I am really struggling with the concept of a "branch point". I understand that, for example, if we take the $\log$ function, by going around $2\pi$ we arrive at a different value, so therefore it is a multivalued function. However, surely this argument holds for all points in the complex plane, so I don't really understand how $z=0$ is the ONLY branch point.
Additionally, the course I am revising for needs no Riemann surfaces or knowledge of that area of mathematics, just what a branch point is and how to find it.
Thanks for any help.
Best Answer
A branch point of a "multi-valued function" $f$ is a point $z$ with this property: there does not exist an open neighbourhood $U$ of $z$ on which $f$ has a single-valued branch.
In the case of $\log$, the only branch point is $0$: indeed, if $z \ne 0$, we could take $$U = \{ w \in \mathbb{C} : \lvert w - z \rvert < \lvert z \rvert \}$$ and define a single-valued branch of $\log$ on $U$. If you want to think in terms of paths, the point is that the value of $\log$ cannot jump if the path does not wind around $0$.