Formulario transformada de laplace

FORMULARIO DE TRANSFORMADA DE LAPLACE
L{f (t)}(s) = F (s) ⇔ L−1 {F (s)}(t) = f (t) I Propiedades de Linealidad 1)L{f (t) ± g(t)} = L{f (t)} ± L{g(t)} 3)L−1 {F (s) ± G(s)} = L−1 {F (s)} ± L−1{G(s)} 2)L{kf (t)} = kL{f (t)} 4)L−1 {kF (s)} = kL−1 {F (s)} II F´rmulas B´sicas o a a) L{A}(s) = A , s > 0 s n! b) L{tn }(s) = sn+1 , n ∈ N, s > 0 c) L{tα }(s) = Γ(α+1) , α > −1 sα+1 1 at d) L{e }(s)= s−a , a ∈ R, s > a −as e) L{µa (t)} = e s , a ∈ R f ) L{µ(t)} = 1 s III Teoremas Importantes a) L{eat f (t)}(s) = L{f (t)}(s−a) = F (s − a) ⇔ L−1 {F (s − a)}(t) = eat L−1 {F (s)}(t) b) L{µa (t)f(t−a)}(s) = e−as ·L{f (t)}s = e−as F (s) ⇔ L−1 {e−as F (s)}(t) = µa (t)L−1 {F (s)}(t−a) c) L{tn f (t)}(s) = (−1)n ·
dn (L{f (t)})(s) dsn F (s) ⇔ L−1 { d dsn }t = (−1)n tn L−1 {F (s)}(t) n F (s) ⇔L−1 {F (s)}(t) = (−1)n t−n L−1 { d dsn }(t) ∞ s
n

g) h) i) j) k) l)

s L{cos (at)}(s) = s2 +a2 , a ∈ R, s > 0 a L{sin (at)}(s) = s2 +a2 , a ∈ R, s > 0 s L{cosh (at)}(s) = s2 −a2 , a ∈ R aL{sinh (at)}(s) = s2 −a2 , a ∈ R L{δ a (t)} = e−as , a ∈ R L{δ(t)} = 1

d ) Si existe l´ + ım
t→0

f (t) , entonces L{ f (t) }(s) = t t

L{f (t)}(s) ds
p −st e f (t) 0 1 − e−ps

e) Si f esuna funci´n peri´dica de periodo p, entonces: L{f (t)}(s) = o o

f ) L{f (n) (t)}(s) = sn L{f (t)}(s) −sn−1 f (0)−sn−2 f (0)−sn−3 f (0)−· · ·−sf (n−2) (0)−f (n−1) (0) En particular para n = 1, 2,se tiene: L{f (t)}(s) = s · L{f (t)}(s) − f (0) L{f (t)}(s) = s2 · L{f (t)}(s) − s · f (0) − f (0) g) L{ a f (t)dt}(s) = 1 · L{f (t)}(s) − 1 · 0 f (t)dt s s En particular para a = 0, se tiene:
t ta

L{
0

f (t)dt}(s) =

1 · L{f (t)}(s) s

IV Producto de Convoluci´n o a) Definici´n. o
t t

f (t) ∗ g(t) =
0

f (u)g(t − u)du =
0

f (t − u)g(u)du

b) Propiedades 1) L{f (t) ∗g(t)}(s) = L{f (t)}(s) · L{g(t)}(s) 2) L−1 {F (s) · G(s)}(t) = L−1 {F (s)}(t) ∗ L−1 {G(s))}(t) = f (t) ∗ g(t)

Formulario de Transformada de Laplace
Sergio Luis Ricardo Barrientos D´ ıaz...