Disclaimer: I have posted this question on mathoverflow.net following the instructions of this topic.
In complex analysis we have well know results about isolated singularities. Poles are characterized by a 'nice' (rational) controlled growth around it and for essential singularities we have the Great Picard's Theorem.
Are there a similar classification for branch points? I mean: a clear list with all possibilities and results that characterizes each case?
For example: if we compare $f(z)=\sqrt z$ and $g(z) = \sin(\ln(z))$, they have very different behavior, one has a well defined limit as we approach $z=0$ in any branch and the other has a accumulation point of zeros. Are there results that characterizes this 'fast oscillations' of $g(z) = \sin(\ln(z))$ and the 'calmly' behavior of $f(z)=\sqrt z$ ? (May be in an appropriate Riemann surface).