Dirac Delta Function

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

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 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 first and second properties are true for any interval that contains c and where c is not 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 is Laplace transform of dirac function (this can be calculated easily using 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 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 definiiton of Heaviside function:

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

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