Square-root branch point ($m=2$)

A square-root branch point has ramification index $2$: one circuit switches local branches and two restore the original branch. Both zero and infinity are square-root branch points of $z^{-1/2}$, since the coordinate $u=1/z$ changes the latter local expression to $u^{1/2}$. A finite limiting value at a branch point does not imply a removable singularity.