HP 50g Graphing Calculator - Manual (Page 531)

Page 16-54
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
where γ is the Euler constant, defined by
and h
m
represents the harmonic series
For the case n = 0, the Bessel function of the second kind is defined as
With these definitions, a general solution of Bessel’s equation for all values of ν
is given by y(x) = K
1
⋅J
ν
(x)+K
2
⋅Y
ν
(x).
In some instances, it is necessary to provide complex solutions to Bessel’s
equations by defining the Bessel functions of the third kind of order
ν as
H
n
(1)
(x) = J
ν
(x)+i ⋅Y
ν
(x), and H
n
(2)
(x) = J
ν
(x) −i⋅Y
ν
(x),
These functions are also known as the first and second Hankel functions of order
ν.
In some applications you may also have to utilize the so-called modified Bessel
functions of the first kind of order
ν defined as I
ν
(x)= i
- ν ⋅
J
ν
(i
⋅
x), where i is the unit
imaginary number. These functions are solutions to the differential equation
x
2 ⋅(d 2 y/dx 2 ) + x ⋅ (dy/dx)- (x 2 + ν 2 ) ⋅y = 0.
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
..., 0 5772156649 . 0 ] ln
1
...
3
1
2
1
1 [ lim ≈ − + + + + =
∞ →
r
r
r
γ
m
h
m
1
...
3
1
2
1
1 + + + + =
.
) ! ( 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