There is a question in my complex analysis textbook whose answer I don't understand, and cannot find a clear explanation online:
Find a branch cut $C_\tau = \{z=re^{i\tau} : r>0,\tau\in\mathbb{R}\}$ where $f(z)=\log_{\tau}(z^3 - 2)$ is holomorphic at $z=0$
The textbook claims that $f$ is holomorphic only if $-2 \notin C_\tau$. So any branch cut that is not $(-\infty, 0)$
But to me, it seems like the branch cut should be any cut not containing $z=2^{1/3}e^{2k\pi i /3}$, $k\in\mathbb{Z}$, since that is where $f(z)=\log_\tau (0)$.
Whichever the case, can someone give an end to end explanation??
We are only interested in $z$ close to $0$. If $-2 \notin C_{\tau}$ then $z^{3}-2 \notin C_{\tau}$ for $|z|$ sufficiently small so $f(z)$ is well defined and analytic.