Environmental Engineering Reference
In-Depth Information
f
e
−
st
f
Delay Property
Let
L
[
f
(
t
)] =
(
s
)
.Then
L
[
f
(
t
−
τ
)] =
(
s
)
, where the real
constant
0.
Proof
. By the definition of the Laplace transformation,
τ
>
+
∞
e
−
st
d
t
L
[
f
(
t
−
τ
)] =
f
(
t
−
τ
)
0
τ
+
∞
e
−
st
d
t
e
−
st
d
t
=
(
−
τ
)
+
(
−
τ
)
f
t
f
t
0
τ
+
∞
+
∞
e
−
st
d
t
e
−
s
(
u
+
τ
)
d
u
=
f
(
t
−
τ
)
=
f
(
u
)
τ
0
e
−
st
f
=
(
s
)
,
Re
(
s
)
>
k
.
0,
0
f
e
−
st
d
t
When
τ
<
(
t
−
τ
)
=
0 is not valid.
Theorems of Initial Value and Final Value
f
s f
Theorem of initial value
.If
L
[
f
(
t
)] =
(
s
)
and lim
s
→
∞
(
s
)
exist, then
s f
s f
t
→
+
0
f
lim
(
t
)=
lim
s
(
s
)
or
f
(
0
)=
lim
s
(
s
)
.
→
∞
→
∞
s f
Proof.
Since lim
s
(
s
)
exists, we have
→
∞
L
f
(
)
=
s f
)
=
s f
lim
s
t
lim
s
(
s
)
−
f
(
0
lim
s
(
s
)
−
f
(
0
)
.
→
∞
→
∞
→
∞
Also, by the existence theorem of Laplace transformations,
+
∞
f
(
e
−
st
d
t
c
β
−
t
)
≤
k
,
0
f
(
c
e
−
kt
,
Re
where
|
t
)
|≤
(
s
)=
β
.As
s
→
∞
,
β
→
+
∞
so that
L
f
(
lim
s
t
)
=
0
.
→
∞
s f
Thus
.
Therefore we may obtain the initial value of
f
f
(
0
)=
lim
s
(
s
)
→
∞
by taking the limit of
s f
(
t
)
(
s
)
as
s
.
Theorem of final value.
If
L
→
∞
f
s f
[
f
(
t
)] =
(
s
)
and lim
s
(
s
)
exist, then
→
∞
0
s f
0
s f
.
Proof
. By the differential property of the Laplace transformation,
lim
f
(
t
)=
lim
s
(
s
)
or
f
(
∞
)=
lim
s
(
s
)
t
→
+
∞
→
→
+
∞
0
L
f
(
)
=
0
s f
f
(
e
−
st
d
t
)
=
(
)
−
(
)
,
lim
s
→
t
lim
s
→
t
lim
s
→
s
f
0
0
0
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