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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 14B / b / Solution

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}} .

Solved by gpt-5.6-sol high.

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