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6.3.10  Dirac distribution

The Dirac δ distribution is the distributional derivative of the Heaviside function. This means that

  
+∞


−∞
 δ(xdx=1

and, in fact,

  
b


a
 δ(xdx


    1if  0 ∈ [a,b],
    0otherwise.

The defining property of the Dirac distribution is that

  
+∞


−∞
 δ(xf(xdx=f(0)

and consequently

  
b


a
 δ(xc)f(xdx=f(c)

as long as c is in [a,b].

The Dirac command represents the Dirac distribution.

Examples

int(Dirac(x)*sin(x),x,-1,2)
     
sin
0
          
int(Dirac(x-1)*sin(x),x,-1,2)
     
sin
1
          

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