= Solution
Cauchy's theorem says that a <holomorphic function> integrates to zero around a closed <contour> homologous to zero in its domain. Thus deforming one <contour> into another without crossing a singularity preserves the <integral>. More generally for a <meromorphic> $f$, the exact condition is
$$
\boxed{\sum_a[n_\Gamma(a)-n_{\Gamma'}(a)]\operatorname{Res}_a f=0.}
$$
Equal <Winding number>[winding numbers] about every singularity are sufficient. They are not necessary for one particular function because <Residue>[residues] can cancel; even arbitrary <Contour>[contours] have equal <Integral>[integrals] for $f=0$. A topological deformation criterion must not be confused with a necessary condition for accidental equality.
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