Rectangular function

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function,^[1] gate function, unit pulse, or the normalized boxcar function) is defined as^[2]

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

Alternative definitions of the function define ${\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}$ to be 0,^[3] 1,^[4][5] or undefined.

Its periodic version is called a rectangular wave.