Logarithmic branch point

The complex logarithm has infinite-sheeted branching: continuing once around its zero changes its value by $2\pi i$. Multiplication by a function vanishing at that point need not remove the branching. For example $z^m\log z$ with positive integer $m$ tends to zero but changes by $2\pi i{z}^m$ after continuation around zero. This remains a branch point, rather than a removable singularity, because no single-valued holomorphic function exists on a full punctured neighbourhood. The coordinate $w=1/z$ tests logarithmic branching at infinity in the same way.