6.8.10 The Dirac distribution: Dirac

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

and, in fact,

The defining property of the Dirac distribution is that

and consequently

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

The Dirac command represents the Dirac distribution.

Examples.

• Input:
  int(Dirac(x)*sin(x),x,-1,2)
  Output:

  sin

  ⎛ ⎝

  0

  ⎞ ⎠

• Input:
  int(Dirac(x-1)*sin(x),x,-1,2)
  Output:

  sin

  ⎛ ⎝

  1

  ⎞ ⎠

If you have Dirac compute a value:

• Dirac it takes one argument:
  x, a real number.
• Dirac(x) returns ∞ if x=0, it returns 0 otherwise.