FORMULARIO DE TRANSFORMADA DE LAPLACE

L{f(t)}(s) = F (s) ⇔ L−1{F (s)}(t) = f(t)

I Propiedades de Linealidad1)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 Formulas Basicasa) L{A}(s) = A

s, s > 0 g) L{cos (at)}(s) = s

s2+a2 , a ∈ R, s > 0

b) L{tn}(s) = n!sn+1 , n ∈ N, s > 0 h) L{sin (at)}(s) = a

s2+a2 , a ∈ R, s > 0

c) L{tα}(s) = Γ(α+1)sα+1 , α > −1 i) L{cosh (at)}(s) = s

s2−a2 , a ∈ Rd) L{eat}(s) = 1

s−a, a ∈ R, s > a j) L{sinh (at)}(s) = a

s2−a2 , a ∈ Re) L{µa(t)} = e−as

s, a ∈ R k) L{δa(t)} = e−as, a ∈ R

f) L{µ(t)} = 1s

l) L{δ(t)} = 1

III Teoremas Importantes

a) L{eatf(t)}(s) = L{f(t)}(s−a) = F (s− a) ⇔ L−1{F (s− a)}(t) = eatL−1{F (s)}(t)

b) L{µa(t)f(t−a)}(s) = e−as·L{f(t)}s = e−asF (s) ⇔ L−1{e−asF (s)}(t) = µa(t)L−1{F (s)}(t−a)

c)L{tnf(t)}(s) = (−1)n · dn

dsn (L{f(t)})(s) ⇔ L−1{dnF (s)dsn }t = (−1)ntnL−1{F (s)}(t)

⇔ L−1{F (s)}(t) = (−1)nt−nL−1{dnF (s)dsn }(t)

d) Si existe lımt→0+

f(t)

t, entonces L{f(t)

t}(s) =

∫∞sL{f(t)}(s)ds

e) Si f es una funcion periodica de periodo p, entonces: L{f(t)}(s) =

∫ p

0e−stf(t)

1− e−ps

f ) L{f (n)(t)}(s) = snL{f(t)}(s)−sn−1f(0)−sn−2f ′(0)−sn−3f ′′(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{∫ t

af(t)dt}(s) = 1

s· L{f(t)}(s) − 1

s·∫ a

0f(t)dt

En particular para a = 0, se tiene:

L{∫ t

0

f(t)dt}(s) =1

s· L{f(t)}(s)

IV Producto de Convolucion

a) Definicion.

f(t) ∗ g(t) =

∫ t

0

f(u)g(t− u)du =

∫ t

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 LaplaceSergio Luis Ricardo Barrientos Dıaz