The Lambert function has two real branches for $x∈[−1/e,0)$: the principal branch $W_0$ and the branch $W_-1$

I am trying to understand Lambert W function. I am new to this special function. What is the actual meaning of the word two real branches of Lambert W function for any real $x$. How to find the branch point and branch cuts for the Lambert W function. When I used $lambertw(0,1)$ in Matlab i got the real value but for all other branches I got complex values. Actually I couldn't understand that why only these two branches gives real values. Can anyone can explain it clearly?

• Did you look at en.wikipedia.org/wiki/Lambert_W_function ? – Claude Leibovici Dec 18 '14 at 7:47
• Yes, I have seen that but things are not crystal clear to me. I didn't get the concept of branches, branch cut and branch point w.r.t Lambert W function. – user3563283 Dec 18 '14 at 9:20
• Euler and Lambert showed that there is no real solution if $x < -\frac 1e$. The main branch is for $-\frac 1e < x <\infty$, the other branch for $-\frac 1e < x <0$. – Claude Leibovici Dec 18 '14 at 9:25
• How the branch cuts of Lambert function are derived? – user3563283 Dec 23 '14 at 10:14