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# The Impulse Function

In each field of engineering, there is a question as to what will occur to the system
if a sudden but brief force is applied to a system. For example, when a charged
body is brought close to ground, at some point the charge will overcome the natural
resistance of the air and produce a spark. This produces a sudden and brief electric
force. Lightning is nothing more than a similar spark caused by the friction between
the air and the ground.

In general, such forces are referred to as *impulse forces* and we must
represent such forces mathematically.

## Derivation

Consider the function
where is a very small number. We can shift this function to start at
an arbitrary point by
. Some examples are shown in Figure 1.

Figure 1. , , and .

The integral of this function for any values of and may be calculated as follows:

.

By our definition, the area under the impulse is independent of
our choice of and therefore we can make
it arbitrarily small. In the limit, as ,
we are left with a function which is defined as

.

We will represent this infinite spike at
as an arrow, as shown in Figure 2.

Figure 2. The representation of an infinite spike at time .

The limit of this sequence of functions does not exist in the strict
definition of a *function*; however, it is still an object where the
integral is defined and that integral is 1.

This *bizarre* function was first conceived by the British
mathematical physicist Paul A.M. Dirac in 1925 in his investigations of quantum mechanics.
He used the symbol which is
today called the *Dirac delta* by physicists and called
the *impulse function* by engineers.

## The Integral of the Impusle Function

Next, ask yourself what is the integral of the impulse function
.
Because the area under the square wave is always one, the integral
must grow linearly from 0 to 1 along the non-zero region of the
square pulse:

.

This is shown in Figure 3 and it becomes clear that the integral
approaches the unit step function.

Figure 3. The integral of the peicewise constant approximations of
the impulse function.

While, in the limit, the value of the integral is undefined
at
, it is conventional to use the
unit impulse function and define

and

.

# The Sifting Property

One of the most significant uses of the impulse function is the ability
to *select* the value of a function at a point. Assuming that a
function is continuous at a point , it follows that

.

# Remark

When Dirac introduced his impulse function, he did not focus on the rigorous
aspects: it gave the correct results. Twenty years later, mathematicians
(e.g.,
Laurent Schwartz)
introduced the theory of distributions of which the impulse function is
an example.

# Remark

In a similar manner, Oliver Heaviside
pioneered the use of the Laplace
transform methods for solving linear differential equations which were
later made rigorous by Thomas Bromwich.

# Remark

It is not necessary to start with the step function used in the
above example. For example, one could use

which is symmetric around the origin, or a *tent* function as demonstrated
in Figure 4.

Figure 4. Three tent functions corresponding to the three functions
shown in Figure 1.