 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-4/14d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

f (t) = \frac{1}{2 πi} \int_{γ - i \infty}^{γ + i \infty} \frac{e ^{s t}}{s ^{2}} d s, γ > 0.

For

t > 0

, close the Bromwich contour in the left half-plane. The only enclosed singularity is the double pole at

s = 0

, whose residue is

Res_{s = 0} \frac{e ^{s t}}{s ^{2}} = \frac{d}{d s} e^{s t}_{s = 0} = t .

The residue theorem therefore yields

L^{- 1} {s^{- 2}} (t) = t .

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