= 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 $2\pi iz^m$ 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.
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