Common Laplace Transform Pairs
Time Domain Function
Name
Definition*
δ (t )
Unit Impulse
Unit Step
γ (t)
Unit Ramp
t
Parabola
t2
Exponential
e − at
†
1
(1 − e −at )
a
Asymptotic
Exponential
(
)
1
e −at − e −bt
b−a
1 
1

1+
be − at − ae − bt ) 
(

ab  a − b

Dual Exponential
Asymptotic Dual
Exponential
Time multiplied
Exponential
te − at
Sine
sin(ω0 t)
Cosine
cos(ω0 t)
Decaying Sine
e − at sin(ωd t)
Decaying Cosine
e − at cos(ωd t)
Generic Oscillatory
Decay
Prototype Second
Order Lowpass,
underdamped
Prototype Second
Order Lowpass,
underdamped
Step Response
Laplace Domain
Function
1
1
s
1
s2
2
s3
1
s+a
1
s( s + a)
1
( s + a )( s + b)
1
s ( s + a )( s + b)
1
( s + a) 2
ω0
2
s + ω02
s
2
s + ω02


C − aB
e − at  Bcos ( ωd t ) +
sin ( ωd t ) 
ωd


ω0
e −ζω0 t sin ω0 1 − ζ 2 t
2
1− ζ
(
1−
1
1− ζ
2
(
)
e −ζω0 t sin ω0 1 − ζ 2 t + φ
 1− ζ2
φ = tan −1 
 ζ





)
ωd
(s + a) 2 + ωd2
s+a
(s + a) 2 + ωd2
Bs + C
(s + a )
2
+ ωd2
ω02
s 2 + 2ζω0s + ω02
ω02
s(s 2 + 2ζω0s + ω02 )
*All time domain functions are implicitly=0 for t<0 (i.e. they are multiplied by unit step, γ(t)).
†u(t) is more commonly used for the step, but is also used for other things. γ(t) is chosen to avoid confusion
(and because in the Laplace domain it looks a little like a step function, Γ(s)).
Common Laplace Transform Properties
Name
Definition of Transform
Illustration
L
f (t) ←
→ F(s)
∞
F(s) = ∫ − f (t)e − st dt
0
Linearity
First Derivative
Second Derivative
nth Derivative
Integral
Time Multiplication
Time Delay
Complex Shift
Scaling
Convolution Property
Initial Value
(Only if F(s) is strictly proper;
order of numerator < order of denominator).
Final Value
(if final value exists;
e.g., decaying exponentials or constants)
L
Af1 (t ) + Bf 2 (t ) ←→
AF1 ( s ) + BF2 ( s )
df (t ) L
←→ sF ( s ) − f (0 − )
dt
2
d f (t ) L
←→ s 2 F ( s ) − sf (0 − ) − f (0 − )
dt 2
n
d n f (t ) L
n
s
F
s
s n−i f (i −1) (0 − )
←
→
−
(
)
∑
n
dt
i =1
t
1
L
∫0 f (λ )dλ ←→ s F (s)
dF ( s )
L
tf (t ) ←→
−
ds
L
f (t − a) γ (t − a) ←→ e − as F(s)
γ(t) is unit step
f (t )e
− at
L
←→
F (s + a)
t L
f   ←→
aF (as )
a
L
f1 (t ) * f 2 (t ) ←→
F1 ( s ) F2 ( s )
lim f (t ) = lim sF ( s )
t →0 +
s →∞
lim f (t ) = lim sF ( s )
t →∞
s →0