 Solution

ID: past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2021/s01/2/a/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

\oint_{Γ} f (z) d z = 2 πi j \sum Res (f, a_{j}) .

(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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