HP 50g Graphing Calculator - Manual (Page 491)

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).
{ }
= = − ∫ )} )( * {( ) ( ) (
0
t g f du u t g u f
t
L L
) ( ) ( )} ( { )} ( { s G s F t g t f ⋅ = ⋅L L
. ) ( ) ( )} ( {
0
⎟⎠
⎞
⎜⎝
⎛
⋅ ⋅ − ⋅ = + ∫ −
a
st as
dt e t f s F e a t f
L
∫
∞
=
⎭
⎬
⎫
⎩
⎨
⎧
s
du u F
t
t f
. ) (
) (
L
∫ ⋅ ⋅ ⋅
−
= −
−
T
st
sT
dt e t f
e
t f
0
. ) (
1
1
)} ( { L
)]. ( [ lim ) ( lim
0
0
s F s t f f
s t
⋅ = =
∞ → →