Branch cuts have been asked about and discussed on MSE extensively. That is, every answer to something along the lines of "What is a branch cut?" is... extensive.
I'm looking for a quick, intuitive description of what a branch cut is.
I know that:
- The logarithm of a complex number can have different values (making it not a function)
- A branch of the logarithm is a chosen set of values such that it is a function
What is the branch cut, though?
Do I say $$\ln{R} + i\theta,~~\theta \in [0, 2\pi) \\ \ln{R} + i\theta, ~~ \theta \in [2\pi, 4\pi)$$ are two distinct branches? Do these two branches have a branch cut between them? Or do they have distinct branch cuts?
I understand that some people know a lot of information regarding this subject, but reading the answers that describe this the same way that my textbook does hasn't helped. (What helped me at least start to get the idea of branches most was someone's comment: "It involves the 'non-function-ness' of the function. I.e. when a function can be mutli-valued. You have to decide which "branch" you are going to work on. — It's simple, to the point, not overly elaborate, etc.
A branch cut itself is a curve in the complex plane where we cut the multivalued function so that we get a single valued branch. In the logarithm example you gave, it is a multivalued function that can be visualized as a spiral:
The vertical dimension is the imaginary part, as a function in the complex plane (horizontal dimensions). The two branches you gave are separated from each other by a cut along the positive real axis. If you imagine taking scissors to the spiral and making cuts everywhere the spiral crosses the positive real axis, then each piece left is a branch, and each piece doesn't overhang itself (it is single valued).