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