Page 145 - Schaum's Outline of Theory and Problems of Signals and Systems

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

3.10. Verify the time differentiation property (3.20), that is,

From Eq. (3.24) the inverse Laplace transform is given by

Differentiating both sides of the above expression with respect to t, we obtain

Comparing Eq. (3.77) with Eq. (3.76), we conclude that h(t)/dt is the inverse Laplace
transform of sX(s). Thus,

R'3R

Note that the associated ROC is unchanged unless a pole-zero cancellation exists at s = 0.

3.11. Verify the differentiation in s property (3.21), that is,

From definition (3.3)
- IIj

Differentiating both sides of the above expression with respect to s, we have

Thus, we conclude that

3.12. Verify the integration property (3.22), that is,

Let

Then