Laplace Transform Pairs

f(t) F(s)

Unit impulse δ(t) 1

Unit step 1(t) s1

t 2

1s

)!1(

1

−

−

ntn

(n = 1, 2, 3, . . .) ns1

nt (n = 1, 2, 3, . . .) 1

!+ns

n

ate− as +

1

atte− ( )2

1as +

atn

ent −

−

− )!1(

1

(n = 1, 2, 3, . . .) ( )nas +1

atnet − (n = 1, 2, 3, . . .) ( ) 1

!++ nas

n

tωsin 22 ω

ω+s

tωcos 22 ω+s

s

tωsinh 22 ω

ω−s

tωcosh 22 ω−s

s

)1(1 atea

−− )(

1ass +

)(1 btat eeab

−− −−

))((

1bsas ++

)(1 atbt aebeab

−− −−

))(( bsas

s++

−

−+ −− )(111 btat aebe

baab

))((1

bsass ++

Laplace Transform Pairs (continued)

)1(12

atat ateea

−− −− 2)(1

ass +

)1(12

ateata

−+− )(

12 ass +

te at ωsin− 22)( ω

ω++ as

te at ωcos− 22)( ω++

+as

as

te ntn n 2

21sin

1ζω

ζω ζω −−

− 22

2

2 nn

n

ss ωζωω

++

ζζφ

φζωζ

ζω

21

22

1tan

)1sin(1

1

−=

−−−

−

−

− te ntn

22 2 nnss

sωζω ++

ζζφ

φζωζ

ζω

21

22

1tan

)1sin(1

11

−=

+−−

−

−

− te ntn

)2( 22

2

nn

n

sss ωζωω

++

tωcos1− )( 22

2

ωω+ss

tt ωω sin− )( 222

3

ωω+ss

ttt ωωω cossin − 222

3

)(2ωω+s

tt ωω

sin21 222 )( ω+s

s

tt ωcos 222

22

)( ωω

+−

ss

)cos(cos1212

122

tt ωωωω

−−

)( 22

21 ωω ≠

))(( 22

221

2 ωω ++ sss

)cos(sin21 ttt ωωωω

+ 222

2

)( ω+ss