Find the Laplace Transforms of e^-t (sinat-at cosat)
Answers
Step-by-step explanation:
128 CHAPTER 5. LAPLACE TRANSFORMS
Solution:
L(1)(s) = Z ∞
0
1e
−st dt = limc→∞
−
1
s
e
−st
¯
¯
¯
¯
c
0
In order for this limit to exist, we must insist that s 6= 0 and that s > 0 so
that e
−sc has a limit (of zero). When s > 0, we obtain
−
1
s
limc→∞
(e
−sc − 1) = 1
s
So
L(1)(s) = 1
s
; s > 0.
¤
Example 5.2 Compute the Laplace transform of f(t) = t
Solution:
L(t)(s) = Z ∞
0
te−st dt
We integrate by Parts (letting u = t and dv = e
−st dt) to obtain:
Z
te−st dt = −
1
s
te−st −
1
s
2
e
−st
,
so
Z ∞
0
te−st dt = limc→∞ µ
−
1
s
te−st −
1
s
2
e
−st¶ ¯
¯
¯
¯
c
0
In order for this limit to exist, we again must insist that s 6= 0 and that s > 0
so that e
−sc has a limit (of zero). We obtain
−
1
s
limc→∞
(ce−sc − 0) −
1
s
2
limc→∞
(e
−sc − 1)
which exists for s > 0 and after L’Hˆopital’s rule yields
L(t)(s) = 1
s
2
; s > 0.
¤
The previous example can be upgraded to find the Laplace transform of
f(t) = t
n
for any positive integer n.