Good afternoon!
I wanted to ask a quick question, as a beginner in complex analysis, I am trying to get my head around branch points. I came upon some lecture notes, but then did not understand something.
The notes define (page 4) that a branch point is a point such that the function does not return to its original value, as you go around a sufficiently small closed curve around it. To motivate that definition, it shows the following example page 3:
And it says that if you go around the first path, the $\log$ function ($z \mapsto \log|z| + i*\arg(z)$) is multi-valued (unless you introduce a branch cut), but that, if you go around the second path, it returns to its original value.
And I don't understand why that is so. Because, in the second case, $z = re^{i\theta}$, exactly as in the first one, so why would a full rotation come back to the same value in one case and not in the other? I don't see any mention of introducing a principal value to the argument or anything; so I don't understand what's so fundamentally different between those two examples.
Thanks for your time! :)