Tabla De Laplace

LAPLACE TRANSFORM TABLES

MATHEMATICS CENTRE

©2000

DEFINITION The Laplace transform f ( s ) of a function f(t) is defined by:
∞

f (s) =

∫e
0

− st

f (t ) dt

TRANSFORMS OFSTANDARD FUNCTIONS

f(t) 1
e−αt
t 1 −T e T

f (s)

1 s 1 s+α 1 1+ sT
α s (s + α )
1

1− e −α t te− α t

(s + α )2
β −α ( s + α )(s + β

e −α t − e − β t

)

t
tn e −α t t n

1 s2n! sn +1
n!

(s + α )n +1
ω s + ω2
2

sin ωt
cosωt

s s + ω2
2

2

f(t)
e −α t sin ω t e −α t cos ω t

f (s)

(s + α )2 + ω 2 (s + α )2 + ω 2
ω2 s s2 +ω s+α

ω

1− cosωt

(2

)

1 2ω
3

(sin ω

t − ω t cos ω t )

(s (s

1
2

+ω s

2 2

) )

t sin ω t 2ω
α   e − α t  cos ω t − sin ω t  ω  

2

+ω
s
2

2 2

(s + α )

+ω

2sin (ω t + φ )
e −α t + α sin ω t − cos ω t ω
sin 2 ωt

s sin φ + ω cos φ s2 +ω 2

( s + α )(s 2 + ω
s s 2 + 4ω

α

2

+ω

2 2

)

( (

2ω

2 2

) )

cos2 ωt

s 2 + 2ω2 2

s s 2 + 4ω

sinh βt cosh βt

β s −β2
2

s s −β2
2

3

f(t) e −α t sinh β t e −α t cosh β t
t sinh β t

f (s)

( s + α )2 − β ( s + α )2 − β
s +α

β

2

2

(s
(s2β
2

s

−β

2 2

)

t cosh β t

s2 + β
2

2 2 2

−β

)

1 2β
3

(β

t cosh β

t − sinh β

t)

(s

1
2

−β

2 2

)

Transforms of Special Functions Unitimpulse : Unit step : Ramp: δ(t) H(t) tH(t) 1

1 s 1 s2
e-sT
e − sT s 1− e − sT s

Delayed Unit Impulse: δ(t-T) Delayed Unit Step: Rectangular Pulse: H(t-T) H(t)-H(t-T)

4

TRANSFORM THEOREMSf(t) Damping: Delay: Time scale: f(kt) Integral: Differentiation e-αt f(t) f(t-T)H(t-T)
f (s) f (s + α ) e − sT f ( s )

1 s f  k k 

∫

t

0

f (t ) dt

1 f ( s) s
sf ( s ) − f (0) s 2 f ( s) − sf ( 0) − f '( 0) s n f ( s) − sn −1 f ( 0) − sn −2 f '( 0) −... − f n −1 ( 0)

d f (t ) dt d2 f (t ) dt 2 dn f (t ) dt n
Initial Value: Final Value: Periodic Functions:

lim {...