Im going through Bracewell: Fourier Transform and its Applications and I'm struggling with one part in particular (Chapter 5 the impluse symbol: pg 89-90 2nd ed).
$\textbf{Definition}$
Consider the class S of functions which possess derivatives of all orders at all points and which together with all the derivatives, die off at least as rapidly as $|x|^{-N}$ as $|x|\rightarrow\infty$, no matter how large $N$ may be. Members of this class are called $\textit{particularly well-behaved functions}$.
I'm struggling to show that the Gaussian is particularly well behaved:
$\frac{1}{2\sqrt{\pi \tau}}e^{-\frac{x^2}{4\tau}}.$
In the book he goes about showing the Fourier transform ($\mathscr{F}(s)$, say) of a particularly well-behaved function (call it $F(x)$) is particularly well behaved by differentiating $\mathscr{F}(s)$ $p$ times and then integrating by parts $N$ times and showing that $|\mathscr{F}(s)|\leq \frac{(2\pi)^{p-N}}{|s|}\int_{-\infty}^{\infty}|\frac{d^N}{dx^N}[x^pF(x)]|dx=O(|s|^{-N})$.
I've tried methods like that but have had no luck, any help would be great.