I am asking this question because I wish to get a feel for how it would be to invent the Lambert W function. I would like to really understand it.
We have the equation: $$xe^x=y \tag{1}$$ and we want to find all the real solution of it.
Of course nowadays we know that the solutions are: $x=W_k(y)$ and that, to be more specific, in $\mathbb{R}$ we only have the branches $k=0$ and $k=-1$.
But suppose instead that we know nothing about the Lambert function; in searching for solution to (1) how can we "invent" it? And in particular: what's up with the branches?
I would prefer answer that don't involve heavy mathematical machinery, the simpler the better.
(Bonus question: If you like this question and want to extend the discussion I would like to see how the reasoning can be extended to the complex numbers, although I would like to see it as an extension and not as the starting point of the answer.)