HP F2226A - 48GII Graphic Calculator - Page 492

Page 16-14
Now, use ‘(-X)^3*EXP(-a*X)’ ` LAP µ. The result is exactly the same.
• Integration theorem
. Let F(s) = L{f(t)}, then
• Convolution theorem
. Let F(s) = L{f(t)} and G(s) = L{g(t)}, then
{
}
==−
∫
)})(*{()()(
0
tgfduutguf
t
LL
)()()}({)}({ sGsFtgtf ⋅=⋅ LL
Example 4
– Using the convolution theorem, find the Laplace transform of
(f*g)(t), if f(t) = sin(t), and g(t) = exp(t). To find F(s) = L{f(t)}, and G(s) = L{g(t)},
use: ‘SIN(X)’ ` LAP µ. Result, ‘1/(X^2+1)’, i.e., F(s) = 1/(s
2
+1).
Also, ‘EXP(X)’ ` LAP. Result, ‘1/(X-1)’, i.e., G(s) = 1/(s-1). Thus, L{(f*g)(t)}
= F(s)⋅G(s) = 1/(s
2
+1)⋅1/(s-1) = 1/((s-1)(s
2
+1)) = 1/(s
3
-s
2
+s-1).
• Shift theorem for a shift to the right
. Let F(s) = L{f(t)}, then
L{f(t-a)}=e
–as
⋅L{f(t)} = e
–as
⋅F(s).
• Shift theorem for a shift to the left
. Let F(s) = L{f(t)}, and a >0, then
• Similarity theorem
. Let F(s) = L{f(t)}, and a>0, then L{f(a⋅t)} =
(1/a)⋅F(s/a).
• Damping theorem
. Let F(s) = L{f(t)}, then L{e
–bt
⋅f(t)} = F(s+b).
• Division theorem
. Let F(s) = L{f(t)}, then
{
}
).(
1
)(
0
sF
s
duuf
t
⋅=
∫
L
.)()()}({
0






⋅⋅−⋅=+
∫
−
a
stas
dtetfsFeatfL