I am trying to find a definitions which link the concept of branches, branch cuts and branch points. Relating to this I have a question concerning the nature of branch cuts: Do they necessary have to be lines? i.e. could they be strips - or any other shaped region (along as it keeps the branch single-valued and continuous)?
Note I am the OP. I have been looking the wrong way wrong. The following is a definition of a branch cut (from these notes):
A branch cut for a multifunction $f$ is a curve in the plane on whose complement we can pick a holomorphic branch of $f$. Thus a branch cut must contain all the branch points.
As you can see from this definition the branch cut defines the holomorphic branch not the other way round. Thus it is true that not all branches of a function have 'branch cuts' but it is true that all branch cuts are associated with a holomorphic branch.