Page 162 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

(a) Using Eq. (3.441, we have
mk(') e-"
SX,(S) -x(o-) = / -
0- dt
Wt)
= /O'd'oe-s'dt +cT e -" dt
0- dt
e-st dt
=x(t)E?+ / -
a
o+ dt
-Wf) e-s,dt
=x(O+) -x(o-) + / -
o+ dt
Thus,

and lirn sX,(s) =x(O+) +
5-07

since lim, ,, e-" = 0.
(b) Again using Eq. (3.441, we have
a&(t)
lirn [sX,(s) - x(0-)] = 1im / - ePS' dt
s-o s-o 0- dt

lirn e-"'

= lirn x(t ) - ~(0~)
t-rm
Since lirn [sX,(s) -x(0-)] = lim [sx,(s)] -x(O-)
s-ro s-ro
we conclude that
limx(t) = lirnsX,(s)
t--t- s-ro

3.36. The unilateral Laplace transform is sometimes defined as

with O+ as the lower limit. (This definition is sometimes referred to as the 0'
definition.)
(a) Show that