= Solution
Let $f$ be <holomorphic> on a neighbourhood of a positively oriented, piecewise smooth <simple closed curve> and its interior except at finitely many <isolated singularities> $a_j$ strictly inside. The <residue theorem> states
$$
\boxed{\oint_\Gamma f(z)\,dz=2\pi i\sum_j\operatorname{Res}(f,a_j).}
$$
The <residue> is the coefficient of $(z-a_j)^{-1}$ in the local <Laurent series>. No <singularity> may lie on the <contour>. For more general closed <contours>, each <residue> is multiplied by the corresponding <winding number>.
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