I am trying to understand exactly what is going on in the picture below:
From what I understand so far, these are two complex planes. The left one is z, and the right one is the image of z under $f(z) = z^2$. I understand that the shaded right half of the z plane maps to all of the w plane, save the real line $(-∞, 0]$. (Please ignore the +/- in the inverse function, I realize there is only supposed to be a '+' there.)
Here are my questions/requests for clarification:
(1) Does the orange circle represents one branch, and the dotted circle represents another branch? Does the direction (clockwise/counterclockwise) matter?
(2) What to the +s and -s signify on the graphs?
(3)Are there infinitely many branches that are sort of 'paired up' by the positive part of the z plane mapping to all of the w plane, and the whole left half of the z plane mapping to the whole w plane?
(4) How would I go about finding these 'branch points' on my own? Would it just be intuition? I don't particularly understand how to do that. It is clear that f(i) = f(-i), but how would I know that these are the branching points, or is that when I know there is a branching point (when two values map to the same point)?
Any extra tips for these kinds of problems is much appreciated! Thank you all.