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The unitary Fourier transforms of the rectangular function arecite-ref-wolfram-2-1[2] ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t ) ⋅ ⋅ e − − i 2 π π f t d t = sin ⁡ ⁡ ( π π f ) π π f = sinc ⁡ ⁡ ( π π f ) = sinc π π ⁡ ⁡ ( f ) , {\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} (\pi f)=\operatorname {sinc} _{\pi }(f),} using ordinary frequency f, where sinc π {\displaystyle \operatorname {sinc} _{\pi }} is the normalized formcite-ref-10[10] of the sinc function and 1 2 π π ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t ) ⋅ ⋅ e − − i ω ω t d t = 1 2 π π ⋅ ⋅ sin ⁡ ⁡ ( ω ω / 2 ) ω ω / 2 = 1 2 π π ⋅ ⋅ sinc ⁡ ⁡ ( ω ω / 2 ) , {\displaystyle {\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),} using angular frequency ω ω {\displaystyle \omega } , where sinc {\displaystyle \operatorname {sinc} } is the unnormalized form of the sinc function.

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