tablice laplasovih transformacija
DESCRIPTION
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1
Tablice Laplasovih transformacija
Broj Original Kompleksni lik
∫ t
0+ x1(τ)x2(t− τ)dτ = x1 ∗ x2
1 X+1 (s)X+
2 (s)∫ t
0+ x1(t− τ)x2(τ)dτ = x2 ∗ x1
2 δ(t) X−(s) = 1, X+(s) = 0, X(s) ne postoji.
3 h(t)1s
4 n(t) = th(t)1s2
5 tn−1h(t)(n− 1)!
sn
6 h(t− α)1se−αs
7 h(t)− h(t− α)1s(1− e−αs)
8 eαth(t)1
s− α
9 eα(t−β)h(t− β)e−βs
s− α
10 tn−1eαth(t)(n− 1)!(s− α)n
11 (1− eαt)h(t)α
s(s + α)
121
αβ
[1− β
β − αe−αt +
α
β − αe−βt
]h(t)
1s(s + α)(s + β)
131
αβ
[γ − β(γ − α)
β − αe−αt +
α(γ − β)β − α
e−βt
]h(t)
s + γ
s(s + α)(s + β)
2
Broj Original Kompleksni lik
141
β − α
[e−αt − e−βt
]h(t)
1(s + α)(s + β)
151
α− β
[αe−αt − βe−βt
]h(t)
s
(s + α)(s + β)
161
β − α
[(γ − α)e−αt − (γ − β)e−βt
]h(t)
s + γ
(s + α)(s + β)
17[
e−αt
(β − α)(γ − α)+
e−βt
(γ − β)(α− β)+
e−γt
(α− γ)(β − γ)
]h(t)
1(s + α)(s + β)(s + γ)
18[
(δ − α)e−αt
(β − α)(γ − α)+
(δ − β)e−βt
(γ − β)(α− β)+
(δ − γ)e−γt
(α− γ)(β − γ)
]h(t)
s + δ
(s + α)(s + β)(s + γ)
19 αh(t) sin(ωt)αω
s2 + ω2
20 αh(t) cos(ωt)αs
s2 + ω2
21
√α2 + ω2
ωh(t) sin(ωt + θ), θ = arctan
ω
α
s + α
s2 + ω2
22 αh(t) sin(ωt + θ) αs sin(θ) + ω cos(θ)
s2 + ω2
23α
ω2h(t)[1− cos(ωt)]
α
s(s2 + ω2)
24[
α
ω2−√
α2 + ω2
ω2cos(ωt + θ)
]h(t), θ = arctan
ω
α
s + α
s(s2 + ω2)
25α
βe−γth(t) sin(βt)
α
(s + γ)2 + β2
26[
e−αt
α2 + ω2+
1ω√
α2 + ω2sin(ωt− θ)
]h(t), θ = arctan
ω
α
1(s + α)(s2 + ω2)
27[
1
ωn
√1− ζ2
e−ζωnt sin(ωnt√
1− ζ2)]h(t), ζ ∈ (0, 1)
1s2 + 2ζωns + ω2
n
28 e−αth(t) cos(βt)s + α
(s + α)2 + β2
3
Broj Original Kompleksni lik
29
√(γ − α)2 + β2
βe−αth(t) sin(βt + θ), θ = arctan
β
γ − α
s + γ
(s + α)2 + β2
1√1− ζ2
[√1− ζ2 cos(ωnt
√1− ζ2)− ζ sin(ωnt
√1− ζ2)
]h(t)e−ζωnt
30s
s2 + 2ζωns + ω2n
ζ ∈ (0, 1)
31[
1α2 + β2
+1
β√
α2 + β2e−αt sin(βt− θ)
]h(t), θ = arctan
β
−α
1s[(s + α)2 + β2]
32[
1ω2
n
− 1
ω2n
√1− ζ2
e−ζωnt sin(ωnt√
1− ζ2 + θ)]h(t), θ = arccos ζ
1s(s2 + 2ζωns + ω2
n)
[γ
α2 + β2+
1β
√(γ − α)2 + β2
α2 + β2e−αt sin(βt + θ)
]h(t)
33s + γ
s[(s + α)2 + β2]
θ = arctanβ
γ − α− arctan
β
−α
34[
e−γt
(γ − α)2 + β2+
e−αt sin(βt− θ)β√
(γ − α)2 + β2
]h(t), θ = arctan
β
γ − α
1(s + γ)[(s + α)2 + β2]
[1
γ(α2 + β2)− e−γt
γ[(γ − α)2 + β2]+
e−αt sin(βt− θ)
β√
α2 + β2√
(γ − α)2 + β2
]h(t)
351
s(s + γ)[(s + α)2 + β2]
θ = arctanβ
−α+ arctan
β
γ − α
[δ
γ(α2 + β2)− (γ − δ)e−γt
γ[(γ − α)2 + β2]+
36 +
√(δ − α)2 + β2
β√
α2 + β2√
(γ − α)2 + β2e−αt sin(ωt + θ)
]h(t)
s + δ
s(s + γ)[(s + α)2 + β2]
θ = arctanβ
δ − α− arctan
β
−α− arctan
β
γ − α
371α2
[αt− 1 + e−αt]h(t)1
s2(s + α)
381α2
[1− e−αt − αte−αt]h(t)1
s(s + α)2
4
Broj Original Kompleksni lik
391α2
[β − βe−αt + α(α− β)te−αt]h(t)s + β
s(s + α)2
40[
γ0
αβ+
α2 − αγ1 + γ0
α(α− β)e−αt − β2 − γ1β + γ0
β(α− β)e−βt
]h(t)
s2 + γ1s + γ0
s(s + α)(s + β)
{δ0
γ2+
1βγ
[(α2 − β2 − δ1α + δ0)2+
41 +β2(δ1 − 2α)2] 1
2
e−αt sin(βt + θ)}
h(t)s2 + δ1s + δ0
s[(s + α)2 + β2]
θ = arctanβ(δ1 − 2α)
α2 − β2 − δ1α + δ0− arctan
β
−α, γ2 = α2 + β2
( 1ω ) sin(ωt + θ1) + 1
β e−αt sin(βt + θ2)[4α2ω2 + (α2 + β2 − ω2)2
] 12
h(t)
421
(s2 + ω2)[(s + α)2 + β2]
θ1 = arctan−2αω
α2 + β2 − ω2, θ2 = arctan
2αβ
α2 − β2 + ω2
[1ω
(δ2 + ω2
γ)
12 sin(ωt + θ1)+
+1β
[(δ − α)2 + β2
γ
] 12
e−αt sin(βt + θ2)]h(t)
43s + δ
(s2 + ω2)[(s + α)2 + β2]γ = (2αω)2 + (α2 + β2 − ω2)2
θ1 = arctanω
δ− arctan
2αω
α2 + β2 − ω2
θ2 = arctanβ
γ − α+ arctan
2αβ
α2 − β2 + ω2
[1γ
(δt + 1− 2αδ
γ) +
[β2 + (δ − α)2]12
βγe−αt sin(βt + θ)
]h(t)
44s + δ
s2[(s + α)2 + β2]
θ = 2arctanβ
α+ arctan
β
δ − α, γ = α2 + β2
[δ1 + δ0t
αβ− δ0(α + β)
(αβ)2− 1
α− β(1− δ1
α+
δ0
α2)e−αt−
45s2 + δ1s + δ0
s2(s + α)(s + β)
− 1α− β
(1− δ1
β+
δ0
β2)e−βt
]h(t)
46 tx1(t) − dds
X1(s)