How should I go about seeking the Fourier transform for the piecewise function:
$$f(x) = \left\{\begin{matrix} 0 ,&|x|>a \\ 1 ,&|x|<a \end{matrix}\right.$$
Is this the correct attempt?
$$\frac{1}{2\pi}\int_{-a}^{0}1e^{i\omega x}.dx+\frac{1}{2\pi}\int_{0}^{a}0e^{i \omega x}.dx$$?
$\endgroup$ 111 Answer
$\begingroup$The function in questions is $1$ on $[-a,a]$ and $0$ elsewhere. So the Fourier transform of this function is $$ \frac{1}{\sqrt{2\pi}}\int_{-a}^{a}e^{-isx}dx = \left.\frac{1}{\sqrt{2\pi}}\frac{e^{-isx}}{-is}\right|_{x=-a}^{x=a} = \frac{e^{isa}-e^{-isa}}{\sqrt{2\pi}is}=\sqrt{\frac{2}{\pi}}\frac{\sin(sa)}{s} $$ This is the "sinc" function, and you'll want to become familiar with this functon.
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