Environmental Engineering Reference
In-Depth Information
Note that by the theory of power series, we can take derivatives of
y
with respect to
x
term by term in the convergent domain of
+
∞
k
=
0
a
k
x
k
. Thus
+
∞
k
=
0
(
c
+
k
)
a
k
x
c
+
k
−
1
+
∞
k
=
0
(
c
+
k
)(
c
+
k
−
1
)
a
k
x
c
+
k
−
2
y
=
y
=
,
.
Substituting
y
,
y
and
y
into Eq. (A.10) yields
+
∞
c
2
2
a
0
+
(
2
a
1
x
k
=
2
(
c
+
k
)
2
a
k
+
a
k
−
2
x
k
2
2
−
γ
c
+
1
)
−
γ
+
−
γ
=
0
.
By the uniqueness of expansion of power series, we thus have
a
0
c
2
2
=
a
1
(
2
=
2
−
γ
0
,
c
+
1
)
−
γ
0
,
2
a
k
+
2
(
+
)
−
γ
a
k
−
2
=
,
=
,
, ··· .
c
k
0
k
2
3
From the first equation, we obtain
c
2
2
−
γ
=
=
±
γ
0, so
c
. Substituting it into the
=
=
γ
second equation yields
a
1
0. A substitution of
c
into the third equation leads
−
a
k
−
2
to
a
k
=
.Since
a
1
=
0, we have
a
1
=
a
3
=
a
5
=
···
=
0.
(
γ
+
)
k
2
k
When
k
is even,
−
a
0
a
0
a
2
=
)
,
a
4
=
)
,
2
(
2
γ
+
2
2
·
4
(
2
γ
+
2
)(
2
γ
+
4
−
a
0
a
6
=
)
··· ,
2
·
4
·
6
(
2
γ
+
2
)(
2
γ
+
4
)(
2
γ
+
6
a
0
m
a
2
m
=(
−
1
)
)
.
2
2
m
m
!
(
γ
+
)(
γ
+
)
···
(
γ
+
1
2
m
Thus Eq. (A.11) becomes
+
∞
m
=
0
(
−
1
)
a
0
x
n
+
2
m
m
y
=
)
,
2
2
m
m
!
(
γ
+
1
)(
γ
+
2
)
···
(
γ
+
m
where
a
0
can be any constant. For a specified
a
0
, we have a particular solution of
Eq. (A.10). For convenience, let
1
a
0
=
)
,
2
n
Γ
(
γ
+
1
where the
Γ
−
function satisfies
Γ
(
x
+
1
)=
x
Γ
(
x
)
. The particular solution
J
γ
(
x
)
of
Eq. (A.10) is thus
x
2
γ
+
2
m
+
∞
m
=
0
m
(
−
1
)
J
γ
(
x
)=
.
(A.12)
m
!
Γ
(
γ
+
m
+
1
)
It is called the
Bessel function of
γ
-th order of the first kind
. It converges, by the
ratio test, over the whole real-axis.
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