Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 4 / a

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Solution

With the paper's Fourier transform convention, the Fourier inversion theorem gives

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

This holds at continuity points under the usual inversion hypotheses; at a jump with one-sided limits it gives their mean. The factor

1/ (2 π)

and the sign change in the exponential are part of this normalization.

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