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)
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