28
PiecewiseDefined Functions and
Periodic Functions
At the start of our study of the Laplace transform, it was claimed that the Laplace transform is
“particularly useful when dealing with nonhomogeneous equations in which the forcing func
tions are not continuous”. Thus far, however, we’ve done precious little with any discontinuous
functions other than step functions. Let us now rectify the situation by looking at the sort of
discontinuous functions (and, more generally, “piecewisedefined” functions) that often arise in
applications, and develop tools and skills for dealing with these functions.
We will also take a brief look at transforms of periodic functions other than sines and cosines.
As you will see, many of these functions are, themselves, piecewise defined. And finally, we
will use some of the material we’ve recently developed to reexamine the issue of resonance in
mass/spring systems.
28.1
PiecewiseDefined Functions
PiecewiseDefined Functions, Defined
When we talk about a “discontinuous function
f
” in the context of Laplace transforms, we
usually mean
f
is a piecewise continuous function that is not continuous on the interval
(
0
,
∞
)
.
Such a function will have jump discontinuities at isolated points in this interval. Computationally,
however, the real issue is often not so much whether there is a nonzero jump in the graph of
f
at a point
t
0
, but whether the formula for computing
f
(
t
)
is the same on either side of
t
0
. So
we really should be looking at the more general class of “piecewisedefined” functions that, at
worst, have jump discontinuities.
Just what is a
piecewisedefined
function? It is any function given by different formulas on
different intervals. For example,
f
(
t
)
=
0
if
t
<
1
1
if
1
<
t
<
2
0
if
2
<
t
and
g
(
t
)
=
0
if
t
≤
1
t
−
1
if
1
<
t
<
2
1
if
2
≤
t
are two relatively simple piecewisedefined functions.
The first (sketched in figure 28.1a) is
discontinuous because it has nontrivial jumps at
t
=
1 and
t
=
2 .
However, the second
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548
PiecewiseDefined and Periodic Functions
(a)
(b)
T
T
1
1
1
1
2
2
0
0
Figure 28.1:
The graphs of two piecewisedefined functions.
function (sketched in figure 28.1b) is continuous because
t
−
1 goes from 0 to 1 as
t
goes
from 1 to 2 . There are no jumps in the graph of
g
.
By the way, we may occasionally refer to the sort of lists used above to define
f
(
t
)
and
g
(
t
)
as
conditional sets of formulas
or
sets of conditional formulas
for
f
and
g
, simply because
these are sets of formulas with conditions stating when each formula is to be used.
Do note that, in the above formula set for
f
, we did not specify the values of
f
(
t
)
when
t
=
1 or
t
=
2 . This was because
f
has jump discontinuities at these points and, as we agreed
in chapter 24 (see page 492), we are not concerned with the precise value of a function at its
discontinuities. On the other hand, using the formula set given above for
g
, you can easily verify
that
lim
t
→
1
−
g
(
t
)
=
0
=
lim
t
→
2
+
g
(
t
)
and
lim
t
→
1
−
g
(
t
)
=
1
=
lim
t
→
2
+
g
(
t
)
;
so there is not a true jump in
g
at these points. That is why we went ahead and specified that
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 Summer '09
 Differential Equations, Equations, Continuous function

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