388 Chapter 13Second-order differential equations. Some special functions
so that
Similarly, withn 1 = 1 − 122 in (13.56),
0 Exercise 26
Series expansions for J
l+ 122(x)and J
−l− 122(x)are given by equations (13.48) and
(13.49), respectively. Thus
(13.57)
(13.58)
EXAMPLE 13.13Forl 1 = 10 ,
Equations (13.57) and (13.58) show that whereasJ
l+ 122(x) 1 = 10 at the origin (x 1 = 10 )
and is therefore finite for all values of x,J
−l− 122(x) 1 → 1 ∞asx 1 → 10.
The functions of half-integral order are the Bessel functions that occur in the
partial wave method in the theory of scattering processes. They occur there in the
forms, forl 1 ≥ 10 ,
(13.59)
jxx
Jx x
x
J
ll lll()=,=() () ( )− (
+/+−−/ππ
2
1
2
12112η xx)
Jx
x
xx
x
x
−= −+−
=
12242
1
24
2
()
!!
cos
ππ
Jx x
xx
x
x
x
121224 32
1
35
2
3
()
!!!
= −+−
=−
ππ
++−
=
x
x
x
55
2
!
sin
π
Jx x
x
l
x
l
ll−−−−=+
−
−
1212242
1
22 1 2 42 1 2
()
π () ()(· ll−
Jx x
x
l
x
ll
ll++=−
++
1212242
1
22 3 2 42 3 2
()
π () ()(· 5 5)
−
Jx
x
JxJx
x
x
x
x
−−=− − =− +
32 12 1212
() () ()
cos
sin
π
Jx
x
JxJ x
x
x
x
x
32 12 1212
() () ()
sin
=−= −cos
−π