Use the definition of the Fourier transform to prove the following results, where F{ f

Chapter 18, Problem 42

(choose chapter or problem)

Use the definition of the Fourier transform to prove the following results, where \(\mathcal{F}\{f(t)\}=\mathbf{F}(j \omega)\): (a) \(\mathcal{F}\left\{f\left(t-t_{0}\right)\right\}=e^{-j \omega t_{0}{ }^{2}} \mathcal{F}\{f(t)\}\); (b) \(\mathcal{F}\{d f(t) / d t\}=j \omega \mathcal{F}\{f(t)\}\); (c) \(\mathcal{F}\{f(k t)\}=(1 /|k|) \mathbf{F}(j \omega / k)\); (d) \(\mathcal{F}\{f(-t)\}=\mathbf{F}(-j \omega)\); (e) \(\mathcal{F}\{t f(t)\}=j d[\mathbf{F}(j \omega)] / d \omega\).

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