 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2008/paper-1/4/a/i/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The even extension is the triangular function

f (x) = (1 - ∣ x ∣)

for

∣ x ∣ < 1

and zero outside. Its Fourier transform is real and even:

f (k) = 2 \int_{0}^{1} (1 - x) cos (k x) d x, f (k) = \frac{2 ( 1 - cos k )}{k ^{2}} = (\frac{sin ( k /2 )}{k /2})^{2} .

At zero the removable limiting value is one, also obtained directly by integrating the triangle. This is the Fourier transform of a triangular function.

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