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Incoming links: Upper-half-plane indentation rule
Past exam of the mathematics course of the University of Cambridge 2021 ii Paper 1 7E Solution Created 2026-09-24 Updated 2026-09-25
Consider the meromorphic functionClose the contour with a large semicircle in the upper half-plane, indenting above the three real pole . The large-arc integral vanishes by Jordan's lemma, while the upper-half-plane indentation rule givesThe three residue areTheir sum is , so the residue theorem yields
The real part of the integrand is an odd function, so its full-line principal value is zero. Its imaginary part,is an even function. Taking imaginary parts and halving the full-line integral therefore gives