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

With the paper's Fourier transform convention, the Fourier inversion theorem givesThis holds at continuity points under the usual inversion hypotheses; at a jump with one-sided limits it gives their mean. The factor $1/(2\pi )$ and the sign change in the exponential are part of this normalization.

Insert the inverse Fourier transforms of $f$ and its complex conjugate, initially under enough decay to justify interchanging integrals:The middle step uses the stated Dirac delta distribution identity, and the last uses its sampling property. This is the Parseval identity for the chosen normalization. Approximation extends the result to square-integrable functions, including the function used below.

The real odd function has a purely imaginary odd Fourier transform. Setting $A=\pi /2$ and using a product-to-sum identity,The apparent singularities are removable. Direct evaluation or limits giveIn particular, the transform stays finite where its numerator and denominator both vanish.

The squared norm of the original function isApplying the Parseval identity to part (c), using the evenness of the squared transform, givesThus the last integral equals ${\pi }^2/8$. Now set $u=\pi k/2$. Sincewe obtainThe apparent singularity at $u=\pi /2$ is removable: the continuous value of the integrand there is $1/4$. No principal-value interpretation is needed.