Page 475 - Mechanics of Asphalt Microstructure and Micromechanics

APPENDIX2

Laplace Transform

TABLE A2.1 Typical operations and relations.
∞
−
st
{(
( )
Laplace transform Lf t)} ≡ F s ( ) ≡ ∫ f t e dt
0
t
∫ ft ( − ξ g ) ( )ξ dξ
Convolution
0
t
{ ( )} {
Convolution for Laplace L{ ∫ f t ( − ξ g ) ( )ξ d } =ξ L f t L g t ( )}
0
df t ()
0
{
Derivative theorem for Laplace L{ } = sL f t)} − f( )
(
dt
{
Linearity (superposition) La f t () + a f t ()} = a F s ( ) + a F s ( )
2 2
22
1 1
11
⎧ df t ()⎫
f( )
Derivative property L ⎨ ⎬ = sF s() − 0
⎩ dt ⎭
⎧ ⎪ t ⎫ ⎪ Fs()
d
Integral property L ⎨ ∫ f()ττ ⎬ =
s ⎩ ⎪ ⎭ ⎪ s
(
Multiplication by time Ltf t)} =− d Fs)
{(
ds
⎧ ft ()⎫ ∞
()
Division by time L ⎨ ⎬ = ∫ Fu du
⎩ t ⎭ s
−
at
Multiplication by an exponential Le f t ( )} = F s ( + a)
{
−
Ts
{(
)
Time shift Lf t − T H t ( − T)} = e F s ( )
s
{(
Scale change Lf at)} = 1 F( )
a a
⎧ ⎪
L ⎨∫ t f() (τ g t τ τ− ⎫ ⎪ ⎬ = F s () •
Convolution theorem d ) G s ()
⎩ ⎪ 0 ⎭ ⎪
467