 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/5/a/solution

Use the Fourier transform convention

h (k) = \int_{R} e^{- ik x} h (x) d x

. Split at zero and integrate the decaying exponentials:

h (k) = \frac{1}{L ^{- 1} - ik} + \frac{1}{L ^{- 1} + ik} = \frac{2 L}{1 + L ^{2} k ^{2}} .

The transform is real and even because

h (x) = e^{- ∣ x ∣/ L}

is real and even.

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