HP 50g User Manual page 531

for non-integer , and for n integer, with n > 0, by
2
Y ( x )
n
where is the Euler constant, defined by
lim
r
and h represents the harmonic series
m
For the case n = 0, the Bessel function of the second kind is defined as
Y x
0
With these definitions, a general solution of Bessel's equation for all values of
is given by
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
H
These functions are also known as the first and second Hankel functions of order
.
In some applications you may also have to utilize the so-called modified Bessel
functions of the first kind of order defined as I (x)= i
imaginary number. These functions are solutions to the differential equation
2 2 2
x (d y/dx ) + x (dy/dx)- (x
Y (x) = [J (x) cos
x
J ( x ) (ln
n
2
n
x
1 1
1 [
2 3
h
m
⎡
J x
⎢
0
⎣
y(x) = K
(1)
(x) = J (x)+i Y (x), and H
n
2
+
– J (x)]/sin
n
(
x
∑
)
m 0
( n m 1 )!
n 1
∑
2 m n
2 m !
m 0
1
... ln r ]
r
1 1
1 ...
2 3
x
∑
m
J (x)+K
1 2
(2)
(x) = J (x) i Y (x),
n
2
) y = 0.
,
m 1
) 1 ( h h
m
2 m n
2 m ( ! m
2 m
x
. 0 5772156649
1
m
m 1
h
m x
2 m 2
m
0
Y (x).
-
J (i x), where i is the unit
)
2 m
m n x
n )!
0 ...,
⎤
2 m
⎥
⎦
as
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