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Rectangular function

<p>The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized <a href="/facts/Boxcar_function/oPj0WQYJ">boxcar function</a>) is defined as
</p><p>
  
    
      
        rect
        ⁡
        
          (
          
            
              t
              a
            
          
          )
        
        =
        Π
        
          (
          
            
              t
              a
            
          
          )
        
        =
        
          {
          
            
              
                
                  0
                  ,
                
                
                  
                    if 
                  
                  
                    |
                  
                  t
                  
                    |
                  
                  >
                  
                    
                      a
                      2
                    
                  
                
              
              
                
                  
                    
                      1
                      2
                    
                  
                  ,
                
                
                  
                    if 
                  
                  
                    |
                  
                  t
                  
                    |
                  
                  =
                  
                    
                      a
                      2
                    
                  
                
              
              
                
                  1
                  ,
                
                
                  
                    if 
                  
                  
                    |
                  
                  t
                  
                    |
                  
                  <
                  
                    
                      a
                      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.}

</p><p>Alternative definitions of the function define 
  
    
      
        rect
        ⁡
        
          (
          
            ±
            
              
                1
                2
              
            
          
          )
        
      
    
    {\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}
  
 to be 0, 1, or undefined.
</p><p>Its periodic version is called a <i><a href="/facts/Rectangular_wave/CEPen7fY">rectangular wave</a></i>.
</p>