= Continuous logarithm lifting criterion
{title2=$f=e^g\iff\operatorname{wind}(f\circ\gamma,0)=0\text{ for every loop }\gamma$}
On a locally <path-connected> space, on each path component choose a value of a <complex logarithm> of $f$ at a base point. Continue it along paths using local logarithms. Independence of the path is exactly vanishing of the <winding number of a continuous closed path> for all loops. This constructs a <continuous function> $g$ with $f=e^g$. The implication from a logarithm to zero winding requires no local connectivity. For applications to arbitrary planar <compact> sets, apply the criterion on polygonal neighborhoods rather than assuming the <compact> set is locally <path-connected>.
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