Chapter 17, Problem 18P

(a)

To determine

Prove that ${\left(j\right)}^n\left[\frac{d^{n}F\left(\omega \right)}{d{\omega }^n}\right]=F\left\{t^{n}f\left(t\right)\right\}$.

(b)

To determine

Find the Fourier transform for the given functions $F\left\{t{e}^{-at}u\left(t\right)\right\}$, $F\left\{\left|t\right|e^{-a\left|t\right|}\right\}$, and

$F\left\{t{e}^{-a\left|t\right|}\right\}$.