TITLE: Inverse of $x\log(x)$ for $x>1$
QUESTION [5 upvotes]: Let $y(x)=x\log(x)$ for $x>1$. Is it possible to write down the inverse function explicitly? Has this inverse function been named? (For example, the Bessel functions are "named" but cannot be defined explicitly using elementary functions.)

REPLY [6 votes]: The Lambert $W$ function is the inverse function of $g(x)=xe^x$, i.e. a function such that $W(x)\,e^{W(x)}=x$ for every $x$ in some range. To solve:
$$ y \log y = x $$
by setting $y=e^{f(x)}$ is the same as solving $f(x) e^{f(x)}=x$, that gives $f(x)=W(x)$.
It follows that:
$$ y = e^{W(x)} = \frac{x}{W(x)}.$$