I'm struggling as to how I'm supposed to do an analysis of how different branch cuts would affect a function f(z). The problem I'm struggling with has
$f(z)=\sqrt[3]{z^3+1}$
I've found the branch points to be $-1=e^{i*pi}$, $e^{\frac{i*pi}{3}}$ and $e^{\frac{-i*pi}{3}}$ - my reasoning behind this is that I think that the expression inside the cube root should be equal to 0, as I figure every other number would yield 3 specific 3th roots? However I'm not completely secure in this. I've inserted an illustration from Wolfram Alpha that matches my own drawing of where the branch points.
The textbook I'm using (Essential Mathematical Methods) is determining the branch cuts by considering what would happen to a function g(z) when a complete circuit is made either enclosing no, all or some of the branch points. However, I'm not sure how the authors calculate what is happening to g(z) as a circuit is made.
Thank you in advance,
