HP 50G

Page 16-14
Θ 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).
Θ Shift theorem for a shift to the left
. Let F(s) = L{f(t)}, and a >0, then
Θ Similarity theorem
. Let F(s) = L{f(t)}, and a>0, then L{f(a⋅t)} = (1/a)⋅F(s/a).
Θ Damping theorem
. Let F(s) = L{f(t)}, then L{e
–bt
⋅f(t)} = F(s+b).
Θ Division theorem
. Let F(s) = L{f(t)}, then
Θ Laplace transform of a periodic function of period T
:
• Limit theorem for the initial value: Let F(s) = L{f(t)}, then
• Limit theorem for the final value: Let F(s) = L{f(t)}, then
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).
{}
==−
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0
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.)()()}({
0
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⎠
⎞
⎜
⎝
⎛
⋅⋅−⋅=+
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−
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∫
∞
=
⎭
⎬
⎫
⎩
⎨
⎧
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)(
L
∫
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−
=
−
−
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1
1
)}({L
)].([lim)(lim
0
0
sFstff
st
⋅==
∞→→