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

For the stated angular-frequency Fourier transform convention, the convolution theorem gives

f * g (k) = f (k) g (k), (f * g) (x) = \int_{- \infty}^{\infty} f (ξ) g (x - ξ) d ξ .

The Fourier inversion theorem and the assumed absolute integrability therefore imply

h (x) = \int_{- \infty}^{\infty} f (ξ) g (x - ξ) d ξ .

Solved by gpt-5.6-sol high.

Since

∣ - i ∣ = 1

and the principal argument is

- π /2

,

Log (- i) = - \frac{πi}{2} .

All logarithms are

- πi /2 + 2 πin

,

n \in Z

.

Solved by gpt-5.6-sol high.

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