Niels Walet 5/14/2020 10.5.1 CC- BY-NC- SA https://math.libretexts.org/link?8329
10.5: Properties of Bessel functions
Bessel functions have many interesting properties:
Let me prove a few of these. First notice from the definition that is even or odd if is even or odd,
Substituting in the definition of the Bessel function gives if , since in that case we have the sum of positive
powers of , which are all equally zero.
Let’s look at :
Here we have used the fact that since , [this can also be proven by defining a recurrence relation
for ]. Furthermore we changed summation variables to .
The next one:
(0)
J
0
(
x
)
J
ν
(
x
)
J
−
n
[
(
x
)
]
d
dx
x
−
ν
J
ν
[ (
x
)]
d
dx
x
ν
J
ν
[ (
x
)]
d
dx
J
ν
x
(
x
)
J
ν
+1
∫
(
x
)
dx
x
−
ν
J
ν
+1
∫
(
x
)
dx
x
ν
J
ν
−1
= 1,
= 0 (if
ν
> 0),
= (−1 (
x
),)
n
J
n
= − (
x
),
x
−
ν
J
ν
+1
= (
x
),
x
ν
J
ν
−1
= [ (
x
)− (
x
)] ,
1
2
J
ν
−1
J
ν
+1
= 2
ν
(
x
)−
x
(
x
),
J
ν
J
ν
−1
= − (
x
)+
C
,
x
−
ν
J
ν
= (
x
)+
C
.
x
ν
J
ν
(
x
)
J
n
n
(
x
) = .
J
n
∑
k
=0
∞
(−1
)
k
k
!(
n
+
k
)!
( )
x
2
n
+2
k
(10.5.1)
x
= 0 0
ν
> 0
0
J
−
n
(
x
)
J
−
n
=
∑
k
=0
∞
(−1
)
k
k
!Γ(−
n
+
k
+1)!
( )
x
2
n
+2
k
=
∑
k
=
n
∞
(−1
)
k
k
!Γ(−
n
+
k
+1)!
( )
x
2
−
n
+2
k
=
∑
l
=0
∞
(−1
)
l
+
n
(
l
+
n
)!
l
!
( )
x
2
n
+2
l
= (−1 (
x
).)
n
J
n
Γ(−
l
) = ±∞ 1/Γ(−
l
) = 0
1/Γ(
l
)
l
= −
n
+
k
[
(
x
)
]
d
dx
x
−
ν
J
ν
=
{ }
2
−
ν
d
dx
∑
k
=0
∞
(−1
)
k
k
!Γ(
ν
+
k
+1)
( )
x
2
2
k
=
2
−
ν
∑
k
=1
∞
(−1
)
k
(
k
−1)!Γ(
ν
+
k
+1)
( )
x
2
2
k
−1
= −
2
−
ν
∑
l
=0
∞
(−1
)
l
(
l
)!Γ(
ν
+
l
+2)
( )
x
2
2
l
+1
= −
2
−
ν
∑
l
=0
∞
(−1
)
l
(
l
)!Γ(
ν
+ 1 +
l
+1)
( )
x
2
2
l
+1
= −
x
−
ν
∑
l
=0
∞
(−1
)
l
(
l
)!Γ(
ν
+ 1 +
l
+1)
( )
x
2
2
l
+
ν
+1
= − (
x
).
x
−
ν
J
ν
+1
