Proving the Dirac Delta Function, Etc.

In this post we prove the Dirac Delta function and its sampling property. Here are the variables we need:

So we ask, is equation 1 below true?

To prove equation 1, we first define the Dirac Delta function:

And it helps if we draw a diagram:

In the diagram above, we have a line segment along the x axis ranging from minus infinity to infinity. If we let the value of epsilon go to infinity (or assume an infinite number of points along any epsilon distance) we can make a substitution and create the diagram below:

At line 3 we create a new integral that is equivalent to the one we started with. From there to line 5 we show that the integral does indeed equal 1.

Now let's prove the sampling property. Suppose a particle is located at position 'a' instead of zero. Is the value of the integral f(a)?

We assume the following are true:

We make some changes to equation 6 to get 7:

We draw a new diagram to account for the fact that x equals 'a' instead of zero:

We can now derive equation 8. From there we derive the calculus difference quotient or derivative formula at line 14.

Line 15 above confirms the Dirac Delta function's sampling property.