Page 16-56
If you want to obtain an expression for J
0
(x) with, say, 5 terms in the series,
use J(x,0,5). The result is
‘1-0.25*x^3+0.015625*x^4-4.3403777E-4*x^6+6.782168E-6*x^8-
6.78168*x^10’.
For non-integer values
ν
, the solution to the Bessel equation is given by
y(x) = K
1
⋅
J
ν
(x)+K
2
⋅
J
-
ν
(x).
For integer values, the functions Jn(x) and J-n(x) are linearly dependent, since
J
n
(x) = (-1)
n
⋅
J
-n
(x),
therefore, we cannot use them to obtain a general function to the equation.
Instead, we introduce the Bessel functions of the second kind
defined as
Y
ν
(x) = [J
ν
(x) cos
νπ
– J
−ν
(x)]/sin
νπ
,
for non-integer
ν
, and for n integer, with n > 0, by
m
m
n
m
n
m
m
m
n
n
n
x
n
m
m
h
h
x
x
x
J
x
Y
2
0
2
1
)!
(
!
2
)
(
)
1
(
)
2
(ln
)
(
2
)
(
⋅
+
⋅
⋅
+
⋅
−
⋅
+
+
⋅
⋅
=
∑
∞
=
+
+
−
π
γ
π
m
n
m
n
m
n
x
m
m
n
x
2
1
0
2
!
2
)!
1
(
⋅
⋅
−
−
⋅
−
∑
−
=
−
−
π
where
γ
is the Euler constant, defined by
...,
0
5772156649
.
0
]
ln
1
...
3
1
2
1
1
[
lim
≈
−
+
+
+
+
=
∞
→
r
r
r
γ
and h
m
represents the harmonic series
m
h
m
1
...
3
1
2
1
1
+
+
+
+
=
For the case n = 0, the Bessel function of the second kind is defined as
.
)
!
(
2
)
1
(
)
2
(ln
)
(
2
)
(
2
0
2
2
1
0
0
⋅
⋅
⋅
−
+
+
⋅
⋅
=
∑
∞
=
−
m
m
m
m
m
x
m
h
x
x
J
x
Y
γ
π
