 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/5/a/i/solution

f (s) = \int_{- \infty}^{\infty} f (x) e^{- i s x} d x .

For

f, f^{'} \in L^{1} (R)

f^{'} (s) = i s f (s) .

For example, this applies when

f

C^{1}

and both displayed functions are integrable. These assumptions make integration by parts valid and the endpoint terms vanish.

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