Spherical Bessel functions

Spherical Bessel Functions
Spherical Bessel functions, jℓ (x) and nℓ (x), are the regular and singular solutions to the
differential equation
d2 fℓ 2 dfℓ
ℓ(ℓ + 1)
fℓ = 0.
+
+ 1
x dx
dx2
x2
(1)
(2)
Also useful are the combinations hℓ (x) = jℓ (x) + i nℓ (x) and hℓ (x) = jℓ (x) i nℓ (x) and
(1)
the modified spherical Bessel functions iℓ (x) = ( i)ℓ jℓ (ix) and kℓ (x) = iℓ hℓ (ix). For
ℓ = 0, the solutions are
sin x
cos x
,
n0 (x) =
.
j0 (x) =
x
x
From the series solution, with the conventional normalization [see Arfken, Mathematical
Methods for Physicists], it can be shown that
fℓ−1 + fℓ+1 =
and
2ℓ + 1
fℓ (x),
x
ℓfℓ−1
i
d h ℓ+1
x fℓ (x) = xℓ+1 fℓ−1(x),
dx
(1)
df
(ℓ + 1)fℓ+1 = (2ℓ + 1) ℓ ,
dx
i
d h −ℓ
x fℓ (x) = x−ℓ fℓ+1 (x),
dx
(2)
where fℓ can be any of jℓ , nℓ , hℓ , hℓ . These recurrence relations in turn lead back to the
differential equation. Induction on ℓ leads to the Rayleigh formulas,
jℓ (x) = ( 1)ℓ xℓ
1 d ℓ
x dx
j0 (x),
nℓ (x) = ( 1)ℓ xℓ
1 d ℓ
x dx
n0 (x).
Applied for ℓ = 1 and ℓ = 2, these give
j1 (x) =
j2 (x) =
3
sin x
x2
1
sin x
x
x3
cos x
,
x
n1 (x) =
3
cos x,
x2
n2 (x) =
cos x
x2
sin x
,
x
1
cos x
x
3
x3
3
sin x.
x2
From the Rayleigh expressions it is easy to extract limiting behaviors: For x
solutions behave as
ℓ
jℓ
ℓ
(2ℓ2 +ℓ!1)! xℓ = (2ℓ x+ 1)!! ,
nℓ
(2ℓ)! −(ℓ+1)
x
=
2ℓ ℓ!
(2ℓ 1)!!
xℓ+1
and for x ℓ,
jℓ
x1 sin
x
ℓπ ,
2
nℓ
1
cos x
x
ℓπ ,
2
(1)
hℓ
(
Plots of j0 through j4 and n0 through n4 appear on the following page.
i)ℓ+1
eix
.
x
l, the
2
Spherical Bessel functions jℓ (x)
Spherical Neumann functions nℓ (x)