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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
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
...,05772156649.0]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
22
1
00
⎥
⎦
⎤
⎢
⎣
⎡
⋅
⋅
⋅−
++⋅⋅=
∑
∞
=
−
m
m
m
m
m
x
m
h
x
xJxY
γ
π