Chapter 62
Properties of Laplace
transforms
62.1 The Laplace transform ofeatf(t)
FromChapter61,thedefinitionoftheLaplacetransform
off(t)is:
L{f(t)}=
∫∞
0
e−stf(t)dt (1)
ThusL{eatf(t)}=
∫∞
0
e−st(eatf(t))dt
=
∫∞
0
e−(s−a)f(t)dt (2)
(where a is a real constant)
Hence the substitution of(s−a)forsin the transform
shown in equation (1) corresponds to the multiplication
of the original functionf(t)by eat. This is known as a
shift theorem.
62.2 Laplace transforms of the form
eatf(t)
From equation (2), Laplace transforms of the form
eatf(t)may be deduced. For example:
(i) L{eattn}
SinceL{tn}=
n!
sn+^1
from (viii) of Table 61.1,
page 584.
thenL{eattn}=
n!
(s−a)n+^1
from equation (2)
above (provideds>a).
(ii) L{eatsinωt}
SinceL{sinωt}=
ω
s^2 +ω^2
from (iv) of Table
61.1, page 584.
then L{eatsinωt}=
ω
(s−a)^2 +ω^2
from equa-
tion (2) (provideds>a).
(iii) L{eatcoshωt}
SinceL{coshωt}=
s
s^2 −ω^2
from (ix) of Table
61.1, page 584.
thenL{eatcoshωt}=
s−a
(s−a)^2 −ω^2
from equa-
tion (2) (provideds>a).
A summary of Laplace transforms of the form
eatf(t)is shown in Table 62.1.
Table 62.1Laplace transforms of the form
eatf(t)
Function eatf(t) Laplace transform
(ais a real constant) L{eatf(t)}
(i) eattn
n!
(s−a)n+^1
(ii) eatsinωt
ω
(s−a)^2 +ω^2
(iii) eatcosωt
s−a
(s−a)^2 +ω^2
(iv) eatsinhωt
ω
(s−a)^2 −ω^2
(v) eatcoshωt
s−a
(s−a)^2 −ω^2