= Solution
One standard form of <Cauchy's residue theorem> is the following. Let $\Gamma$ be a <positively oriented contour> that is a <simple closed curve>, and let $f$ be a <holomorphic function> on an <open set> containing $\Gamma$ and its interior except for finitely many interior <isolated singularities> $z_1,\ldots,z_m$. There are no <singularities> on $\Gamma$. Then
$$
\boxed{\int_\Gamma f(z)\,dz=2\pi i\sum_{j=1}^m\operatorname{Res}(f,z_j).}
$$
The <residue> $\operatorname{Res}(f,z_j)$ is the coefficient of $(z-z_j)^{-1}$ in the <Laurent series> at $z_j$. Each interior <isolated singularity> contributes once because the <winding number> of this <positively oriented contour> about each such point is one. Reversing the <orientation of a contour> reverses the sign. This form of <Cauchy's residue theorem> applies to the rectangle below.
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