= Solution
The <residue> $\operatorname{res}(f;a)$ is \b[the coefficient $c_{-1}$ of $(z-a)^{-1}$] in the <Laurent series> at the isolated singularity. Equivalently it is $(2\pi i)^{-1}$ times the integral around a sufficiently small counterclockwise circle enclosing $a$.
The <residue theorem> states that if $f$ is holomorphic on a neighborhood of a positively oriented simple closed contour and its interior except for finitely many isolated singularities $a_j$ inside, then
$$
\boxed{\int_\gamma f(z)\,dz=2\pi i\sum_j\operatorname{res}(f;a_j).}
$$
No singularity may lie on the integration contour. A more general contour weights each residue by its winding number.
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