Tabla de propiedades de la Transformada de Laplace
[ ] )()( saFtaf =! Teorema del valor inicial )()(0
ssFlimtflimst
=
Linealidad [ ] )()()()( 2121 sFsFtftf +=+! Teorema del valor final )()(0
ssFlimtflimst
=
Desplazamiento en el tiempo [ ] )()()( sFetutf s =! Tiempo por una funcin [ ]ds
sdFttf )()( =!
donde [ ])()( tfsF !=Impulso [ ] 1)( =t! [ ]
=
asF
aatf 1)(!
Desplazamiento de frecuencia [ ] )()( asFtfe at +=! [ ] ( ) nn
nn
dssFdtft )(1)( =!
Derivada )0()()( fssFdttdf
=
! )(asaF
atf =
!
Integrals
dttf
ssFdttf t
t
at
a
0
)()()( =
+=
!
= s dssFttf )()(!
Pares de Transformadas de Laplacef(t) F(s) f(t) F(s)
1 Impulso unitario 1( )!1
1
nt n
ns1
)(tu Escaln unitarios1
( )!11
net atn " Rampa amortiguada ( )nas
1
a Escalnsa ( )ate
a
11 ( )ass +1
at Rampa2sa ( )ateat
a+1
12 ( )ass +2
1
ate " Exponencialas
1 ( )btat eeab
1( )( )bsas ++
1
tsen Seno22
+s( )atbt aebe
ab
1( )( )bsas
s++
tcos Coseno22 +s
s ( )
+ btat aebebaab
111 ( )( )bsass ++1
te at sen Seno amortiguado( ) 22
++ astsh
22
s
te at cos Coseno amortiguado( ) 22 ++
+
asas tch
22 ss
nt1
!+nsn te n
tn n 22
1sen1
22
2
2 nss nn
++
atn et
( ) 1!
++ nasn
2
22
11sen
11 arctante n
tn 22 2 nsssn ++
tt cos
( )22222
+
ss
+
2
22
11sen
111 arctante n
tn ( )222
2 nsss nn
++
tt
sen2 ( )222 +s
s ( ) + teK t cos2 K es un n complejo = Kjs
Kjs
K ++++
*
( ) + teKt t cos2 K es un n complejo = K( ) ( )2
*
2 jsK
jsK
++++
Tabla de Transformadas de Laplace
L {f (t)} = 0 es t f (t) dtf(t) F (s) f(t) F (s)
1. 11
s2. tn, n = 1, 2, 3, ..
n!
sn+1
3. t, 1 < (+ 1)s+1
4. ea t1
s a
5. tn ea t, n = 1, 2, 3, ..n!
(s a)n+1 6. sin ( t)
s2 + 2
7. cos ( t)s
s2 + 28. sinh ( t)
s2 2
9. cosh ( t)s
s2 2 10. ea t sin ( t)
(s a)2 + 2
11. ea t cos ( t)s a
(s a)2 + 2 12. t sin ( t)2 s
(s2 + 2)2
13. t cos ( t)s2 2
(s2 + 2)214. sin ( t) t cos ( t) 2
3
(s2 + 2)2
15. sin ( t) + t cos ( t)2 s2
(s2 + 2)216.
1
a b(ea t eb t
) 1(s a) (s b)
17.1
a b(a ea t b eb t
) s(s a) (s b) 18.
1
a2(1 cos (a t)) 1
s (s2 + a2)
19.1
a3(a t sin (a t)) 1
s2 (s2 + a2)20. f(t) + g(t) F (s) +G(s)
21. c f(t) c F (s) 22. f (t) s F (s) f(0)
23. f (t) s2 F (s) s f(0) f (0) 24. f (n)(t) sn F (s) sn1 f(0) f (n1)(0)
25. ea t f(t) F (s a) 26. tn f(t) (1)n dn
dsnF (s)
27. Ua(t) = U(t a) ea s
s28. f(t a)Ua(t) ea s F (s)
29. f g = t0
f(t )g()d F (s)G(s) 30. (t c) ec s
31. f(t+ T ) = f(t)
T0
es t f(t) dt
1 es T 32. t0
f() d1
sF (s)
33.f(t)
t
+s
F () d 34. f(a t)1
aF(s
a
)
lims+ s F (s) = f(0) lims0+
s F (s) = limt+ f(t)
(c) Departamento de Matematicas. ITESM, Campus Monterrey
1
Tabla de transformada v1.pdfTabla de transformada v2.pdf
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