¡Descarga Transformadas de Laplace: Propiedades y Ejemplos y más Apuntes en PDF de Matemáticas Aplicadas solo en Docsity!
TRANSFORMADAS DE LAPLACE
No. (^) f ( s ) F ( t )
1 (^1) Impulso unitario δ(t)
s
(^1) Medida Unitaria 1(t)
3 e − as^ δ (^ t^ − a )
4^12
s
t
(^5) n s
(^1) (n = 1,2,3, ...)
1 −
− n
tn , 0! = 1
s + a
(^1) at e −
s + a te − at
( s + a ) n
(^1) n = 1,2,3,...
1 −
− − n
t n e at , 0! = 1
ω
ω s +
sen ωt
s + ω
s (^) cos ωt
( ) 2 ω^2
ω s + b + e^ sen t
− bt (^) ω
( + ) 2 + ω^2
s b
s b e − bt^ cos ωt
ω
ω s −
senh ωt
s − ω
s (^) cosh ωt
( ) 2 ω^2
ω s + b − e^ senh t
− bt (^) ω
( + ) 2 − ω^2
s b
s b e − bt^ cosh ωt
18 ( s + a )(^1 s + b ) a^ ≠^ b
b a
e at e bt −
19 ( s + a )( s s + b ) a^ ≠^ b
b a
be bt ae at −
2 (^2) n n
n s ξω s ω
ω
n (^) 2 e n tsen 1 2 t 1
ξ ξ
ω (^) ξω (^) − −
−
s ns n
s
φ ξ
ω ξ φ ξ
ξω
1 2
2 2
tan^1
( 1 ) 1
1
=^ −
− − −
−
−
e −^ ntsen n t
2
n n
n s s ξω s ω
ω
ξ
φ ξ
ω ξ φ ξ
ξω
1 2
2 2
tan^1
( 1 ) 1
1 1
=^ −
− + −
−
−
e −^ ntsen n t
(^23) s ( s + a^1 )( s + b )
(^1 1) + 1 ( be − at (^) ae − bt ) ab a b
(^24) 2 ( )
s s + a (^1 )
2 e at^ ate^ at a
− −^ −^ −
s^2 s + a (^1 )
2 at e^ at a
− +^ −
2 ω
ω s s +
1 −cos ωt
3 ω
ω s s +
ω t − senωt
( 2 2 )^2
ω
ω s +
sen ωt − ωt cos ωt
( s 2 + ω^2 )^2
s ω
ω 2
tsen t
( 2 2 )^2
2 s + ω
s ω
ω ω ω 2
sen t + t cos t
( 2 2 )^3
ω
ω
s
s s 21 t^^2 cos ωt
( 2 2 )^4
ω
ω ω
s
s s 61 t^^3 cos ωt
( 2 2 )^4
3 2 ω
ω
s
s s ω
ω 24
t^3 sen t
( 2 2 )^3
s − ω
( ) 5
22 8
3 3 cosh ω
( s 2 − ω^2 )^3
s 3
2 8
cosh ω
ωt ωt − tsenhωt
( 2 2 )^3
2 s − ω
s (^) ( ) 3
22 8
cosh 1 ω
ωt ωt + ωt − senhωt
( 2 2 )^3
3 s − ω
s ω
ω ω ω 8
3 tsenh t + t^2 cosh t
( 2 2 )^3
4 s − ω
s (^) ( ) ω
ω ω ω ω 8
3 +^2 t^2 senh t + 5 t cosh t
( 2 2 )^3
5 s − ω
s (^) ( ) 8
8 + ω^2^ t^2 cosh ωt + 7 ωtsenhωt
( 2 2 )^3
ω
ω −
s
s ω
ω 2
t^2 senh t
( 2 2 )^3
ω
ω −
s
s s 21 t^^2 cosh ωt
( 2 2 )^4
ω
ω ω −
s
s s 61 t^^3 cosh ωt
( 2 2 )^4
3 2 ω
ω −
s
s s ω
ω 24
t^3 senh t
s + ω
2
2 2
cos^3 2
e ωt^ sen ωt ωt e ω t ω
(^61) 3 3 s + ω
s
cos^3 3
e ωt^ ωt sen ωt e ω t ω
(^62) 3 3
2 s + ω
s
2 cos^3 3
