Formulario Transformadas De Laplace 1w1l4

Transformadas de Laplace

Transformadas inversas de Laplace

Sea F(t), t > 0, dada llamaremos Transformada de Laplace de F (t) a:

Sea {F(t)} = f (s), entonces, F(t) se llama Transformada inversa de Laplace de f (s),y se expresa como:

∞

−st ⋅ F ( t ) ⋅ dt f (s) = { F (t )} = ∫ e

F (t ) =

-1

0

Algunas funciones elementales:

Algunas funciones elementales:

F (t)

f (s) = {F(t)}

f (s )

 -1{ f (s) }=F(t)

1

1 s

1 s

1

eat

1 s −a

1 s −a

eat

ωt

sen

ω s +ω

ωt

cos

sen a

cos a

ωt ωt

t

t

−

2

s +ω

2

s s +ω2 ω 2 s −ω2 s 2 s −ω2 1 s2

π

( s + a) 2

e −bs

Y’(t)

s ⋅ y 2 −Y (0 )

(s − a) n

Y’’(t)

s ⋅ y 2 −s ⋅Y (0 ) −Y ' (0 )

1

e cos

( s) − a ) s 3 ⋅ y −s 2 ⋅Y (0 ) −s ⋅Y (0 ) −Y ' ' (0 ) −Y ' (0

ωt

( s − a) 2 + ω 2

(s

2⋅s 2

)

+1

{ a ⋅ F (t ) + b ⋅ G(t )} = a ⋅ { F (t )} + b ⋅ 

}

{ f ( s)} + b ⋅  -1





n

{ a ⋅ f (s) + b ⋅ g (s)} = a ⋅

}

{



}

 g (s )

{ f ( s −a )} = e a⋅t F (t )

{f

( n)

}

( s ) = ( −1) n ⋅ t n ⋅ F (t ) -1

(t ) = s n ⋅ y −s n −1Y ( 0) +.... + s ⋅Y (0 ) +....{+ ) ( u ) ⋅ G (t − u ) ⋅ du = F ⋅ G f (Ys')' ⋅(0 g ()s+ )}Y=' (0F (t>a)

}

 e a ⋅t ⋅ F ( t ) = f ( s − a )

{tn}=

t n −1 ⋅ e at ( n − 1)!

t

e −s⋅a  { H ( t − a )} = s n

(t − b) n ⋅ e a ⋅( t −b ) ( n − 1)!

-1



 -1

{  {t

ωt

2

Algunos teoremas de funciones y propiedades de T.L.I:

 G (t ) ( n)

e –a cos

2

Algunos teoremas de funciones:

{Y

t

s−a

t sen t

{

ωt

( s + a ) 2 +ω2

y

Y’’’(t)

cos a

ω⋅ e −at ⋅ sen ω⋅ t

Y(t)

at

ω

ω2

π s5

3 ⋅ 4

ωt

sena ωt

( s + a) +ω

s

3 2 t

cos

2

=

1 2

ω

2

ω s 2 2 s +ω s +ω2 ω 2 s −ω2 s 2 s −ω2 1 s2 2

-

sen ωt

1

2



{ f (s )}

}

⋅ F (t ) = ( −1) n ⋅ f

Γ( n +1) s

n +1

=

( n)

( s)

n +1

(n=1,2,..)

Algunas integrales y propiedades: dx / x =  x –1 = ln|x| +C a x dx = ax / ln a + C e x dx = e x + C

(propiedad de convolución)



n! s

∫o

sec x dx = ln|sec + tg x| +C sec2 x dx = tg x +C cosec x dx = ln|cosec x – cotg x|

-1

{e

−as

}

 F (t − a ) ,t > a  ⋅ f ( s ) = G (t ) ⇒ G (t ) =    0, t < 0 