$\newcommand{\const}{\mathrm{const}}$ $\newcommand{\erf}{\operatorname{erf}}$
Some of the problems in this assignment could be solved based on the other problems and properties of Fourier transform (see Lecture 16) and such solutions are much shorter than from the scratch; seeing and exploiting connections is a plus.
Let $\alpha>0$. Find Fourier transforms of
Let $\alpha>0$. Find Fourier transforms of
Let $\alpha>0$. Based on Fourier transform of $e^{-\alpha x^2/2}$ find Fourier transforms of
Find Fourier transforms of
$f(x)=\left\{\begin{aligned} & 1&& |x|\le a,\\ & 0 && |x|\ge a;\end{aligned}\right.$
$f(x)=\left\{\begin{aligned} & x && |x|\le a,\\ & 0 && |x|\ge a;\end{aligned}\right.$
Using (a) calculate $\int_{-\infty}^\infty \frac{\sin (x)}{x}\,dx$.