 Solution

ID: past-exam-of-the-algebra-and-analysis-ii-for-mathematics-and-philosophy-course-of-the-university-of-oxford/2013/5/b/solution

The residue theorem states that if a function is holomorphic on and inside a positively oriented simple closed contour except at finitely many isolated singularities

a_{1}, \dots, a_{N}

in its interior, then

\int_{γ} f (z) d z = 2 πi j = 1 \sum N res (f; a_{j}) .

No singularity may lie on the contour. More generally, a closed contour has a winding-number-weighted sum of residues, provided it and its enclosed winding regions lie in the domain on which the isolated singularities are specified.

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