The Dirac distribution: Dirac

### 5.9.10 The Dirac distribution: Dirac

The Dirac command takes as input a number.
Dirac returns infinity if the number is 0, it returns 0 otherwise.

Dirac represents the distribution which is the derivative of the Heaviside function. This means that

∫ |
∞
−∞
| Dirac(x) dx = 1

and, in fact, ∫ab Dirac(x) dx is 1 if [a,b] contains 0 and the integral is 0 otherwise. The defining property of the Dirac distribution is that

∫ |
∞
−∞
| Dirac(x) f(x) dx = f(0)

and consequently

∫ |
b
a
| Dirac(x−c) f(x) dx = f(c)

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