Montonicity of Lambert W

Question

Is Lambert $W(x)$ function, an increasing function from $0\rightarrow\infty$? How about in negative axis and complex plane? Note $W(x)$ is given by $$W(x)e^{W(x)}=x.$$ Charts could help understand.

Answer 1

On $[0,\infty)$, it is true that $W(x)$ is a strictly increasing function. This follows from the fact that it is the inverse of a strictly increasing function, $f(x)=x\cdot e^x$ - and the fact that this is increasing is obvious since both $x$ and $e^x$ are. More firmly, from the derivative $f'(x)=(x+1)\cdot e^x$, we can see that $f$ is increasing whenever $x\geq -1$ - which means that $W$ is increasing on the range $[f(-1),\infty)$, or $[\frac{-1}e,\infty)$. This happens to be the entire range for which the $W$ function takes on real values.

The question is not meaningful in the complex plane; there is no notion of order (i.e. is $i>-i$?), so there is no notion of "strictly increasing". So, the only thing we can say is that $W$ is strictly increasing on the domain where it is real (when we choose the principal value of $W$ to be in $[-1,\infty)$).

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As was pointed out in comments, since $f$ is decreasing in the interval $(-\infty,1]$, where its image is $(-\frac{1}e,0)$, in this domain there is a branch of $W$ which is strictly decreasing. To plot this, here is the plot of the (multivalued) relation $W$: The principal branch that I primarily address is in red, and is increasing, as you can see. The other branch is in blue, and is decreasing (towards the vertical asymptote of the $y$-axis).