Disclaimer: This question was originally posted in [math.stackexchange.com](https://math.stackexchange.com/questions/4272335/are-there-a-in-depth-classification-of-branch-points-in-complex-analysis) and, after 30 days with no answers, I followed the instructions of [this topic](https://meta.mathoverflow.net/questions/2637/cross-posts-to-math-se/2638#2638).


In complex analysis we have well known results about isolated singularities. Poles are characterized by ‘nice’ (rational) controlled growth around them and for essential singularities we have the [Great Picard's Theorem](https://en.wikipedia.org/wiki/Picard_theorem).


**Question:** Is 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 an accumulation point of zeros. Are there results that characterize the ‘fast oscillations’ of $g(z) = \sin(\ln(z))$ and the ‘calm’ behavior of $f(z)=\sqrt z$? (Maybe in an appropriate Riemann surface.)



Is there a, in depth, classification of branch points in complex analysis?
Yes, there is a classification. An isolated branch point can be algebraic or logarithmic. If the branch point is at 0, algebraic means that $f(z^n)$ has a pole or removable singularity at 0. It can also have an essential singularity, but this does not have an accepted name. In the case of a logarithmic point
$f(e^z)$ is an (arbitrary) meromorphic function in some left half-plane. Anyway, plugging $z^n$ or $e^z$ (this is called local uniformization) reduces any classification at an isolated branch point to the case of a single valued function.


Literature: on singularities of analytic functions in general,
and their classification, there is a book


P. Dienes,
Leçons sur les singularités des fonctions analytiques; professées à l’université de Budapest,
Paris: Gauthier-Villars, VIII + 172 S. 8∘ (1913),


which is somewhat out of date.


"Isolated branch points" by themselves are
not a subject of a special
study, but "functions whose all singularities are isolated and
may include isolated branch points" play an important role in Ecalle's theory
of "resurgent functions". Jean Ecalle wrote several books on them, but they are very hard to read. Of the many expositions of Ecalle's theory I can mention


C. Mitschi and D. Sauzin,
Divergent series, summability and resurgence I. Monodromy and resurgence.
Lecture Notes in Mathematics 2153, Springer (ISBN 978-3-319-28735-5/pbk; 978-3-319-28736-2/ebook). xxi, 298 p. (2016).


See also arXiv:1405.0356.


