 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-1/14d/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Split the exponential kernel at zero. For

a > 0

f (k) = \int_{0}^{\infty} e^{- (a + ik) x} d x + \int_{- \infty}^{0} e^{(a - ik) x} d x = \frac{1}{a + ik} + \frac{1}{a - ik} .

Thus its Fourier transform is

f (k) = \frac{2 a}{a ^{2} + k ^{2}} .

Both half-line integrals converge absolutely because

a > 0

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