 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-4/14b/b/solution

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

The standard contour transform

\int_{- \infty}^{\infty} \frac{x}{x ^{2} + α ^{2}} e^{- ik x} d x = - iπ sgn (k) e^{- α ∣ k ∣}

holds for

Re α > 0

, with the appropriate upper or lower pole chosen according to the sign of

k

. Therefore

m (x) = \frac{x}{x ^{2} + α ^{2}} .

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