3. Definition of Laplace Transform
• Let f(t) be a given function of t defined for all
then the Laplace Transform ot f(t) denoted by L{f(t)}
or F(s) or is defined as
provided the integral exists, where s is a parameter real or
complex.
0t
)(sf )(s
dttfessFsftfL st
)()()()()}({
0
4. Laplace Transform of some
Elementary Functions
asif
a-s
1
)(
e.)e(
Definition-By:Proof
a-s
1
)L(e(2)
)0(,
s
1
1.)1(
Definition-By:Proof
s
1
L(1)(1)
0
)(
0
)(
0
atat
at
00
as
e
dtedteL
s
s
e
dteL
tas
tasst
st
st
7. n!1n0,1,2...n
n!
)(or
0,n-1n,
1
)(
1
ust,.)-L(:Proof
n!
or
1
)()8(
1
0
1
1
0
1)1(
1
0
0
11
n
n
nx
n
n
nu
n
n
u
nstn
nn
n
S
tL
ndxxe
S
n
tL
duue
S
s
du
s
u
e
puttingdttet
SS
n
tL