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→Demonstration of validity: use tfrac |
→Fourier transform of the rectangular function: When using ordinary frequency f, the convention is to use the normalized sinc() function. And vise versa. |
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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 \mathrm{rect}(t)\cdot e^{-i 2\pi f t} \, dt
=\frac{\sin(\pi f)}{\pi f} = \mathrm{sinc}{(
using ordinary frequency {{mvar|f}}, where [[sinc function|<math>\mathrm{sinc}</math>]] is the normalized form of the [[sinc function]] and
<math display=block>\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \mathrm{rect}(t)\cdot e^{-i \omega t} \, dt
=\frac{1}{\sqrt{2\pi}}\cdot \frac{\mathrm{sin}\left(\omega/2 \right)}{\omega/2}
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