 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-2/12a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Close the contour in the upper half-plane. The degree assumption and Jordan lemma make the integral over the large semicircle tend to zero, while the absence of real zeros avoids indentations of the contour. The residue theorem therefore gives

\int_{- \infty}^{\infty} R (x) e^{i x} d x = 2 πi Q (z_{k}) = 0 Im z_{k} > 0 \sum Res_{z = z_{k}} (R (z) e^{i z}) .

Repeated roots are handled by the usual higher-order residue formula.

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