Spherical Bessel function of the third kind (Hankel), $h_\nu(x)$, and its respective derivatives

Computes the spherical Hankel function of the first kind ($h^{(1)}_\nu(z)$) and its first-th derivative.

Usage

hs(l, n)

hsdk(l, n, k)

Arguments

l: Numeric. The order of the spherical Hankel function. Can be integer or fractional.
n: Numeric. The argument ($z$) at which to evaluate the function.
k: Non-negative integer. The order of the derivative for hsdk.

Value

A complex vector containing:

hs: $h^{(1)}_\nu(z)$
hsdK: $\frac{d}{dz^k}h^{(1)}_l(z)$ (k-th derivative)

Details

The spherical Hankel function of the first kind is defined as: $$h^{(1)}_\nu(z) = j_\nu(z) + i y_\nu(z)$$

where $j_\nu(z)$ is the spherical Bessel function of the first kind and $y_\nu(z)$ is the spherical Bessel function of the second kind.

It is related to the cylindrical Hankel function by: $$h^{(1)}_\nu(z) = \sqrt{\frac{\pi}{2z}} H^{(1)}_{\nu+1/2}(z)$$

The spherical Hankel functions are used extensively in scattering theory to represent outgoing spherical waves.

Derivative: $$ \frac{d}{dz}h^{(1)}_\nu(z) = \frac{\nu}{z} h^{(1)}_\nu(z) - h^{(1)}_{\nu+1}(z) $$

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 Hankel functions: Eq. 10.47.5 at https://dlmf.nist.gov/10.47

Examples

# Spherical Hankel function
hs(0, 1)
#> [1] 0.841471-0.5403023i
hs(1, 2.5)
#> [1] 0.416213-0.1112059i

# Fractional order
hs(0.5, 3)
#> [1] 0.04704+0.3299975i

# Vector input
hs(0, c(1, 2, 3))
#> [1] 0.8414710-0.5403023i 0.4546487+0.2080734i 0.0470400+0.3299975i

# First derivative
hsdk(1, 2, 1)
#> [1] 0.01925094+0.5586854i

Spherical Bessel function of the third kind (Hankel), \(h_\nu(x)\), and its respective derivatives