HP 48gII - Manual (Page 534)

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