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, for c∈[a,b],

| ∫ |
b
a
| δ(x−c)f(x) dx=f(c).

The Dirac command represents the Dirac distribution.

- Dirac takes one mandatory argument and one optional argument:
  - x, a symbol or an expression.
  - Optionally, n, a nonnegative integer.
- Dirac(x ⟨,n ⟩) returns δ(n)(x), where δ(n) is the nth derivative of δ.

Note that x can be a real number, for which Dirac returns 0 if x≠ 0 and ∞ otherwise. However, since δ is a distribution, not a function, computing its value at a point makes little sense.

#### Examples

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

sin | ⎛ ⎝ | 1 | ⎞ ⎠

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

−cos | ⎛ ⎝ | 1 | ⎞ ⎠