We define the singlesided Laplace transform as
X
(
s
)
=
∫
0

∞
x
(
t
)
e

s
t
d
t
X
(
s
)
=
∫
0

∞
x
(
t
)
e

s
t
d
t
(10)where the lower limit of integration tacitly includes the point t=0t=0. That is, 00 represents a point just to the negative side of t=0t=0. This allows for the integral to take into account signal features that occur at t=0t=0 such as a step or impulse function. The singlesided Laplace transform is motivated by the fact that most signals are turned on at some point. If the region of convergence is not specified then different signals can yield identical doublesided Laplace transforms, as the following example illustrates.
Example 3.1 Consider the signal
x
1
(
t
)
=
e

α
t
u
(
t
)
x
1
(
t
)
=
e

α
t
u
(
t
)
(11)The doublesided Laplace transform is given by
X
1
(
s
)
=
∫
0
∞
e

α
t
e

s
t
d
t
=
∫
0
∞
e

(
s
+
α
)
t
d
t
=

1
s
+
α
e

(
σ
+
j
Ω
+
α
)
t
0
∞
=

1
s
+
α
e

(
σ
+
α
)
∞
e
j
Ω
∞

1
X
1
(
s
)
=
∫
0
∞
e

α
t
e

s
t
d
t
=
∫
0
∞
e

(
s
+
α
)
t
d
t
=

1
s
+
α
e

(
σ
+
j
Ω
+
α
)
t
0
∞
=

1
s
+
α
e

(
σ
+
α
)
∞
e
j
Ω
∞

1
(12)the magnitude of e(σ+α)∞e(σ+α)∞ is zero only if σ>ασ>α which establishes the region of convergence. Therefore we have
e

α
t
u
(
t
)
↔
1
s
+
α
,
σ
>

α
e

α
t
u
(
t
)
↔
1
s
+
α
,
σ
>

α
(13)Now consider the Laplace transform of the signal
x
2
(
t
)
=

e

α
t
u
(

t
)
x
2
(
t
)
=

e

α
t
u
(

t
)
(14)We have
X
2
(
s
)
=

∫

∞
0
e

α
t
e

s
t
d
t
=

∫

∞
0
e

(
s
+
α
)
t
d
t
=
1
s
+
α
e

(
σ
+
j
Ω
+
α
)
t

∞
0
=
1
s
+
α
1

e
(
σ
+
α
)
∞
e
j
Ω
∞
X
2
(
s
)
=

∫

∞
0
e

α
t
e

s
t
d
t
=

∫

∞
0
e

(
s
+
α
)
t
d
t
=
1
s
+
α
e

(
σ
+
j
Ω
+
α
)
t

∞
0
=
1
s
+
α
1

e
(
σ
+
α
)
∞
e
j
Ω
∞
(15)Here, the quantity e(σ+α)∞e(σ+α)∞ is zero only if σ<ασ<α so we have

e

α
t
u
(

t
)
↔
1
s
+
α
,
σ
<

α

e

α
t
u
(

t
)
↔
1
s
+
α
,
σ
<

α
(16)which is identical to X1(s)X1(s) except for the region of convergence.
To avoid scenarios where two different signals have the same doublesided Laplace transform, we restrict our signals to those which are assumed to be zero for t<0t<0. Such signals are called causal signals.
The region of convergence for the singlesided Laplace transform is a region in the ssplane satisfying σ>σminσ>σmin as shown in Figure 2.
To see this, we observe that if
∫
0

∞
x
(
t
)
e

σ
m
i
n
t
d
t
<
∞
∫
0

∞
x
(
t
)
e

σ
m
i
n
t
d
t
<
∞
(17)then it must be the case that
∫
0

∞
x
(
t
)
e

σ
t
d
t
<
∞
∫
0

∞
x
(
t
)
e

σ
t
d
t
<
∞
(18)for σ>σminσ>σmin, since eσteσt decreases faster than eσminteσmint. Finally, the inverse singlesided Laplace transform is the same as the inverse doublesided Laplace transform (see Equation 7), since a single sided Laplace transform can be interpreted as the doublesided Laplace transform of a signal satisfying x(t)=0,t<0x(t)=0,t<0. From here on, we will work exclusively with the singlesided Laplace transform. Unless we need to specifically differentiate between the single or doublesided transforms, we will refer to the singlesided Laplace transform as simply the “Laplace transform".