I’m asking this question because I’m trying to write the series expansion of $\sqrt{\cos(z)}$ about $0$, and it seems problematic because its series expansions are not unique (if I’m doing the calculations correctly).
I guess this problem is linked to branch points. Since $\sqrt{z}$ has branch point at $0$ and infinity, I think the branch points of $\sqrt{\cos(z)}$ should be $\cos(z) = 0$, therefore don’t include $0$. Then why does $\sqrt{\cos(z)}$ have a non-unique series expansion about $0$, the point at which it’s analytic?