Question:
Derive the relationship
\(\int^t_{- \infty} f(\tau) d \tau \Leftrightarrow \frac{F(\omega)}{j \omega} + \pi F(0) \delta (\omega)\)
(where \(\Leftrightarrow\) means "Fourier transforms into").
Attempt:
I have already proved the relationship
\(\frac{dg(t)}{dt} \Leftrightarrow j \omega G( \omega)\)
so define \(h(t) = \frac{dg(t)}{dt}\). Then we have
\(\int^t_{- \infty} h(\tau) d \tau = \int^t_{- \infty} \frac{dg}{d \tau} d \tau = [g(\tau)]^t_{- \infty} = g(t) - g(-\infty)\)
so
\(g(t) = \int ^t _{- \infty} h(\tau) d \tau + g(- \infty)\)
Using the Fourier differentiation relationship above we get
\(\mathcal{F}[h(t)] = j \omega \mathcal{F}[g(t)] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau + g(- \infty) \right ] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega \mathcal{F}[g(- \infty)] \)
\(= j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega 2 \pi g(-\infty) \delta(\omega)\)
so
\(\mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] = \frac{\mathcal{F}[h(t)]}{j \omega} - 2 \pi g(-\infty) \delta(\omega)\)
I'm not sure if i've done something wrong here or if somehow this is equivalent to the correct relationship? Could someone help please
Thanks!
Jon.
Derive the relationship
\(\int^t_{- \infty} f(\tau) d \tau \Leftrightarrow \frac{F(\omega)}{j \omega} + \pi F(0) \delta (\omega)\)
(where \(\Leftrightarrow\) means "Fourier transforms into").
Attempt:
I have already proved the relationship
\(\frac{dg(t)}{dt} \Leftrightarrow j \omega G( \omega)\)
so define \(h(t) = \frac{dg(t)}{dt}\). Then we have
\(\int^t_{- \infty} h(\tau) d \tau = \int^t_{- \infty} \frac{dg}{d \tau} d \tau = [g(\tau)]^t_{- \infty} = g(t) - g(-\infty)\)
so
\(g(t) = \int ^t _{- \infty} h(\tau) d \tau + g(- \infty)\)
Using the Fourier differentiation relationship above we get
\(\mathcal{F}[h(t)] = j \omega \mathcal{F}[g(t)] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau + g(- \infty) \right ] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega \mathcal{F}[g(- \infty)] \)
\(= j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega 2 \pi g(-\infty) \delta(\omega)\)
so
\(\mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] = \frac{\mathcal{F}[h(t)]}{j \omega} - 2 \pi g(-\infty) \delta(\omega)\)
I'm not sure if i've done something wrong here or if somehow this is equivalent to the correct relationship? Could someone help please
Thanks!
Jon.
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