← Lecture 16 | Lecture 18 →

We have already used the unit step function which
can be used to act as a switch for a single or as a switch, double-switch, or delay.

Using it as a switch, we turn a function *on* at some time by using
multiplication; for example, Figure 1 shows a function
and
.

Figure 1. The function .

If , we can create a double switch which turns the function
on at time
and off again at time
; for example, Figure 2 shows a function
and
.

Figure 2. The function .

To delay a signal, both the function and the unit step function must be delayed; for
example, Figure 3 shows a function and
.

Figure 3. The function .

Because these functions arise so often in engineering, we require their Laplace transforms:

Finally, the *second shift theorem* states that if , it follows
that

.

The proofs are straight forward and are given in AMEM.

# Periodic Functions

It is possible to use the unit step functions to define periodic functions other than the usual trigonometric
functions. For example, Figure 4 shows the function

.

We may now take the Laplace transform of this infinite series simply by applying it term-by-term and each term is simply a shifted
unit step function. Therefore, we have

.

If we factor out , we can write this sum as

or

.

This is a geometric series where in this case
and therefore, we have that

.

This is a special case of the following more general theorem:

# Theorem

If is defined for all and it is a periodic function
with period , then

.

## Example

Give the example we previously looked at, with
.
We must therefore calculate

and therefore the Laplace transform is

as before.

Further examples are in the assignments.