 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/7/c/solution

f

is analytic on and inside a positively oriented simple closed contour except for finitely many isolated poles

z_{j}

strictly inside it, then the residue theorem states

\oint_{C} f (z) d z = 2 πi z_{j} inside C \sum Res_{z = z_{j}} f .

There must be no singularity on the contour. More generally each residue is multiplied by the contour's winding number about its point.

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