 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/4/ii/solution

Differentiate under the Fourier transform integral:

\frac{d ^{n} f}{d k ^{n}} = \int_{- \infty}^{\infty} (- i x)^{n} f (x) e^{- ik x} d x = (- i)^{n} g (k) .

g (k) = i^{n} \frac{d ^{n} f ( k )}{d k ^{n}} .

Absolute integrability of the required moments justifies differentiation under the integral. The identity also extends to tempered distributions.

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