Finding Principal Branch Value of Lambert W function

Question

I have this negative number $x<-1$ (thus in the lower branch of the Lambert W function). Is it possible to find $W_{0}(xe^x)$ in terms of $x$ in a useful/non-trivial form?

Answer 1

By the definition of the Lambert W function, the value $y = W_0\left(x \mathrm{e}^{x}\right)$ satisfies: $$ y \mathrm{e}^{y} = x \mathrm{e}^{x} $$ There are two real solutions to this equation for each $x<-1$. One $y\leqslant -1$ and another $-1 \leqslant y < 0$. The solutions are $W_{-1}\left(x \mathrm{e}^{x}\right) = x$ and $W_0\left( x \mathrm{e}^{x} \right)$. The latter solution is a non-trivial function of $x$.