A.1 The Dirac Impulse

Here we describe the Dirac impulse function (also known as the Dirac delta function).

Let us consider a symmetric function which decreases quickly, has its maximal valueat x = 0 and is such that:

In other words, the area under the graph is equal to 1. The functions ^[1]:

satisfy these conditions. They are depicted in Figure A.1. Let us now consider the following transformations:

Figure A.1: Surrounding functions of the Dirac impulse

For the function h ( x), the effect of these transformations is shown in Figure A.2.

Figure A.2: Effect of the transformations mh( x) and h( mx) on h( x)

In this case the integral (the area between the graphical representation of this function and the x axis) remains unchanged:

A question now arises. What happens when a ? ?? For the function ag ( ax), we obtain:

which decreases indefinitely as a increases. If we now consider the transformed function h ( x), we obtain:

This function also decreases as a increases, but it decreases more quickly than ag ( ax). Therefore, by increasing the parameter a indefinitely, we obtain a function which is equal to zero for all values of x such that x <...