0195136047.pdf

3.3 LAPLACE TRANSFORM 143

f(t)=0 fort<0 and allf(t) exist fort≥0. Also note that in Table 3.3.1, functions 8 through
20 can be considered as being multiplied byu(t).
From Table 3.3.1 of Laplace transform pairs one can see that

L

[

df (t)

dt

]

=sF(s)−f

(

0 +

)

(3.3.4)

By recalling thatvL=LdiL/dtandiC=CdvC/dtand the principle of continuity of the inductor
current and the capacitor voltage, the significance of the termf( 0 +)in Equation (3.3.4) is that the
initial condition is automatically included in the Laplace transform of the derivative, and hence

TABLE 3.3.1Laplace Transform Pairs

f(t) in time domain=L-1[F(s)] ⇔ F(s) in frequency domain=Lf[(t)]

(1)

df (t)

dt sF (s)−f(^0

+)

(2)

d^2 f(t)

dt^2

s^2 F(s)−sf ( 0 +)−

df

dt

( 0 +)

(3) d

nf(t)

dtn

snF(s)−sn−^1 f( 0 +)−sn−^2 df

dt

( 0 +)−···d

n− (^1) f
dtn−^1
( 0 +)
(4) g(t)=
∫t
0
f(t)dt
F(s)
s

• g( 0 +)
  s
  (5)
  ∫t
  0
  f(τ)g(t−τ ) dτ F (s)G(s)
  (6) u(t), unit-step function^1
  s
  (7) δ(t), unit-impulse function 1
  (8) t
  1
  s^2
  (9) t
  n− 1
  (n− 1 )!
  ,ninteger^1
  sn
  (10) ε−at^1
  s+a
  (11) tε−at
  1
  (s+a)^2
  (12) tn−^1 ε−at
  (n− 1 )!
  (s+a)n
  (13) sinωt
  ω
  s^2 +ω^2
  (14) cosωt s
  s^2 +ω^2
  (15) sin(ωt+θ)
  ssinθ+ωcosθ
  s^2 +ω^2
  (16) cos(ωt+θ)
  scosθ−ωsinθ
  s^2 +ω^2
  (17) ε−atsinωt ω
  (s+a)^2 +ω^2
  (18) ε−atcosωt
  s+a
  (s+a)^2 +ω^2
  (19) tε−atsinωt
  2 ω(s+a)
  [(s+a)^2 +ω^2 ]^2
  (20) tε−atcosωt (s+a)
  (^2) −ω 2
  [(s+a)^2 +ω^2 ]^2