The rectangular function (also known as the rectangle function or
the normalized boxcar function
Rectangular function Rectangular function
r
e
c
t
(
t
)
=
{
0
if
|
t
|
>
1
2
1
2
if
|
t
|
=
1
2
1
if
|
t
|
<
1
2
{\displaystyle \mathbf {rect} (t)=\left\{{\begin{matrix}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{matrix}}\right.}
or in terms of the Heaviside step function
r
e
c
t
(
t
)
=
H
(
t
+
1
/
2
)
−
H
(
t
−
1
/
2
)
{\displaystyle \mathbf {rect} (t)=\mathbf {H} (t+1/2)-\mathbf {H} (t-1/2)}
The rectangular function is normalized:
∫
−
∞
∞
rect
(
x
)
d
x
=
1
{\displaystyle \int _{-\infty }^{\infty }{\textrm {rect}}(x)\,dx=1}
The Fourier transform of the rectangular function is
1
2
π
∫
−
∞
∞
rect
(
x
)
e
−
i
k
x
d
x
=
sinc
(
k
/
2
)
2
π
{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\textrm {rect}}(x)e^{-ikx}\,dx={\frac {{\textrm {sinc}}(k/2)}{\sqrt {2\pi }}}}
where "sinc" is the sinc function . Viewing the rectangular function as a probability distribution function, its characteristic function is therefore written
φ
(
k
)
=
sinc
(
k
/
2
)
{\displaystyle \varphi (k)={\textrm {sinc}}(k/2)\,}
and its moment generating function is:
M
(
k
)
=
sinh
(
k
/
2
)
k
/
2
{\displaystyle M(k)={\frac {{\textrm {sinh}}(k/2)}{k/2}}\,}
where "sinh" is the hyperbolic sine function.
See also 
![{\displaystyle \mathbf {rect} (t)=\left\{{\begin{matrix}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{matrix}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15501a740986e0b04ab275d33fa63b3054a733e1)
