I have a question about branch cuts. Suppose you have $f(z) = \sqrt{z^2 -1} $. Then the branch points are $ \pm 1$, so we can make a branch cut from $ (- \infty , 1]$ in order to define $f$ continuously. Do we also say that $f$ has a branch point at $ \infty$? Isn't so that a branch cut need to be taken between two branch points or have I misunderstood?
Another thing, if $p(x) $ is a polynomial, and $f(z) = \sqrt{p(z)} $ is it true the the branch points of $f$ is the same as the zeros of the polynomials? If not, what is the connection?
Thanks