I can't seem to find a good definition for a branch point. I have several thoughts and have found various definitions but would like to hear MHFs opinions on this one!

I'm thinking something along the lines of...

A branch point is a point of a (analytic?) function that is undefined/discontinuous?

Or perhaps...

A branch point(s?) $\displaystyle z$ is a point(s?) of an analytic function $\displaystyle f$ such that $\displaystyle f$ is holomorphic everywhere except at $\displaystyle z$? (or perhaps near z?)