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