 Real branches of Lambert W

On $-e^{-1}<z<0$, the Lambert W function has two real branches: $-1<W_0(z)<0$ and $W_{-1}(z)<-1$. The derivative of $w{e}^w$ is $e^w(1+w)$, so this function decreases on $(-\infty ,-1)$ and increases on $(-1,\infty )$, proving the two-branch structure. As $u\to \infty$, series reversion givesConsequently the two positive roots of $x{e}^{1/x}=e^u$ are $-1/W_0(-e^{-u})$ and $-1/W_{-1}(-e^{-u})$.