Understanding Engineering Mathematics

We have already foundL[t] in Problem 17.1, and as a reminder:

L[t]=

∫∞

0

te−stdt=

[

te−st

−s

]∞

0

+

1

s

∫∞

0

e−stdt(by parts)=

1

s^2

ForL[t^2 ] the integration is more lengthy, but follows the same pattern – first integrate
with the limits 0,athen leta→∞.

L[t^2 ]=

∫∞

0

t^2 e−stdt

=

2

s^3

by integrating by parts twice

L[t^3 ] is even more of a slog, but with care and patience you should find that

L[t^3 ]=

6

s^4

=

3!

s^4

So, summarising, we have

L[1]=

0!

s

, L[t]=

1!

s^2

, L[t^2 ]=

2!

s^3

, L[t^3 ]=

3!

s^4

where we have used the convention 0!=1(16

➤
), which we also used in the binomial
theorem. These results lead us tosuspectthat in general

L[tn]=

n!

sn+^1

forna positive integer. In fact this is a correct generalisation, although we will not prove it.
Proceeding as in the above, exercising your integration, you can if you wish verify the
results given in Table 17.1 for the Laplace transforms of the elementary functions (see
Exercises on this section and page 285). There are a number of points to note:

• the restrictions ons, which we have assumed to be real


• results involving the exponential function, which clearly has the effect
  of replacingsbys−a–i.e.ofshiftingor translatings


• the results fortsinωt,tcosωtcan be obtained by differentiation of
  those for sinωt,cosωtwith respect toω.

Note an important implication of the fact that the Laplace transform is defined by an
integral. Specifically, recall that in any mathematical operation, such as differentiation for
example, there are precise conditions under which the operations are allowed – for example
a function must be continuous at a point if it is to be differentiable there. On the other hand
anintegrablefunction does not have to be continuous in order to integrate it, meaning that
we can apply the Laplace transform to a wider range of functions than continuous ones.
A large and important class of functions to which we can apply the Laplace transform
includes the class of piecewise continuous functions. Apiecewise continuous function
f(t)on an intervala≤t≤bis a function which consists of continuous sections separated
by a finite number of isolated points at which the function may not be continuous, but the