 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 1

Since
$$
\mathcal F^{-1}[e^{ika}]=\delta(x+a),
\qquad
\mathcal F^{-1}[e^{-ika}]=\delta(x-a),
$$
the <Inverse Fourier transforms of phase factors> give
$$
\boxed{
\mathcal F^{-1}[\cos(ka)]
=\frac12\bigl[\delta(x+a)+\delta(x-a)\bigr]
}
$$
and
$$
\boxed{
\mathcal F^{-1}[\sin(ka)]
=\frac1{2i}\bigl[\delta(x+a)-\delta(x-a)\bigr].
}
$$

From $z=\tan w=-i(e^{2iw}-1)/(e^{2iw}+1)$ one obtains
$$
e^{2iw}=\frac{1+iz}{1-iz},
\qquad
w=\frac1{2i}\log\frac{1+iz}{1-iz}.
$$
For $z=(2\sqrt3-3i)/7$ the quotient is $1+i\sqrt3=2e^{i\pi/3}$. Its principal logarithm is $\log2+i\pi/3$, so the principal value is
$$
\boxed{\tan^{-1}z=\frac\pi6-\frac i2\log2.}
$$

Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)