Algebraic branch point ($w(z)={\sum }_{j=j_0}^{\infty }{c}_j(z-a{)}^{j/k}$)

At an algebraic branch point, a local solution becomes meromorphic after the finite substitution $z-a=t^k$. The displayed convergent expansion has only finitely many negative terms. A square-root branch point corresponds to $k=2$; an essential singularity or logarithmic singularity need not become meromorphic under any such finite substitution.

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.