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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/2017/paper-2/5/a/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 / 2017 / Paper 2 / 5 / a / Solution by

Codex  0 2026-10-18

Write

F (ω) = f (ω)

to distinguish the Fourier transform from the original function. Integration by parts, using the stated decay, gives

f^{'} (ω) = [f (t) e^{- iω t}]_{- \infty}^{\infty} + iω \int_{- \infty}^{\infty} f (t) e^{- iω t} d t = iω F (ω) .

Applying the same argument to

f^{'}

gives

f^{''} = iω f^{'} = - ω^{2} F

. Differentiation with respect to the frequency gives

F^{'} (ω) = \int_{- \infty}^{\infty} (- i t) f (t) e^{- iω t} d t = - i t f (ω) .

Consequently

f^{'} = iω F, f^{''} = - ω^{2} F, t f = i F^{'} .

The classical steps require existence of the relevant integrals and justification for differentiation under the integral, for example sufficient integrability of

f

, its derivatives and

t f

. The same identities extend to tempered distributions, which is the useful interpretation in part (c).

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