Chapter 17, Problem 12P

(a)

To determine

Prove that $F\left[\frac{df\left(t\right)}{dt}\right]=j\omega F\left(\omega \right)$, where $F\left(\omega \right)=F\left\{f\left(t\right)\right\}$.

(b)

To determine

Mention the restriction on $f\left(t\right)$, if the result shown in Part (a) is valid.

(c)

To determine

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