Spherical Bessel function of the first kind, $j_\nu(z)$, and its respective derivatives

Computes the spherical Bessel function of the first kind ($j_\nu(z)$) and its k-th derivative.

Usage

js(l, n)

jsdk(l, n, k)

Arguments

l: Numeric. The order of the spherical Bessel function. Can be integer or fractional.
n: Numeric or matrix. The argument ($z$) at which to evaluate the function. If a matrix is provided, the function is applied column-wise.
k: Non-negative integer. The order of the derivative for jsdk.

Value

A numeric vector or matrix (matching the input structure) containing:

js: $j_\nu(z)$
jsdk: $j^{(k)'}_l(z)$ (k-th derivative)

Details

The spherical Bessel function of the first kind is related to the cylindrical Bessel function by: $$j_\nu(z) = \sqrt{\frac{\pi}{2z}} J_{\nu+1/2}(z)$$

where $J_\nu(z)$ is the cylindrical Bessel function of the first kind.

The spherical Bessel functions satisfy the differential equation: $$ z^2 \frac{d^2 j_\nu}{dz^2} + 2z \frac{dj_\nu}{dz} + [z^2 - l(l+1)] j_\nu = 0 $$

Special cases:

$j_\nu(0) = 0$ for all $\nu$.
$j_0(z) = \frac{\sin(z)}{z}$
$j_1(z) = \frac{\sin(z)}{z^2} - \frac{\cos(z)}{z}$

Derivatives:

First derivative: $ j'_\nu(z) = j_{\nu-1}(z) - \frac{\nu+1}{z} j_\nu(z) $
Second derivative: $j''_\nu(z) = \frac{(\nu+1)(\nu+2) - z^2}{z^2} j_\nu(z) - \frac{2}{z} j_{\nu-1}(z)$

References

Abramowitz, M. and Stegun, I.A. (Eds.). (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55. Chapter 10.

NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/

Spherical Bessel functions: https://dlmf.nist.gov/10.47
Relation to cylindrical Bessel functions: Eq. 10.47.3 at https://dlmf.nist.gov/10.47

Examples

# Spherical Bessel function
js(0, 1)
#> [1] 0.841471
js(1, 2.5)
#> [1] 0.416213

# Fractional order
js(0.5, 3)
#> [1] 0.04704

# Vector input
js(0, c(1, 2, 3))
#> [1] 0.8414710 0.4546487 0.0470400

# Matrix input (applied column-wise)
js(1, matrix(1:6, nrow = 2))
#> [1]  0.30116868  0.43539777  0.34567750  0.11611075 -0.09508941 -0.16778992

# First derivative
jsdk(1, 2, 1)
#> [1] 0.01925094

# Second derivative
jsdk(1, 2, 2)
#> [1] -0.2369498

Spherical Bessel function of the first kind, \(j_\nu(z)\), and its respective derivatives