Understanding Engineering Mathematics

Another well known piecewise continuous function is the square wave, shown in
Figure 17.2 and defined by

f(t)=− 1 − 1 <t< 0

= 10 <t< 1

f(t+ 2 )=f(t)

− 2 − 1 0 1 2 3

1

− 1

t

Figure 17.2The square wave.

This is discontinuous att=1, 2, 3, etc. and as the name suggests is a periodic function.
The ideal tool for dealing with such functions is Fourier series, which again involves an
integration that smoothes things out.
Integral transforms can also deal with functions that may be continuous but have discon-
tinuous slope, such as thesaw-tooth waveshown in Figure 17.3. In this case the derivative
fails to exist at the ‘corners’, and yet it still has a Fourier series. Such functions are called
piecewise smooth.

0 t

Figure 17.3The sawtooth wave.

Note that, as mentioned above, the discontinuities must befinitejumps. Thus, 1/tis
notpiecewise continuous – see Figure 17.4.

Problem 17.4

Evaluate the Laplace transform of the function

f.t/= 00 <t< 1

= 11 <t< 2

= 02 <t