Time Differenciation property of the Fourier Transform

Electrical Circuit Analysis > Fourier Transform > Properties of the Fourier Transform

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Given that $ F(\omega)=\mathcal{F}[f(t)] $, then $$\mathcal{F}\left[f^{\prime}(t)\right]=j \omega F(\omega)$$ In other words, the transform of the derivative of $ f(t) $ is obtained by multiplying the transform of $ f(t) $ by $ j \omega $. By definition,

$$f(t)=\mathcal{F}^{-1}[F(\omega)]=\frac{1}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} d \omega$$

Taking the derivative of both sides with respect to $ t $ gives

$$f^{\prime}(t)=\frac{j \omega}{2 \pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} d \omega=j \omega \mathcal{F}^{-1}[F(\omega)] \tag{1}$$

or $$\mathcal{F}\left[f^{\prime}(t)\right]=j \omega F(\omega)$$ Repeated applications of Eq. (1) give $$\mathcal{F}\left[f^{(n)}(t)\right]=(j \omega)^{n} F(\omega)$$ For example, if $ f(t)=e^{-a t} $, then $$f^{\prime}(t)=-a e^{-a t}=-a f(t)$$ Taking the Fourier transforms of the first and last terms, we obtain

$$j \omega F(\omega)=-a F(\omega) \quad \Longrightarrow \quad F(\omega)=\frac{1}{a+j \omega}$$

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