Common Laplace Transform Pairs

Time Domain Function Laplace Domain Function Name Definition*

Unit Impulse )(tδ 1

Unit Step (t)γ † s1

Unit Ramp t 2

1s

Parabola t2 3

2s

Exponential ate− as +1

Asymptotic Exponential )1(1 ate

a−−

)(1

ass +

Dual Exponential ( )btat eeab

−− −−1

))((

1bsas ++

Asymptotic Dual Exponential ( )at bt1 11 be ae

ab a b− − + − −

))((

1bsass ++

Time multiplied Exponential

atte− 2)(1as +

Sine 0sin( t)ω 02 2

0sω+ ω

Cosine 0cos( t)ω 2 20

ss +ω

Decaying Sine atde sin( t)− ω d

2 2d(s a)

ω+ + ω

Decaying Cosine atde cos( t)− ω 2 2

d

s a(s a)

++ +ω

Generic Oscillatory Decay ( ) ( )at

d dd

C aBe Bcos t sin t− −ω + ω ω

( )2 2d

Bs Cs a

+

+ +ω

Prototype Second Order Lowpass, underdamped

( )0t 2002

e sin 1 t1

−ζωωω −ζ

−ζ

20

2 20 0s 2 s

ω+ ζω +ω

Prototype Second Order Lowpass, underdamped

- Step Response

( )0t 202

21

11 e sin 1 t1

1tan

−ζω

−

− ω −ζ + φ−ζ

− ζ φ = ζ

20

2 20 0s(s 2 s )

ω+ ζω +ω

*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 Illustration

Definition of Transform

L

st

0

f (t) F(s)

F(s) f (t)e dt−

∞ −

←→

= ∫

Linearity )()()()( 2121 sBFsAFtBftAf L +→←+

First Derivative )0()()( −−→← fssFdt

tdf L

Second Derivative )0()0()()( 22

2−− −−→← fsfsFs

dttfd L

nth Derivative )0()()( )1(

1

−−

=

−∑−→← in

i

innLn

n

fssFsdt

tfd

Integral )(1)(0

sFs

df Lt→←∫ λλ

Time Multiplication ds

sdFttf L )()( −→←

Time Delay L asf (t a) (t a) e F(s)−− γ − ←→

γ(t) is unit step

Complex Shift )()( asFetf Lat +→←−

Scaling )(asaFatf L→←

Convolution Property )()()(*)( 2121 sFsFtftf L→← Initial Value

(Only if F(s) is strictly proper; order of numerator < order of denominator).

)(lim)(lim0

ssFtfst ∞→→

=+

Final Value (if final value exists;

e.g., decaying exponentials or constants) )(lim)(lim

0ssFtf

st →∞→=