 Solution

ID: 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/solution

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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