How do we know how many branches the inverse function of an elementary function has?

Question

How do we know how many branches the inverse function of an elementary function has ? For instance Lambert W function. How do we know how many branches it has at e.g. $z=-0.5$ , $z=0$ , $z=0.5$ or $z=2i$ ?

Answer 1

Suppose your elementary function $f$ is entire and has an essential singularity at $\infty$ (as in the case you mention, with $f(z) = z e^z$). Then Picard's "great" theorem says that $f(z)$ takes on every complex value infinitely often, with at most one exception. Thus for every $w$ with at most one exception, the inverse function will have infinitely many branches at $w$. Determining whether that exception exists (and what it is) may require some work. In this case it is easy: $f(z) = 0$ only for $z=0$ because the exponential function is never $0$, so the exception is $w=0$.