The Dirac distribution: Dirac

### 5.8.10 The Dirac distribution: Dirac

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

∫ |
∞
−∞
| δ(x) dx = 1

and, in fact,

∫ |
b
a
| δ(x) dx = |
⎧ ⎨ ⎩ |
1 | if 0 ∈ [a,b]
1 | otherwise

The defining property of the Dirac distribution is that

| ∫ |
∞
−∞
| δ(x) f(x) dx = f(0)

and consequently

| ∫ |
b
a
| δ(x−c)f(x) dx = f(c)

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.