Editing Branch point (section)

==Transcendental and logarithmic branch points==
Suppose that ''g'' is a global analytic function defined on a [[Annulus (mathematics)|punctured disc]] around ''z''<sub>0</sub>.  Then ''g'' has a '''transcendental branch point''' if ''z''<sub>0</sub> is an [[essential singularity]] of ''g'' such that [[analytic continuation]] of a function element once around some simple closed curve surrounding the point ''z''<sub>0</sub> produces a different function element.<ref>{{harvnb|Solomentsev|2001}}; {{harvnb|Markushevich|1965}}</ref>  

An example of a transcendental branch point is the origin for the multi-valued function

:<math>g(z) = \exp \left( z^{-1/k}\right)\,</math>

for some integer ''k''&nbsp;>&nbsp;1.  Here the [[monodromy]] group for a circuit around the origin is finite. Analytic continuation around ''k'' full circuits brings the function back to the original.

If the monodromy group is infinite, that is, it is impossible to return to the original function element by analytic continuation along a curve with nonzero winding number about ''z''<sub>0</sub>, then the point ''z''<sub>0</sub> is called a '''logarithmic branch point'''.<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Logarithmic_branch_point|title=Logarithmic branch point - Encyclopedia of Mathematics|website=www.encyclopediaofmath.org|access-date=2019-06-11}}</ref> This is so called because the typical example of this phenomenon is the branch point of the [[complex logarithm]] at the origin.  Going once counterclockwise around a simple closed curve encircling the origin, the complex logarithm is incremented by 2{{pi}}''i''.  Encircling a loop with winding number ''w'', the logarithm is incremented by 2{{pi}}''i&nbsp;w'' and the monodromy group is the infinite cyclic group <math>\mathbb{Z}</math>.

Logarithmic branch points are special cases of transcendental branch points.

There is no corresponding notion of ramification for transcendental and logarithmic branch points since the associated covering Riemann surface cannot be analytically continued to a cover of the branch point itself.  Such covers are therefore always unramified.