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Laplace Transform Table
Largely modeled on a table in D’Azzo and Houpis, Linear Control Systems Analysis and Design , 1988
F ( s ) f ( t ) 0 ≤ t
- (^1) δ ( t ) unit impulse at t = 0
s
(^1) 1 or u ( t ) unit step starting at t = 0
1
s
t ⋅ u ( t ) or t ramp function
- (^) n
s
n t n
n = positive integer
as
e
s
1 − u ( t − a ) unit step starting at t = a
6. (^1 )
1 as
e
s
−
u ( t )− u ( t − a ) rectangular pulse
s + a
1 at e
− exponential decay
- (^) n ( s a )
1
n at t e n
− −
−
1
( 1 )!
n = positive integer
( )
1
s s + a
(^1) at e a
− −
s s + a s + b
(^1) at bt e b a
a e b a
b
ab
− −
−
s ( s a )( s b )
s
+α ]
[
(^1) at bt e b a
a b e b a
b a
ab
− −
−
α α α
s + a s + b
(^1) at bt e e b a
− − − −
( s a )( s b )
s
( )
(^1) at bt ae be a b
− − − −
F(s) f(t) 0 ≤ t
( s a )( s b )
s
+α
[( ) ( ) ]
1 at bt
a e b e
b a
− −
s + a s + b s + c ( )( ) ( )( ) ( a c )( b c )
e
c b a b
e
b a c a
e
at bt ct
− − −
( s a )( s b )( s c )
s
+α
a c b c
ce
c b a b
be
b a c a
a e
at bt ct
− − − α α α
ω
ω
s +
sin ω t
s + ω
s cos^ ω t
s
s
sin( )
2 2 ω φ ω
α ω
t (^) φ =atan2( ω, α)
sin cos
ω
θ ω θ
s
s (^) sin( ω t + θ)
2 2
s s + ω
( 1 cos )
2
ω t
2 2 ω
α
s s
s
2 cos( )
2 2
2 ω^ φ ω
α ω
ω
α
− t φ =atan2( ω, α)
2 2 s + a s + ω
sin( )
(^2222)
− t a a
e
at
φ =atan2( ω, α )
s + a + b
sin( )
e bt b
− at
24a. (^2 ) 2
1
s + ζω (^) n s + ω n
sin( 1 ) 1
(^1 )
2
e (^) n t
t
n
n ω ζ
ω ζ
ζω − −
−
( s a ) b
s a
F(s) f(t) 0 ≤ 1
( )[( ) ]
2 2
s s + c s + a + b
(^22222222)
sin( )
( ) [( ) ]
b a b c a b
e bt
c c a b
e
ca b
ct at
− − φ
φ=atan2( b , − a )+atan2( b , c − a )
( )[( ) ]
2 2 ss c s a b
s
+α
sin( ) ( )
[( ) ]
2 2 2 2
2 2
2 2 2 2
−
−
e bt b a b c a b
a b
c c a b
c e
ca b
at
ct
φ=atan2( b , α− a )−atan2( b ,− a )−atan2( b , c − a )
( )
1 2 s s + a
( 1 )
1 2
at at e a
− − +
s s + a
( 1 )
1 2
at at e ate a
− − − −
s ( s a )
s
[ ( ) ]
2
at at
e aa te
a
− − α−α + −α
( )( )
1 0
2
s s a s b
s s
+α +α at bt e ba b
b b e a a b
a a
ab
− −
−
− + − −
− +
( ) ( )
1 0
2 1 0
2 α 0 α α α α
[( ) ]
2 2
1 0
2
s s a b
s s
+α +α 2 1 0
2 2 2
0 [( )
1 α α
α
( 2 ) ] sin( )
2
1 2 1
2
−
b a e bt
at
atan2[ ( 2 ), 1 0 ] atan2( , )
2 2 φ= b α 1 − a a − b −α a +α − b − a
2 2 2
c = a + b
F(s)
f(t)^0 ≤^1
( )[( ) ]
1
2 2 2 2
s +ω s + a + b
2
1 2 2 2 2 2 2
1 2
[ 4 ( ) ]
( 1 / )sin( ) ( 1 / ) sin( )
ω ω
ω ω φ φ
−
a a b
t b e bt
at
atan2( 2 , )
2 2 2
φ 1 = − a ω a + b − ω
atan2( 2 , )
2 2 2 φ 2 = ab a − b + ω
( )[( ) ]
2 2 2 2 s s a b
s
α ( ) sin( )
1 1
2
2 2 1
t c
] sin( )
( ) [
1 2
2
2 2 1
− +
− e bt c
a b
b
at
2 2 2 2 2 c = ( 2 a ω) +( a + b − ω )
atan2( , ) atan2( 2 , )
2 2 2 φ 1 = ω α − a ω a + b + ω
atan2( , ) atan2( 2 , )
2 2 2 φ 2 = b α− a + aba − b − ω
[( ) ]
2 2 2 s s a b
s
sin( )
[ ( ) ]
1 2 2
− e bt bc
b a
c
a t c
at
2 2 c = a + b
φ=2atan2( b , a )+atan2( b , α− a )
( )( )
2
1 0
2
s s a s b
s s
+α +α at e ab a b a a
a b
ab
t (^) − − + −
−
−
( 1 )
1
( )
( )
2
1 0 2
α 1 α 0 α 0 α α
bt e b a b b
− − + −
− ( 1 )
1
2
α 1 α 0
