How do I recognize branch points?

Question

For instance, $z^2$ - 1 has branch points at i and -i, but that doesn't seem obvious at all - and writing this function using the exponential and complex logarithm functions doesn't seem to help either.

Thanks,

Answer 1

$z^2-1$ does not have branch points; it has zeroes at $\pm1$. They are branch points of $\log(z^2-1)$, $\sqrt{z^2-1}$, ...

In general, $\log(f(z))$ will have branch points at the zeroes of $f$.