where
h
0
= 0
,
h
k
= 1 +
1
2
+
1
3
+
· · ·
+
1
k
,
while
γ
=
lim
k
→ ∞
(
h
k
−
log
k
)
≈
.
5772156649
. . .
(12
.
102)
is known as
Euler’s constant
. All Bessel functions of the second kind have a singularity at
the origin
x
= 0; indeed, by inspection of (12.101), we find that the leading asymptotics
as
x
→
0 are
Y
0
(
x
)
∼
2
π
log
x,
Y
m
(
x
)
∼ −
2
m
(
m
−
1)!
π x
m
,
m >
0
.
(12
.
103)
Finally, we show how Bessel functions of different orders are interconnected by two
important recurrence relations.
Proposition 12.10.
The Bessel functions are related by the following formulae
:
dJ
m
dx
+
m
x
J
m
(
x
) =
J
m
−
1
(
x
)
,
−
dJ
m
dx
+
m
x
J
m
(
x
) =
J
m
+1
(
x
)
.
(12
.
104)
Proof
: Let us differentiate the power series
x
m
J
m
(
x
) =
∞
summationdisplay
k
=0
(
−
1)
k
x
2
m
+2
k
2
2
k
+
m
k
! (
m
+
k
)!
.
We find
d
dx
[
x
m
J
m
(
x
) ] =
∞
summationdisplay
k
=0
(
−
1)
k
2 (
m
+
k
)
x
2
m
+2
k
−
1
2
2
k
+
m
k
! (
m
+
k
)!
=
x
m
∞
summationdisplay
k
=0
(
−
1)
k
x
m
−
1+2
k
2
2
k
+
m
−
1
k
! (
m
−
1 +
k
)!
=
x
m
J
m
−
1
(
x
)
.
(12
.
105)
Expansion of the left hand side of this formula leads to
x
m
dJ
m
dx
+
mx
m
−
1
J
m
(
x
) =
d
dx
[
x
m
J
m
(
x
) ] =
x
m
J
m
−
1
(
x
)
,
which proves the first recurrence formula (12.104).
The second is proved by a similar
manipulation involving differentiation of
x
−
m
J
m
(
x
).
Q.E.D.
For example, using the second recurrence formula (12.104) along with (12.98), we can
write the Bessel function of order
3
2
in elementary terms:
J
3
/
2
(
x
) =
−
dJ
1
/
2
(
x
)
dx
+
1
2
x
J
1
/
2
(
x
)
=
−
radicalbigg
2
π
bracketleftbigg
cos
x
x
1
/
2
−
sin
x
2
x
3
/
2
bracketrightbigg
+
radicalbigg
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Boundary value problem, Recurrence relation, Bessel function, Bessel Functions, Jm

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