 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-4/30a/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Insert the uniformly convergent Fourier series of

g

into the interval formula and integrate termwise. The constant mode contributes

t g_{0}

, while for

k \neq = 0

\frac{1}{2} \int_{x - t}^{x + t} e^{ik s} d s = e^{ik x} \frac{s i n ( k t )}{k} .

The sum is exactly the expression from part (b), proving agreement of the transform and Fourier-series formulas.

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