364 Chapter 6 Laplace Transform
Theorem. Let f(t)be sectionally continuous in every finite interval 0 ≤t<T. If,
for some constant k, it is true that
tlim→∞e−ktf(t)=^0 ,then the Laplace transform of f exists forRe(s)>k.
A function that satisfies the limit condition in the hypotheses of the theorem
is said to be ofexponential order.
The Laplace transform inherits two important properties from the integral
used in its definition:
L(
cf(t))
=cL(
f(t))
, cconstant, (2)
L(
f(t)+g(t))
=L(
f(t))
+L(
g(t))
. (3)
By exploiting these properties, we easily determine that
L(
cosh(at))
=L[ 1
2
(
eat+e−at)]
=
1
2
( 1
s−a+1
s+a)
=
s
s^2 −a^2 ,L(
sin(ωt))
=L[
1
2 i(
eiωt−e−iωt)]
= 21 i(
1
s−iω−1
s+iω)
=s (^2) +ωω 2.
Notice that the linearity properties work with complex constants and func-
tions.
Because of the factore−stin the definition of the Laplace transform, expo-
nential multipliers are easily handled by the “shifting theorem”:
L
(
ebtf(t))
=
∫∞
0e−stebtf(t)dt=
∫∞
0e−(s−b)tf(t)dt=F(s−b),whereF(s)=L(f(t)). For instance, sinceL(sin(ωt))=ω/(s^2 +ω^2 ),
L(
ebtsin(ωt))
=
ω
(s−b)^2 +ω^2 =ω
s^2 − 2 sb+b^2 +ω^2.
The real virtue of the Laplace transform is seen in its effect on derivatives.
Supposef(t)is continuous and has a sectionally continuous derivativef′(t).