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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/4/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 4 / b / Solution by

Codex  0 2026-10-18

Use the trigonometric identity

2 cos^{2} (a x) = 1 + cos (2 a x)

and express the cosine as a sum of two complex exponentials. The Fourier transform of each exponential is a shifted Dirac delta distribution:

F [1] = 2 π δ (k), F [e^{\pm 2 ia x}] = 2 π δ (k \mp 2 a) .

(1)

By linearity,

f (k) = 2 π δ (k) + π δ (k - 2 a) + π δ (k + 2 a) .

(2)

The original function does not decay and is not absolutely integrable on the real line, so this answer is a tempered distribution, rather than an ordinary convergent Fourier integral. Its three lines represent the constant part and the two opposite spatial frequencies.

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