Inverse of $y=xe^x$

http://en.wikipedia.org/wiki/Lambert_W_function, see the section Applications.

Inversion of $xe^{-x}$ is a problem with combinatorial structure, from which one can also invert $xe^x$ and derive most of the properties at the Wikipedia page. At least for theoretical understanding it is easier to treat the inverse of $x\exp(-x)$ as the basic object and then translate it into "Lambert W" terms.

Inverse of $f(x) = x \exp(-x)$ is the generating function for rooted trees $\Sigma n^{n-1} X^n/n!$. That the series represents the inverse is equivalent to a simple combinatorial fact ("removing the root and its edges from a tree, gives a collection of rooted trees"). The inverse of $xe^x$ is then $W(x)= -f^{-1}(-x)$, which is the same series with minus signs. The radius of convergence $1/e$ follows from Stirling's approximation to $n!$.

The coefficients of $W(x)^r$ given in the Wikipedia article are, up to some signs, Abel polynomials. This also follows from the combinatorial interpretation: $(f^{-1}(x))^r$ is the generating function whose $x^n$ coefficient is the number maps from a set of size $n$ to a set of $r$ elements with a rooted tree on each fiber of the map.

Asymptotically, $x \approx \log y - \log\log y$.