Rectangular function

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

[\mathrm(x) = \sqcap(x) = \begin0 & \mbox |x| > \frac \\[3pt]\frac & \mbox |x| = \frac \\[3ptΣ & \mbox |x| < \frac\end ]

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

[\mathrm(x) = u \left( x + \frac \right) - u \left( x - \frac \right) ]

or, alternatively:

[\mathrm(x) = u \left( x + \frac \right) \cdot u \left( \frac - x \right) ]

The rectangular function is normalized:

[\int_^\infty \mathrm(x)\,dx=1]

The unitary Fourier transforms of the rectangular function are:

[\frac}\int_^\infty \mathrm(t)\cdot e^ \, dt=\frac}\cdot \mathrm\left(\frac\right)],

in terms of the normalized sinc function.

[\int_^\infty \mathrm(t)\cdot e^ \, dt= \mathrm(f)]

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

[\varphi(k) = \frac\,]

and its moment generating function is:

[M(k)=\frac(k/2)}\,]

where "sinh" is the hyperbolic sine function.

See also