(^1) e ωt (^) eωt 2 ωt
s − ω
cos^3 3
3 2 2
(^2) t e (^) e t t sen t (^) ω ω ω
ω ω
s − ω
s
− (^232) 2
cos^3 2
e ωt^ sen ωt ωt eω t ω
2 s − ω
s
2 cos^3 3
(^1) e ωt (^) eωt 2 ωt
s + ω
( sen ωt ωt ωtsenhωt )
ω
cosh cos 4
s + 4 ω
s 2 ω^2
senωtsenh ωt
2 s + 4 ω
s ( senω t ωt ωtsenhωt )
ω
cosh cos 2
3 s + 4 ω
s (^) cos ωt cosh ωt
s − ω
( senhω t senωt )
ω
s − ω
s (cosh t-cos t)
a^2 ω ω (^72) 4 4
2 s − ω
s ( t+sen t)
(^1) senhω ω a (^73) 4 4
3 s − ω
s (cosh t+cos t)
(^1) ω ω
74^1
s + b + s + b ( )
e e b a t
− bt (^) − − at 2 − π^3
75^1 s s + a
fer at a 76
s s − a
e fer at a
at
77^1
s − a + b (^ )
e t
at^1 be b t^2 fcer b t π
s^2^ + a^2 J^^0 (^ at )
s^2^ − a^2 I^^0 (^ at )
94 e s
a − s 3 2
sen at a
π
95 e s
a s n
−
+ 1 n > -1^ (^ )
t a
at
n ^
^
2 J (^) n 2
96 e s
− a s
t
e t
a
π
4 −^2
(^97) e − a^^ s a t
a t 2 3
4
2
π
e
−
98 e s
− a s fer a^ t 2
^
99 e s
− a s fcer a^ t 2
^
e s s b
− a s
e^ (^ ) fcer b t a t
b bt + a + ^
101 e s
a s n
−
+ 1 n > -1^ (^ )
2 42 πt
e u du
u a t a 2n+1 0 u^ n J2n
∞ − ∫
102 ln s^ a s b
^
^
e e t
− bt (^) − − at
103 ln^
s a a s
2 2 2 2
^
Ic at ( )
104 ln^
s a a s
^ +
^
^
^ Ie at (^ )
− (^ y^ +ln s)
s y = constante de Euler = 0,
ln t
106 ln s^ a s b
2 2 2 2
^
^
2 cos at - cos bt( )
t
π^2 ( )^2
6 s
y s y
ln s
constante de Euler = 0,
ln^2 t
108 ln s s
ln t y y constante de Euler = 0,
109 ln
(^2) s s
( ln^ t^ y )
y
π constante de Euler = 0,
110 Γ^ (^ )^ Γ(^ )
' (^) n n ln s s n n
1 1 t^ n^ ln t
111 tg a s
^
1 sen^ at t
112 tg^
a s s
^
1
Is at ( )
113 e s
fcer a s
a s (^)
e t
− 2 at π
(^114) e fcer s a
s a
2 42 2
^
^
2 a e a t 2 2 π
−
115 e^ fcer^
s a s
s a
2 42 2
^
^ fer at (^ )
116 e^ fcer^ as s
as
π t + a
117 e as^ Ie as ( )^1
t + a
118^1 ( ) ( )
a
cos as Is as sen 2
^ π^ − as Ic as
^
^
t^2 + a^2
119 sen as Is as ( ) ( )
^ π^ − as Ic as
120 cos as^ (^ )^ as Ic as(^ )
π 2
Is as (^) − sen
s
tg t a
^
1
121 sen^ as^2 Is as (^ )^ as Ic as(^ )
s
π (^) −
2 2 ln (^2) t a a
^
122^ π ( ) ( )
2 − 2 ^
^
Is as + Ic as^1
2 2 t^2
t a a
ln + ^
123 0 N t ( )= funcion nula
124 1 δ ( t )= funcion delta
