 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2004/paper-1/3/a/solution

Use the Fourier transform convention

f (k) = \int_{- \infty}^{\infty} f (x) e^{- ik x} d x, f (x) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (k) e^{ik x} d k .

These hold directly under suitable integrability and inversion hypotheses, and extend to square-integrable functions through the Plancherel theorem. At a jump, ordinary Fourier inversion uses the mean of the one-sided limits under the usual piecewise smooth hypotheses.

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