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
nm
nmm
m
n
nn
x
nmm
hh
xx
xJxY
2
0
2
1
)!(!2
)()1(
)
2
(ln)(
2
)( ⋅
+⋅⋅
+⋅−
⋅++⋅⋅=
∑
∞
=
+
+
−
π
γ
π
m
n
m
nm
n
x
m
mnx
2
1
0
2
!2
)!1(
⋅
⋅
−−
⋅−
∑
−
=
−
−
π
where γ is the Euler constant
, defined by
...,05772156649.0]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
22
1
00
⋅
⋅
⋅−
++⋅⋅=
∑
∞
=
−
m
m
m
m
m
x
m
h
x
xJxY γ
π
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