# What is a simple way of describing branch cuts?

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.

• You can not explain it simple if you do not understand it well. – science Mar 11 '15 at 4:37
• I would say that a branch cut for a multivalued function (like a logarithm or fractional power of $x$) is a subset $S$ of the complex plane such that the function in question is a well-defined analytic function on $\Bbb C\setminus S$. Typical branch cuts for the mentioned functions are rays emanating from $0$, but a spiral $\{re^{ir}\colon r\ge0\}$ would also be a branch cut for these functions. – Greg Martin Mar 11 '15 at 4:46