0195136047.pdf

LAPLACE TRANSFORMS 853

sin(ωt+θ)
ssinθ+ωcosθ
(s^2 +ω^2 )
cosωt
s
s^2 +ω^2
cos(ωt+θ)
scosθ−ωsinθ
(s^2 +ω^2 )
e−atsinωt ω
(s+a)^2 +ω^2
te−atsinωt
2 ω(s+a)
[(s+a)^2 +ω^2 ]^2
e−atcosωt s+a
(s+a)^2 +ω^2

te−atcosωt
[(s+a)^2 −ω^2 ]
[(s+a)^2 +ω^2 ]^2
sinhat a
s^2 −a^2
coshat s
s^2 −a^2
[k 1 e−atcosωt+k^2 −k^1 a
ω
e−atsinωt] k^1 s+k^2
(s+a)^2 +ω^2
√ω
1 −a^2

e−aωtsinω

√

1 −a^2 t ω

2
s^2 + 2 aω+ω^2
1
2 ω
tsinωt
s
(s^2 +ω^2 )^2
1
2 ω
(sinωt+ωtcosωt)
s^2
(s^2 +ω^2 )^2
Note that allf(t)should be thought of as being multiplied byu(t), i.e.,f(t)=0 fort<0.