Couple of Important Mathematical Functions (and their Fourier transforms)

• Introducing the rect function
  

  $${\rm rect}(z) \equiv \left\{\begin{array}{cl} 0 & z<-1/2 \\ 1 & -1/2 \leq z \leq1/2 \\ 0 & z>1/2 \end{array}\right.$$ (9.14)

  
  
  and its Fourier transform the sinc function
  

  $$sinc(z) =\frac{\sin z}{z}$$ (9.15)

  
  

  
  
  
  
  
  
  
  

  and the delta function defined in terms of its Fourier transform
  

  $$\delta(z) = \int_{-\infty}^{\infty} dke^{i2\pi kz} = \left\... ...; z \neq 0 \\ 1\; \mbox{if}\; z = 0 \\ \end{array} \right.$$ (9.16)