Page 476 - Mechanics of Asphalt Microstructure and Micromechanics

468 A p p e n d i x Tw o — Laplace T ransform

TABLE A2.2 Laplace transformations of typical functions.
f(t) F(s)
d (t) 1
d (t − a) e –as
H(t) 1/s
1 −
H(t − a) e as
s
tH(t) 1/s 2
1
e –at
s + a
t n−1 e − at 1
( n − 1)! (s + ) a n
( 1 e )
1 1
−
−
at
(
a ss + a)
⎡ ⎣ be − ae ⎤ ⎦ 1 for a ≠ b
at
bt
( b − a) (s − a )(s − ) b
⎡ ⎣ e − e ⎤ ⎦ s for a ≠ b
bt
at
)
(
( b − a) s ( − a s − b)
s
cosh(at)
s − a 2
2
a
sinh(at)
s − a 2
2
s
cos(at)
s + a 2
2
a
sin(at)
s + a 2
2
sinat
−1
tan ( / ) s
a
t
s − a 2
2
( s + )
tcos(at) 2
2
2
a
2as
tsin(at)
s + a 2
2
ω
( s + a) + ω 2
e –at sin w t
2
s + a
( s + a) + ω 2
e –at cos w t
2
n
t n Γ(n + 1 )s −−1
n
t e –qt Γ(n + 1 )(s + ) q −−1
n