Page 171 - Schaum's Outline of Theory and Problems of Signals and Systems

160 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

3.45. Show that if X(I) is a left-sided signal and X(s) converges for some value of s, then the ROC
of X(s) is of the form

where amin equals the minimum real part of any of the poles of X(s).
Hint: Proceed in a manner similar to Prob. 3.4.

3.46. Verify Eq. (3.21), that is,

Hint: Differentiate both sides of Eq. (3.3) with respect to s.

3.47. Show the following properties for the Laplace transform:
(a) If x(t) is even, then X( -s) = X(s); that is, X(s) is also even.
(b) If ~(t) isodd, then X(-s)= -X(s); that is, X(s)is alsoodd.
(c) If x(t) is odd, then there is a zero in X(s) at s = 0.
Hint:
(a) Use Eqs. (1.2) and (3.17).
(b) Use Eqs. (1.3) and (3.17).
(c) Use the result from part (b) and Eq. (1.83~).

3.48. Find the Laplace transform of

x(t) = (e-'cos21- Se-*')u(t) + :e2'u(-t)

s+l 5 1 1
Ans. X(s)= , -1 < Re(s)<2
(~+1)~+4 S+2 2s-2

3.49. Find the inverse Laplace transform of the following X(s);

1
(b) X(s) = -
s(s + l)? '

s+ l
(d) X(s) = , Re(s) > -2
s2+4s+ 13