Chapter 18, Problem 18P

Given that $F\left(\omega \right)=\mathcal{F}\left[f\left(t\right)\right]$, prove the following results, using the definition of Fourier transform

1. (a) $\mathcal{F}\left[f\left(t-t_0\right)\right]=e^{-jw{t}_0}F\left(\omega \right)$
2. (b) $\mathcal{F}\left[\frac{df\left(t\right)}{dt}\right]=j\omega F\left(\omega \right)$
3. (c) $\mathcal{F}\left[f\left(-t\right)\right]=F\left(-\omega \right)$
4. (d) $\mathcal{F}\left[tf\left(t\right)\right]=j\frac{d}{d\omega }F\left(\omega \right)$