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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 $z=2i$ ?

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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 $w=0$.

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How do we go up the branch ? What is the invariant of $W(z)$ ? $W(z) = W(f(z))$ ? I seem to have trouble ' visualizing '. @Robert Israel: Thanks for the answer. Could you help me with the questions above ? I seem to have read a bad/confusing paper apparantly. –  mick Sep 19 '12 at 16:51
    
$W$ is supposed to be the (multivalued) inverse of $f$, so $f(W(z)) = z$. I'm not sure what you mean by "go up the branch", but I suppose you may be asking how to connect the values at one point with those at another. In brief, the answer is: analytic continuation. –  Robert Israel Sep 19 '12 at 18:13
    
Im aware of analytic continuation. I see now I asked my question incorrect. I did not type what i meant. Sorry. Before i rephrase my question i will consider it again first and maybe i can find the answer myself. –  mick Sep 20 '12 at 15:24

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