Page 485 - Applied Numerical Methods Using MATLAB

474 LAPLACE TRANSFORM
Table D.2 Properties of Laplace Transform
∞

(0) Definition X(s) = L{x(t)}= x(t)e −st dt
0
(1) Linearity αx(t) + βx(t) → αX(s) + βY(s)

0
(2) Time shifting x(t − t 1 )u s (t − t 1 ), t 1 > 0 → e −st 1 X(s) + x(τ)e −sτ dτ
−t 1
s 1 t
(3) Frequency shifting e x(t) → X(s − s 1 )
(4) Real convolution g(t) ∗ x(t) → G(s)X(s)

(5) Time derivative x (t) → sX(s) − x(0)
t 1 1 0
(6) Time integral x(τ) dτ → X(s) + x(τ) dτ
s s
−∞ −∞
d
(7) Complex derivative tx(t) →− X(s)
ds
1 σ 0 +∞
(8) Complex convolution x(t)y(t) → X(v)Y(s − v) dv
2πj
σ 0 −∞
(9) Initial value theorem x(0) → lim sX(s)
s→∞
(10) Final value theorem x(∞) → lim sX(s)
s→0