics. The exponential generating function of rooted labeled trees,

$$T(x) \;=\; \sum_{n\ge1} n^{\,n-1}\,\frac{x^n}{n!},$$

satisfies $T = x\,e^{T}$, which is to say $T(x) = -W(-x)$. Its radius of convergence is $1/e$, and at the boundary $T(1/e) = 1$ — the identity $\sum n^{n-1}e^{-n}/n! = 1$ falls out, with Cayley’s $n^{n-2}$ tree count standing just behind it. Random graphs remember this: the phase transition of the Erdős–Rényi giant component is the same singularity wearing different clothes.

Like its neighbors in the catalog, $\omega$ is transcendental for Lindemann’s reason: were $\omega$ algebraic and nonzero, $e^{-\omega}$ would be transcendental — but $e^{-\omega}$ is $\omega$. The definition refutes the alternative by itself.