1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.5 • THE LAPLACE TRANSFORM 541

8. Use the time differentiation property and the result of Exercise 4 to show that
   -w•
   (
   J° te--Z ••) = - iwet= -r.
   y21r


9. Use the symmetry and linearity properties and the results of Exercise 3 to show
   that

;j ( si:: ~) = {

0

1

.~;wl, for lwl ~ 1;

for lwl > 1.

12.5 THE LAPLACE TRANSFORM

In this section we investigate a very powerful tool for engineering applications.

12.5.1 From the Fourier Transform to the Laplace

Transform

We have shown that certain real-valued functions f (t) have a Fourier transform
and that the integral

g (w) = 1-: f (t) e - iwtdt

defines the complex function g (w) of the real variable w. If we multiply the

integrand f (t) e -iwt by C"t, then we create a complex function G (u + iw) of

the complex variable u + iw:

G (u + iw) = 1-: I (t) e-ute-iwtdt = 1-: J (t) e-<u+iw}tdt.

The function G (u + iw) is called the two-sided Laplace transfor m off (t),
and it exists when the Fourier transform of the function f (t) e-ut exists. From
Fourier transform theory, a. sufficient condition for G ( u + iw) to exist is that

For a function f (t), this integral is finite for values of u that lie in some

interval a < u < b.

The two-sided La.place transform has the lower limit of integration t = -oo

and hence requires a. knowledge of the pa.st history of the function f (t) (i.e.,

when t < 0). For most physical applications, we are interested in the behavior

of a system only for t 2: 0. The initial conditions f (0), f' (0), f" (0), ... are a

consequence of the pa.st history of the system and are often all that we know.

For this reason, it is useful to define the one-sided Laplace transform of f ( t),