 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/4/c/solution

Since

γ > 0

, the causal damped sine has an absolutely convergent Fourier transform for real

ω

. Resolving the sine into complex exponentials gives

f (γ, p, ω) = \frac{1}{2 i} \int_{0}^{\infty} [e^{- [γ + i (ω - p)] t} - e^{- [γ + i (ω + p)] t}] d t = \frac{1}{2 i} [\frac{1}{γ + i ( ω - p )} - \frac{1}{γ + i ( ω + p )}] .

Combining the two fractions yields

f (γ, p, ω) = \frac{p}{( γ + iω ) ^{2} + p ^{2}} .

The formula also includes

p = 0

, for which both the original function and its transform vanish. Its sign reverses when the real parameter

p

reverses.

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