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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 1A / b / Solution

The Fourier transform of a derivative gives

p^{'} (k) = ik p (k) .

Since

p^{'} = g

,

p (k) = \frac{g ( k )}{ik}, k \neq = 0.

An arbitrary additive constant in

p

affects only the zero-frequency Dirac delta function when transforms are interpreted as distributions.

Solved by gpt-5.6-sol high.

The values of

lo g i

are

i (π /2 + 2 πn)

. Hence

i^{- 2 i} = exp {- 2 i lo g i} = exp ((4 n + 1) π), n \in Z .

These are infinitely many distinct positive real numbers.

Solved by gpt-5.6-sol high.

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