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Properties of Laplace Transform
.

Example 2.7,

2
−t
L[e sin 2t] = L[sin 2t] = 2 .
s+1 (s + 1) + 4
. . . . .
.

Example 2.8,

s − 2
2t
L[e cosh 5t] = L[cosh 5t] = 2 .
s−2 (s − 2) − 25
. . . . .
.

Example 2.9, Since

1
L(t) = (Re(s) > 0),
s 2

then
1
at
L(te ) = (Re(s) > a),
(s − a) 2
and, in general,

n!
n at
L(t e ) = , n = 0, 1, 2, . . . (Re(s) > a).
(s − a) n+1
. . . . .
.

Example 2.10, Since

ω
L(sin ωt) = ,
2
s + ω 2
then
2t
L(e sin 3t) =
In general,

s − a
at
L(e cos ωt) = (Re(s) > a),
2
(s − a) + ω 2
ω
at
L(e sin ωt) = (Re(s) > a),
2
(s − a) + ω 2
s − a
at
L(e cosh ωt) = (Re(s) > a),
2
(s − a) − ω 2
ω
at
L(e sinh ωt) = (Re(s) > a).
2
(s − a) + ω 2
. . . . .

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