 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/iii/solution

Let

A (k)

e^{- x^{2}}

. Since

(e^{- x^{2}})^{'} = - 2 x e^{- x^{2}}

F [x e^{- x^{2}}] = - (ik /2) A (k)

. Differentiation under the integral then yields

A^{'} (k) = - (k /2) A (k)

. The Gaussian integral supplies

A (0) = π

, so

A (k) = π e^{- k^{2} /4} .

Applying part (ii) with

n = 1

gives

F [x e^{- x^{2}}] (k) = i A^{'} (k) = - \frac{i π}{2} k e^{- k^{2} /4} .

It is imaginary and odd, as expected for the transform of a real odd function.

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