# Difference between ramification point and branch point

I'm trying to distinguish between the two. According to wiki:

Let $$Ω$$ be a connected open set in the complex plane $$\mathbb C$$ and $$ƒ:Ω → \mathbb C$$ a holomorphic function. If $$ƒ$$ is not constant, then the set of the critical points of $$ƒ$$, that is, the zeros of the derivative $$ƒ'(z)$$, has no limit point in $$Ω$$. So each critical point $$z_0$$ of $$ƒ$$ lies at the center of a disc $$B(z_0,r)$$ containing no other critical point of $$ƒ$$ in its closure.

Let $$γ$$ be the boundary of $$B(z_0,r)$$, taken with its positive orientation. The winding number of $$ƒ(γ)$$ with respect to the point $$ƒ(z_0)$$ is a positive integer called the ramification index of $$z_0$$. If the ramification index is greater than 1, then $$z_0$$ is called a ramification point of ƒ, and the corresponding critical value $$ƒ(z_0)$$ is called an (algebraic) branch point. Equivalently, $$z_0$$ is a ramification point if there exists a holomorphic function $$φ$$ defined in a neighborhood of $$z_0$$ such that $$ƒ(z) = φ(z)(z − z_0)^k$$ for some positive integer $$k > 1$$.

What confuses me are the following:

1) the usage in complex analysis where I think we do not distinguish between the ramification and the branch point.

2) In the wiki definition, what happens if $$f(z_0)$$ is not defined? Do we still say that $$f$$ has a ramification point at $$z_0$$ even though the branch point $$f(z_0)$$ is undefined?

And another question:

3) Is there an easy computational tool to determine the ramification points? I feel like it has to be related to $$f$$ having a multiple root at a ramification point, but I don't think that's precise enough.

• If $f(z) = f(a)+ C(z-a)^n+O(z-a)^{n+1}, C \ne 0$ is holomorphic then $f^{-1}$ has a branch point at $f(a)$ and $m =n$ is the least integer such that $f^{-1}(f(a)+s^m)$ is holomorphic around $s=0$ – reuns Nov 23 '18 at 18:34

4.3 Definition Soppose $$X$$ and $$Y$$ are Riemann surfaces and $$p:Y \to X$$ is a non-constant holomorphic map. A point $$y\in Y$$ is called a branch point or ramification point of $$p$$, if there is no neighborhood $$V$$ of $$y$$ s.t. $$p|_V$$ is injective.