Page 113 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

INTEGRAL TRANSFORMS: THE LAPLACE TRANSFORM 103
and from Eq. (6.8)

1 1
Y = −
s + 1 s + 3 (6.42)
y = e −t − e −3t

6.4 SOME IMPORTANT THEOREMS
6.4.1 Initial Value Theorem

∞

lim f (t)e −st dt = sF(s ) − f (0) = 0 (6.43)

s →∞
t=0
Thus

lim sF(s ) = lim f (t) (6.44)
s →∞ t→0

6.4.2 Final Value Theorem
As s approaches zero the above integral approaches the limit as t approaches infinity minus
f (0). Thus

lim sF(s ) = lim f (t)
(6.45)
s → 0 t →∞

6.4.3 Convolution
A very important property of Laplace transforms is the convolution integral. As we shall see
later, it allows us to write down solutions for very general forcing functions and also, in the
case of partial differential equations, to treat both time dependent forcing and time dependent
boundary conditions.
Consider the two functions f (t)and g(t). F(s ) = L[ f (t)] and G(s ) = L[g(t)]. Because
of the time shifting feature,
∞

e −s τ G(s ) = L[g(t − τ)] = e −st g(t − τ)dt (6.46)
t=0
∞

F(s )G(s ) = f (τ)e −s τ G(s )dτ (6.47)
τ=0