1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

542 CHAPTER 12 • FOURJER SERJES AND THE LAPLACE TRANSFORM

which is commonly referred to simply as the Laplace transform off (t), which

is also defined as an integral:

C (! (t)) = F (s) = 1

00

f (t) e-•tdt, (^12 - 28)

where s = u + iw. If the integral in Equation (12-28) for the Laplace transform

exists for so = uo + iw, then values of u with u > uo imply that e-O't < e-O'ot

and so

from which it follows that F (s) exists for s = u + iw. Therefore, the Laplace

transform [, (! ( t)) is defined for all points s in the right half-plane Re ( s) > uo.

Another way to view the relationship between the Fourier transform and the

Laplace transform is to consider the function U (t) given by

U(t)={ f(t),

0,

for t 2:: 0;

fort< O.

Then the Fourier transform theory shows that

U (t) = _!_loo [loo U (t) e- iwtdt] eiwtdw,

27r -oo -oo

and, because the integrand U (t) is zero fort< 0, we can write this equation as

If we use the change of variable s = u + iw and dw = (ds/i), holding u > u 0

fixed, then the new limits of integration are from s = u -iw to s = u + iw. The

resulting equation is

f (t) = -^1 1"+ioo. [loo f (t) e-•tdt ] e•tds.

27r u-too 0

Therefore, the Laplace transform is

C (! (t)) = F (s) = 1

00

f (t) e-•tdt, wheres= u + iw,

and the inverse Laplace transform is

1 1 u+ioo

.c-^1 (F (s)) = f (t) = -

2

. F (s) e^81 ds.
11" u-ioo

(12-29)