Dirac Delta Function

The Heaviside function represents switches from one value to another at some point, but what if we need an instant change to a very big value? This is what the Dirac delta function means.

The Dirac delta function is defined as follows: `delta(t-c)={(+oo if t=c),(0 if t!=c):}`,

with the following properties:

  1. `int_-oo^oo delta(t-c)dt=1`
  2. `int_-oo^oo f(t)delta(t-c)dt=f(c)`

This is a very strange function. It is zero everywhere except one point, but anyway the integral containing this one point equals 1. The Dirac delta is not a function in the traditional sense. It is instead an example of something called a generalized function or distribution. However, it is very good for modelling instant big changes.

Note that the first and second properties are true for any interval that contains `c` and on which `c` is not an endpoint: `int_(c-tau)^(c+tau) delta(t-c)dt=1`, and `int_(c-tau)^(c+tau) f(t)delta(t-c)dt=f(c)`, where `tau>0`.

Now, let's see what the Laplace transform of the Dirac function is (this can be calculated easily using the second property):

`L(delta(t-c))=int_0^oo e^(-st)delta(t-c)dt=e^(-ct)`, provided `c>0`.

Let's see another fact about the Dirac function. For this, use the fact that `int_-oo^t delta(tau-c)dtau={(1 if t>c),(0 if t<c):}` and note that this is exactly the definition of the Heaviside function:

`int_-oo^t delta(tau-c)dtau=u_c(t)`

Now, using the fundamental theorem of calculus: `d/(dt)u_c(t)=delta(t-c)`.