| course |
course_year |
question_number |
tags |
title |
year |
Complex Methods |
IB |
13 |
|
Paper 4, Section II, A |
2012 |
State the convolution theorem for Fourier transforms.
The function $\phi(x, y)$ satisfies
$$\nabla^{2} \phi=0$$
on the half-plane $y \geqslant 0$, subject to the boundary conditions
$$\begin{gathered}
\phi \rightarrow 0 \text { as } y \rightarrow \infty \text { for all } x \\
\phi(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\
0, & |x|>1\end{cases}
\end{gathered}$$
Using Fourier transforms, show that
$$\phi(x, y)=\frac{y}{\pi} \int_{-1}^{1} \frac{1}{y^{2}+(x-t)^{2}} \mathrm{~d} t$$
and hence that
$$\phi(x, y)=\frac{1}{\pi}\left[\tan ^{-1}\left(\frac{1-x}{y}\right)+\tan ^{-1}\left(\frac{1+x}{y}\right)\right]$$