Editing Rectangular function (section)

{{Short description|Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way}}
{{Redirect|Box function|the Conway box function|Minkowski's question-mark function#Conway box function}}
{{Use American English|date = March 2019}}
[[Image:Rectangular function.svg|300px|thumb|right|Rectangular function with a = 1]]

The '''rectangular function''' (also known as the '''rectangle function''', '''rect function''', '''Pi function''', '''Heaviside Pi function''',<ref>{{cite web |url=https://reference.wolfram.com/language/ref/HeavisidePi.html |title=HeavisidePi, Wolfram Language function |author=Wolfram Research |date=2008 |access-date=October 11, 2022}}</ref> '''gate function''', '''unit pulse''', or the '''normalized [[boxcar function]]''') is defined as<ref name="wolfram">{{MathWorld |title=Rectangle Function |id=RectangleFunction}}</ref>

<math display="block">\operatorname{rect}\left(\frac{t}{a}\right) = \Pi\left(\frac{t}{a}\right) =
\left\{\begin{array}{rl}
 0, & \text{if } |t| > \frac{a}{2} \\
 \frac{1}{2}, & \text{if } |t| = \frac{a}{2} \\
 1, & \text{if } |t| < \frac{a}{2}.
\end{array}\right.</math>

Alternative definitions of the function define <math display="inline">\operatorname{rect}\left(\pm\frac{1}{2}\right)</math> to be 0,<ref>{{Cite book |last=Wang |first=Ruye |title=Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis |pages=135–136 |publisher=Cambridge University Press |year=2012 |url=https://books.google.com/books?id=4KEKGjaiJn0C&pg=PA135 |isbn=9780521516884 }}</ref> 1,<ref>{{Cite book |last=Tang |first=K. T. |title=Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models |page=85 |publisher=Springer |year=2007 |url=https://books.google.com/books?id=gG-ybR3uIGsC&pg=PA85 |isbn=9783540446958 }}</ref><ref>{{Cite book |last=Kumar |first=A. Anand |title=Signals and Systems |publisher=PHI Learning Pvt. Ltd. |pages=258–260 |url=https://books.google.com/books?id=FGGa6BXhy3kC&pg=PA258 |isbn=9788120343108 |year=2011 }}</ref> or undefined.

Its periodic version is called a ''[[rectangular wave]]''.