(a) Given a function $f(x)$, the fourier transform of the function is defined by
$\tilde{f}(k) = \int dx e^{ikx} f(x)$ (4)
Given this definition, show our original function is given by
$f(x) = \int \frac{dk}{2\pi} \tilde{f}(k)e^{-ikx}$ (5)
1
I.e. show we can apply reproduce the original function from $\tilde{f}(k)$.
(b) [challenge] Show that our $\delta$-function, $\delta(x)$ obeys
$\delta(ax) = \frac{1}{|a|}\delta(x)$ (6)
for any constant a. Hint: the $\delta$-function is always understood to be integrated with a test
function.
(c) [challenge] If $\tilde{f}(k)$ is the fourier transform of $f(x)$, write an expression for fourier trans-
form of
$g(x) = f(ax)$, (7)
where $a \neq 0$ is a constant, in terms of $f(x)$. I.e. what is $\tilde{g}(k)$ in terms of a and $\tilde{f}(k)$.
