Dirac distribution

### 6.3.10 Dirac distribution

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],
0 | 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.

- Dirac it takes x, a real number.
- Dirac(x) returns ∞ if x=0, it returns 0 otherwise.

#### Examples

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

sin | ⎛ ⎝ | 0 | ⎞ ⎠

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

sin | ⎛ ⎝ | 1 | ⎞ ⎠