Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 1A (Vectors and Matrices)

For the stated angular-frequency Fourier transform convention, the convolution theorem givesThe Fourier inversion theorem and the assumed absolute integrability therefore imply

Since $|-i|=1$ and the principal argument is $-\pi /2$,All logarithms are $-\pi i/2+2\pi in$, $n\in \mathbb{Z}$.

The Fourier transform of a derivative givesSince $p^{\prime }=g$,An arbitrary additive constant in $p$ affects only the zero-frequency Dirac delta function when transforms are interpreted as distributions.

The values of $\log i$ are $i(\pi /2+2\pi n)$. HenceThese are infinitely many distinct positive real numbers.

Sincethe Inverse Fourier transforms of phase factors giveand

From $z=\tan w=-i(e^{2iw}-1)/(e^{2iw}+1)$ one obtainsFor $z=(2\sqrt{3}-3i)/7$ the quotient is $1+i\sqrt{3}=2e^{i\pi /3}$. Its principal logarithm is $\log 2+i\pi /3$, so the principal value is