9.2 Spherical Bessel functions of the first kind
363
sin x/x and j1 (x) = sin x/x2 cos x/x. Rayleigh’s formula leads to the
recursion relation (exercise 9.22)
`
j` (x) j`0 (x)
(9.69)
x
with which one can show (exercise 9.23) that the spherical Bessel functions
as defined by Rayleigh’s formula do satisfy their di↵erential equation (9.66)
with x = kr.
The spherical Bessel functions j` (kr) satisfy the self-adjoint Sturm-Liouville
(6.333) equation (9.66)
j`+1 (x) =
r2 j`00
2rj`0 + `(` + 1)j` = k 2 r2 j`
(9.70)
with eigenvalue k 2 and weight function ⇢ = r2 . If j` (z`,n ) = 0, then the
functions j` (kr) = j` (z`,n r/a) vanish at r = a and form an orthogonal basis
Z
a
0
j` (z`,n r/a) j` (z`,m r/a) r2 dr =
a3 2
j (z`,n )
2 `+1
n,m
(9.71)
for a self-adjoint system on the interval [0, a]. Moreover, since as n ! 1
2 = z 2 /a2 ⇡ [(n + `/2)⇡]2 /a2 ! 1, the eigenfunctions
the eigenvalues k`,n
`,n
j` (z`,n r/a) also are complete in the mean (section 6.35).
On an infinite interval, the analogous relation is
Z 1
⇡
j` (kr) j` (k 0 r) r2 dr = 2 (k k 0 ).
(9.72)
2k
0
If we write the spherical Bessel function j0 (x) as the integral
Z
sin z
1 1 izx
j0 (z) =
=
e dx
z
2 1
(9.73)
and use Rayleigh’s formula (9.68), we may find an integral for j` (z)
✓
◆ ✓
◆
✓
◆
Z
1 d ` sin z
1 d ` 1 1 izx
` `
` `
j` (z) = ( 1) z
= ( 1) z
e dx
z dz
z
z dz
2 1
Z
Z
z ` 1 (1 x2 )` izx
( i)` 1 (1 x2 )` d` izx
=
e dx =
e dx
2
2
2` `!
2` `!
dx`
1
1
Z
Z
( i)` 1 izx d` (x2 1)`
( i)` 1
=
e
dx =
P` (x) eizx dx (9.74)
2
2
dx` 2` `!
1
1
(exercise 9.24) that contains Rodrigues’s formula (8.8) for the Legendre polynomial P` (x). With z = kr and x = cos ✓, this formula
Z
1 1
`
i j` (kr) =
P` (cos ✓)eikr cos ✓ d cos ✓
(9.75)
2 1