1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.5 • THE LAPLACE TRANSFORM 547

Solution Using linearity and lines 6 and 7 of Table 12.2, we obtain

c-1 (3s + 6)

s^2 + 9

f(t)

1

tn

= 3c-1 (-s ) + zc-1 (-3 )

s^2 + 9 52 + 9

= 3cos3t + 2sin3t.

F(s) = J; f(t) e-•^1 dt

1

s

n!

8 n+1

Uc (t) unit step

e -cs

s

eat^1

s - a

t"eat

n!

(s -at+1

cosbt

s

s2 +bz

sinbt

b

s2 +bz

e"^1 cos bt

s - a

(s - a.}2 + b^2

e"^1 sin bt

b

(s - a)^2 + b^2

tcosbt

52 - b2

(s2 + b2)2

tsinbt

2bs

(s2 + b2)2

coshat

s

52 - a2

sinhat

a

s2 -a2

Table 12.2 Table of Laplace Transforms