Editing Rectangular function (section)

==Fourier transform of the rectangular function==
[[File:Sinc_function_(normalized).svg|thumb|400px|right|Plot of normalized <math>\operatorname{sinc}(x)</math> function (i.e. <math>\operatorname{sinc}(\pi x)</math>) with its spectral frequency components.]]

The [[Fourier transform#Tables of important Fourier transforms|unitary Fourier transforms]] of the rectangular function are<ref name="wolfram"/>
<math display="block">\int_{-\infty}^\infty \operatorname{rect}(t)\cdot e^{-i 2\pi f t} \, dt
=\frac{\sin(\pi f)}{\pi f} = \operatorname{sinc}(\pi f) =\operatorname{sinc}_\pi(f),</math>
using ordinary frequency {{mvar|f}}, where [[sinc function|<math>\operatorname{sinc}_\pi</math>]] is the normalized form<ref>Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html</ref> of the [[sinc function]] and
<math display="block">\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \operatorname{rect}(t)\cdot e^{-i \omega t} \, dt
=\frac{1}{\sqrt{2\pi}}\cdot \frac{\sin\left(\omega/2 \right)}{\omega/2}
=\frac{1}{\sqrt{2\pi}} \cdot \operatorname{sinc}\left(\omega/2 \right),
</math>
using angular frequency <math>\omega</math>, where [[sinc function|<math>\operatorname{sinc}</math>]] is the unnormalized form of the [[sinc function]].

For <math>\operatorname{rect} (x/a)</math>, its Fourier transform is<math display="block">\int_{-\infty}^\infty \operatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\pi f t} \, dt
=a \frac{\sin(\pi af)}{\pi af} = a\ \operatorname{sinc}_\pi{(a f)}.</math>