How many real roots does the Lambert W function has?

The Lambert W function of an argument $x$ is defined as the real root of the equation $y=ye^y$, and is denoted by $W(x)$.

How many real roots does the Lambert W function have?

A:

I haven’t found a rigorous proof for this, but a reasonable approach is to use series expansions.
The Lambert W-function is usually discussed for non-negative arguments:
$$W(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!} \tag{1}$$
Now choose $x=e^{ -\tau^2/2}$, so that the exponent is negative. Then
$$x = e^{ -e^{ -\tau^2/2}} = e^{ -\left(\frac{e^{\tau^2/2}}{e^{\tau^2/2}} \right)} = e^{ -1} = 0. \tag{2}$$
Hence, the Lambert W-function satisfies $W(-1) = 0$, and using (1) we may infer that
$$W(-1) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k!} = e^{ -1} \tag{3}$$
From (2) and (3), we conclude that the Lambert W-function has two real roots: $W(0) = 0$ and $W(-1) = e^{ -1}$.
In this sense, the Lambert W-function is a branch point of the real solution set.
Another approach is to look at the representation
$$W(x) = \frac{2}{e}\left( \int_0^x \frac{1}{1+t} \, dt + C \right) \tag{4}$$
By definition, the Lambert W-function has a unique solution in every non-negative real half-line: $W(x)$ is strictly increasing and completely monotone on $x > 0$. Furthermore, it

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