The rectangular function (also known as the rectangle function, rect function or the normalized boxcar function) is defined as

$\backslash mathrm\{rect\}(x)\; =\; \backslash sqcap(x)\; =\; \backslash begin\{cases\}$

0 & \mbox{if } |x| > \frac{1}{2} \\^* \frac{1}{2} & \mbox{if } |x| = \frac{1}{2} \\^* 1 & \mbox{if } |x| < \frac{1}{2} \end{cases}

or in terms of the Heaviside step function, u(t):

$\backslash mathrm\{rect\}(x)\; =\; u\; \backslash left(\; x\; +\; \backslash frac\{1\}\{2\}\; \backslash right)\; -\; u\; \backslash left(\; x\; -\; \backslash frac\{1\}\{2\}\; \backslash right)$

or, alternatively:

$\backslash mathrm\{rect\}(x)\; =\; u\; \backslash left(\; x\; +\; \backslash frac\{1\}\{2\}\; \backslash right)\; \backslash cdot\; u\; \backslash left(\; \backslash frac\{1\}\{2\}\; -\; x\; \backslash right)$

The rectangular function is normalized:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash mathrm\{rect\}(x)\backslash ,dx=1$

The unitary Fourier transforms of the rectangular function are:

$\backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash mathrm\{rect\}(t)\backslash cdot\; e^\{-i\; \backslash omega\; t\}\; \backslash ,\; dt$

=\frac{1}{\sqrt{2\pi}}\cdot \mathrm{sinc}\left(\frac{\omega}{2\pi}\right), in terms of the normalized sinc function.

$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash mathrm\{rect\}(t)\backslash cdot\; e^\{-i\; 2\backslash pi\; f\; t\}\; \backslash ,\; dt$

= \mathrm{sinc}(f)

Viewing the rectangular function as a probability distribution function, its characteristic function is therefore written

$\backslash varphi(k)\; =\; \backslash frac\{\backslash sin(k/2)\}\{k/2\}\backslash ,$

and its moment generating function is:

$M(k)=\backslash frac\{\backslash mathrm\{sinh\}(k/2)\}\{k/2\}\backslash ,$

where "sinh" is the hyperbolic sine function.

See also

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• Fourier transform
• Square wave
• Triangular function

Elementary special functions