Tabela Laplace

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TABELA DE TRANSFORMADA DE LAPLACE ( Professor: Ariosvaldo)
1.
f(t) F (s) = L{f(t)} f(t) F (s) = L{f(t)}
f(at) 1
a
F ( s
a
) tp Γ(p+1)
sp+1
eatf(t) F (s− a) sen(at) a
s2 + a2
y′(t) sY (s)− y(0) cos(at) s
s2 + a2
y′′(t) s2Y (s)− sy(0)− y′(0) ebtsen(at) a
(s− b)2 + a2
y′′′(t) s3Y (s)− s2y(0)− sy′(0)− y′′(0) ebt cos(at) s− b
(s− b)2 + a2
tn
n!
sn+1
senh(at)
a
s2 − a2
eattn
n!
(s− a)n+1 e
at s
s− a
tn · f(t) (−1)n d
n
dsn
F (s) cosh(at)
s
s2 − a2
f(t)
t
∞∫
s
F (u) du sen(at)− at cos(at) 2a
3
(s2 + a2)2
t∫
0
f(τ) dτ
F (s)
s
tsen(at)
2as
(s2 + a2)2
µa(t) = µ(t− a) e
−as
s
sen(at) + at cos(at)
2as2
(s2 + a2)2
µa(t) · f(t− a) e−asF (s) t cos(at) s
2 − a2
(s2 + a2)2
µa(t) · g(t) e−asL{g(t+ a)} (f ∗ g)(t) F (s) ·G(s)
f(t+ T ) = f(t)
T∫
0
f(t)−stdt
1− e−sT
√
t
√
pi√
s
δ(t− t0) e−t0s δ(t) 1
Tranformada de Laplace. F (s) = L{f(t)} =
∞∫
0
e−stf(t) dt = lim
b−→∞
b∫
0
e−stf(t) dt.
ay′′ + by′ + cy = f(t), y′(0) = y1, y(0) = y0
Enta˜o y(t) = φ(t) + ψ(t), onde φ(t) = L−1
{
(as+ b)y0 + ay1
as2 + bs+ c
}
e ψ(t) = L−1
{
F (s)
as2 + bs+ c
}
,
Teorema de Convoluc¸a˜o: Se F (s) = L{f(t)} e G(s) = L{g(t)}, enta˜o L{(f ∗ g)(t)} = F (s)G(s). onde
(f ∗ g)(t) =
t∫
0
f(t− τ)g(τ) dτ =
t∫
0
f(τ)g(t− τ) dτ = (g ∗ f)(t).
L−1{F (s)G(s)} = (f ∗ g)(t) =
t∫
0
f(t− τ)g(τ) dτ =
t∫
0
f(τ)g(t− τ) dτ
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