I'd like todetermine a branch cut for the function $w=\left(\dfrac{z+1}{z-1}\right)^{1/3}$ that allows to construct analytic branches defined on $|z|>1 \;,\; \forall z\in \mathbb C$. How can I do this?
I noticed: branch points are $z=1$ and $z=-1$. So I'd say: $[-1,1]$ is the branch cut needed. Is this correct?
Note that the quotient $$\frac{z+1}{z-1} $$ is real and nonpositive only on the interval $[-1,1]$. Thus , it is possible to define an analytic branch of $$\log \left( \frac{z+1}{z-1} \right) $$ in $\mathbb C \setminus [-1,1]$ where the imaginary part lies in $(-\pi,\pi)$.
Then $$\exp \left( \frac{1}{3} \log \left(\frac{z+1}{z-1} \right) \right) $$ provides the desired analytic branch.
In short - you are right.