Environmental Engineering Reference
In-Depth Information
Remark.
An irrational number is regarded as the limit of a sequence of rational
numbers. Similarly, a generalized function is defined as the weak limit of a sequence
of classical functions in a specified function space. Equation (B.8) shows that
{
f
n
}
is weakly convergent to
f
. If a sequence
{
r
n
}
of rational numbers satisfies
lim
n
r
n
=
α
(irrational number)
,
→
∞
we define
α
=
{
r
n
}
. Similarly, we define
f
=
{
f
n
}
if
{
f
n
}
is weakly convergent to
f
. We can also write
f
n
→
f
as
weak
===
lim
n
f
n
(
x
)
f
(
x
)
.
→
∞
The
f
(
x
)
is called the
weak limit
of
{
f
n
(
x
)
}
as
n
→
+
∞
.Forafixed
α
,
{
r
n
}
is not
unique. Similarly,
{
f
n
}
is not unique either for a fixed
f
.
Analytical Definition of the
δ
-Function
Definition 5.
The
Dirac function
(or
δ
-function) is a functional in
K
whose values
are
ϕ
(
0
)
for all
ϕ
∈
K
such that
δ
(
ϕ
)=
ϕ
(
0
)
.The
δ
(
ϕ
)
is denoted by
δ
(
x
)
.
-function is a continuous linear functional; therefore it is a
generalized function. Let the functional in
K
defined by Eq. (B.7) be
It can be shown that
δ
ϕ
(
0
)
such that
+
∞
(
f
,
ϕ
)=
f
(
x
)
ϕ
(
x
)
d
x
=
ϕ
(
0
)
,
∀
ϕ
∈
K
.
(B.9)
−
∞
Clearly, the
f
in Eq. (B.9) cannot be a classical integrable function. The generalized
function
f
in Eq. (B.9) is in fact the
δ
-function. Equation (B.9) is often written as
+
∞
(
δ
,
ϕ
)=
δ
(
x
)
ϕ
(
x
)
d
x
=
ϕ
(
0
)
,
∀
ϕ
∈
K
.
−
∞
Similarly,
δ
(
x
−
x
0
)
represents
δ
(
ϕ
)=
ϕ
(
x
0
)
such that
+
∞
(
δ
,
ϕ
)=
δ
(
x
−
x
0
)
ϕ
(
x
)
d
x
=
ϕ
(
x
0
)
,
∀
ϕ
∈
K
.
−
∞
δ
(
−
,
−
)
δ
(
−
,
−
,
−
)
Also,
x
x
0
y
y
0
and
x
x
0
y
y
0
z
z
0
represent
δ
(
ϕ
)=
ϕ
(
,
)
,
δ
(
ϕ
)=
ϕ
(
,
,
)
,
x
0
y
0
x
0
y
0
z
0
respectively.
Definition 6.
If the equivalent classes
f
=
{
}
f
n
of basic sequences in
K
satisfy
+
∞
lim
n
f
n
(
x
)
ϕ
(
x
)
d
x
=
ϕ
(
0
)
,
∀
ϕ
∈
K
,
(B.10)
→
∞
−
∞
the generalized function
f
=
{
f
n
}
is called the
δ
-function
. Denote
δ
=
{
f
n
}
,i.e.
+
∞
(
δ
,
ϕ
)=
lim
n
f
n
(
x
)
ϕ
(
x
)
d
x
=
ϕ
(
0
)
,
∀
ϕ
∈
K
.
(B.11)
→
∞
−
∞
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