Fourier transform of the characteristic function
My qustion is about the Fourier transform of the characteristic function $\chi_{[0,1]}$. How can I find what it is? The problem is I got something really messy, so I think I didn't get it right.
$\endgroup$ 21 Answer
$\begingroup$Did you get this?
$$ \mathcal{F} \chi_{[0,1]} (\xi)= \int_{-\infty}^\infty \chi_{[0,1]}(x) e^{-2\pi ix\xi}dx = \int_{[0,1]} e^{-2\pi ix \xi} dx = \left[ \frac{e^{-2\pi ix \xi }}{-2\pi i\xi} \right]_0^1 = \frac{e^{-2\pi i \xi }}{-2\pi i\xi} - \frac{1}{-2\pi i\xi} = \frac{1 - e^{-2\pi i \xi}}{2\pi i\xi}$$
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