Environmental Engineering Reference
In-Depth Information
m
2
,
m
By
Φ
(
ϕ
)
-equation, we obtain
η
=
=
0
,
1
,
2
, ···
. Therefore,
n
d
2
m
2
sin
2
Θ
d
2
+
cot
θ
θ
+
(
n
+
1
)
−
Θ
=
0
.
(A.7)
d
θ
θ
It is called the
associated Legendre equation
.Let
x
=
cos
θ
(
−
1
<
x
<
1
)
and
P
(
x
)=
Θ
(
θ
)
. Equation (A.7) becomes
1
n
P
x
2
d
2
P
m
2
2
x
d
P
−
d
x
2
−
d
x
+
(
n
+
1
)
−
=
0
.
(A.8)
1
−
x
2
It is called
another form of the associated Legendre equation
.
When
u
(
,
θ
,
ϕ
)
η
=
=
r
is independent of
ϕ
, in particular,
0and
m
0. Equa-
tion (A.8) thus reduces to
1
x
2
d
2
P
2
x
d
P
−
d
x
2
−
d
x
+
n
(
n
+
1
)
P
=
0
,
(A.9)
which is called the
Legendre equation
.
A.2 Bessel Functions
The Bessel function is the series solution of the Bessel equation. We have the fol-
lowing theorem regarding series solutions.
Theorem.
Suppose that
a
(
x
)
and
b
(
x
)
are expandable into power series at
x
=
0.
The equation
a
(
x
)
b
(
x
)
y
+
y
+
y
=
0
x
2
x
x
r
+
∞
k
=
0
c
k
x
k
,where
r
is a constant.
Let
x
and
y
be the independent and dependent variables, respectively. A general
form of Bessel equations of
has at least one series solution
y
=
-th order is
x
2
y
+
γ
xy
+(
x
2
2
−
γ
)
y
=
0
,
γ
≥
0
(A.10)
or
1
y
2
1
−
γ
y
+
x
y
+
=
0
,
x
2
where
γ
can be any real or complex constant. Suppose that the solution of Eq. (A.10)
is
x
c
+
∞
+
∞
k
=
0
a
k
x
c
+
k
k
=
0
a
k
x
k
=
=
,
=
.
y
a
0
0
(A.11)
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