course |
course_year |
question_number |
tags |
title |
year |
Methods |
IB |
37 |
|
1.II.14D |
2007 |
Define the Fourier transform $\tilde{f}(k)$ of a function $f(x)$ that tends to zero as $|x| \rightarrow \infty$, and state the inversion theorem. State and prove the convolution theorem.
Calculate the Fourier transforms of
Hence show that
$$\int_{-\infty}^{\infty} \frac{\sin (b k) e^{i k x}}{k\left(a^{2}+k^{2}\right)} d k=\frac{\pi \sinh (a b)}{a^{2}} e^{-a x} \quad \text { for } \quad x>b$$
and evaluate this integral for all other (real) values of $x$.