Question

What does the residue theorem state?

Answer 1

Suppose that \(C\) is a simple closed contour contained in the interior of a simply connected domain \(D\). If \(f\) is analytic everywhere on and within \(C\), except possibly for at a finite number of isolated singularities (call them \(\left.z_{1}, \ldots, z_{n}\right)\) inside \(C\), then
\[
\oint_{C} f(z) d z=2 \pi i \sum_{k=1}^{n} \operatorname{Res}\left(f(z), z_{k}\right)
\]