Could someone please explain the concept of branch points to me? I have tried searching online and had a read of the textbook Visual Complex Analysis by T. Needham, but I am still not very clear how they work.
An excerpt I found online from Introduction to Complex Analysis by H. Priestly says that
$a$ is a branch point for [$w(z)$] if, for all sufficiently small $r>0$, it is not possible to choose $f(z)\in[w(z)]$ so that $f$ is a continuous function on $\gamma(a;r)^*$.
Firstly, I couldn't find what $\gamma(a;r)^*$ is ... I presume it is an open ball around the point $a$ with radius $r$?
Secondly, I just don't understand what it is saying. Why is there no continuous function? How when asked to find branch points would I know which points in $\mathbb C$ have this property?
Needham's book basically says a branch point is one which if we circle it once we don't get back to the same point... but I still don't get it!
Then I read something about branch cuts and Riemann spheres which really don't help to clarify anything at all!
Thank you for your time.
[Added] For example if I have a map of the form $f(z) =[(z-a)(z-b)...(z-n)]^{1\over m}$ how may branch points are there?