Spherical Bessel Functions. Spherical Bessel functions, jl(x) and nl(x), are
solutions to the differential equation d. 2 fl dx. 2. +. 2 x dfl dx. +. [. 1 − l(l + 1) x. 2. ]
fl = 0.
Spherical Bessel Functions Spherical Bessel functions, j` (x) and n` (x), are solutions to the differential equation d2 f` 2 df` `(` + 1) f` = 0. + + 1− x dx dx2 x2 (1)
(2)
Also useful are the combinations h` (x) = j` (x) + in` (x) and h` (x) = j` (x) − in` (x) = (1) (1) [h` (x)]∗ and the modified spherical Bessel functions i` (x) = i` j` (ix) and k` (x) = −i` h` (ix). For ` = 0, the solutions are j0 (x) =
sin x , x
n0 (x) = −
cos x . x
From the series solution, with the conventional normalization [see George Arfken, Mathematical Methods for Physicists (1985)], it can be shown that f`−1 + f`+1 = or
2` + 1 f` (x), x
i d h `+1 x f` (x) = x`+1 f`−1 (x), dx (1)
df `f`−1 − (` + 1)f`+1 = (2` + 1) ` , dx i d h −` x f` (x) = x−` f`+1 (x), dx
(2)
where f` can be any of j` , nl , h` , h` . These two recurrence relations in turn lead back to the differential equation. Induction on ` leads to the Rayleigh formulas, 1 d ` 1 d ` ` ` ` ` j` (x) = (−1) x j0 (x), n` (x) = (−1) x n0 (x). x dx x dx Applied for ` = 1 and ` = 2, these give j1 (x) = j2 (x) =
3 1 − x3 x
sin x cos x − , x x2
3 sin x − 2 cos x, x
cos x sin x − , x x2 3 1 3 n2 (x) = − − sin x. cos x − x3 x x2 n1 (x) = −
From the Rayleigh expressions it is easy to extract limiting behaviors: For x l, the solutions behave as j` ≈
2` `! x` x` = , (2` + 1)! (2` + 1)!!
(2`)! (2` − 1)!! n` ≈ − l x−(`+1) = − 2 `! x`+1
and for x `, 1 `π j` ∼ sin x − , x 2
1 `π n` ∼ − cos x − , x 2
(1)
h`
∼ (−i)`+1
Plots of j0 through j4 and n0 through n4 appear on the following page.
eix . x
2 Spherical Bessel functions j` (x)
Spherical Neumann functions n` (x)
