Branch cut for multi valued complex functions

Question

This question is about the so called 'branch cut' for defining multivalued function continuously on complex plane.For example to define inverse of the function $f(z)=z^2$ we consider the domain of the inverse function as the slit plane $C\setminus(\infty,0]$. The case is also similar when defining Log function.

So my question, is there any sort of 'rule of thumb' which guarantees that removing this sort slit assures you that the domain is right for defining inverse function? Also why we are removing straight lines always? Would we get same results if we have removed any arbitary type of curves from C?

Answer 1

The need of introducing branch cuts usually arises from points which causes some trouble. That's the case with the point $0$ when you try to define $\log(z)$ or $\sqrt z$. By eliminating a ray whose origin is that point, we get a simply connected set. And in a simply connected set, all sorts of good things happen. For instance, every holomorphic functions has a primitive and every holomorphic functions without zeros has a holomorphic logarithm.

Therefore, the choice of a ray is just because that's the simplest option. You could define a holomorphic function in $\mathbb{C}\setminus\bigl\{x+i\sin x\,|\,x\in(-\infty,0]\bigr\}$.