Environmental Engineering Reference
In-Depth Information
is also a particular solution of Eq. (A.16) and is called the
modified Bessel function
of the first kind
.When
γ
is not an integer, the particular solution
2
π
I
−
γ
(
)
1
x
)
−
I
γ
(
x
K
γ
(
x
)=
sin
γπ
is linearly independent of
I
γ
(
and is called the
modified Bessel function of the
second kind
. To demonstrate its linear independence with
I
γ
(
x
)
x
)
, consider
γ
>
0(
γ
is not an integral) so that
I
γ
(
0
)=
0,
I
−
γ
(
0
)=
∞
and
K
γ
(
0
)=
∞
. Therefore
K
γ
(
x
)
is
linearly independent of
I
γ
(
x
)
.When
γ
is an integer
n
,define
2
π
I
−
γ
(
)
1
x
)
−
I
γ
(
x
K
n
(
x
)=
lim
γ
→
n
K
γ
(
x
)=
lim
γ
→
.
sin
γπ
n
(
)
(
)
It can be shown that
K
n
x
so defined is linearly independent of
I
n
x
. Therefore the
general solution of Eq. (A.6) is, regardless of the value of
γ
y
=
C
1
I
γ
(
x
)+
C
2
K
γ
(
x
)
,
where
C
1
and
C
2
are constants.
The Kelvin function of
n
-th
ord
er of the first kind has two forms: the real part
and the imaginary part of
J
n
x
√
−
i
termed by
ber
n
(
x
)
and
be
i
n
(
x
)
, respectively.
Re
J
n
x
√
−
i
Im
J
n
x
√
−
i
ber
n
(
x
)=
,
be
i
n
(
x
)=
.
The
ber
0
(
x
)
,
be
i
0
(
x
)
,
ber
1
(
x
)
and
be
i
1
(
x
)
appear frequently in applications.
A.3 Properties of Bessel Functions
Differential Property and Recurrence Formula
Based on the series expression of
J
1
(
x
)
, the Bessel function of the first kind, a dif-
ferentiation term by term yields
d
d
x
x
γ
J
γ
(
)
=
d
x
x
−
γ
J
γ
(
)
=
−
d
x
−
γ
J
γ
+
1
(
x
γ
J
γ
−
1
(
x
x
)
,
x
x
)
,
2
x
γ
2
J
γ
(
J
γ
−
1
(
x
)+
J
γ
+
1
(
x
)=
J
γ
(
x
)
,
J
γ
−
1
(
x
)
−
J
γ
+
1
(
x
)=
x
)
.
When
γ
=
n
(natural numbers), in particular, we have the recurrence formula
2
x
nJ
n
(
J
n
−
1
(
x
)+
J
n
+
1
(
x
)=
x
)
.
Therefore we may obtain values of
J
N
(
x
)
based on the function tables of
J
0
(
x
)
and
J
1
(
x
)
for any natural number
N
>
1. We may also obtain
J
−
N
(
x
)
by
N
J
N
(
J
−
N
(
x
)=(
−
1
)
x
)
.
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