 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/5/a/i/solution

With the stated Fourier transform convention, the exponential transform is a shifted Dirac delta distribution:

f_{1} (ω) = 2 π δ (ω - ω_{0}) .

(e^{i ω_{0} t} - e^{- i ω_{0} t}) / (2 i)

then gives

f_{2} (ω) = \frac{π}{i} [δ (ω - ω_{0}) - δ (ω + ω_{0})] .

These are tempered distributions, since the time-domain functions are not integrable on the full real line. The second formula also correctly vanishes when

ω_{0} = 0

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