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

Computes the cylindrical Hankel function of the first kind ($H^{(1)}_\nu(z)$) and its derivatives through hcdk().

Usage

hc(l, n)

hcdk(l, n, k)

Arguments

l: Numeric. The order ($\nu$) of the Hankel function. Must be purely real; complex orders are not supported.
n: Numeric or complex. The argument ($z$) at which to evaluate the function. Supports purely real or purely imaginary values. General complex arguments are not supported.
k: Non-negative integer. The order of the derivative for hcdk.

Value

A complex vector containing:

hc: $H^{(1)}_\nu(z)$
hcdk(..., k = 1): $\frac{d}{dz}H^{(1)}_\nu(z)$ (first derivative)
hcdk(..., k = 2): $\frac{d^2}{dz^2}H^{(1)}_\nu(z)$ (second derivative)
hcdk: $\frac{d^k}{dz^k}H^{(1)}_\nu(z)$ (k-th derivative)

Details

The 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 Bessel function of the first kind and $Y_\nu(z)$ is the Bessel function of the second kind.

Supported argument types: Since $H^{(1)}_\nu(z)$ is computed from $J_\nu(z)$ and $Y_\nu(z)$, the same restrictions apply:

Purely real arguments ($z = x$): Fully supported.
Purely imaginary arguments ($z = iy$): Supported.
General complex arguments ($z = x + iy$): Not supported.

Derivatives:

First derivative: $\frac{d}{dz}H^{(1)}_\nu(z) = \frac{\nu}{z} H^{(1)}_\nu(z) - H^{(1)}_{\nu+1}(z)$
Second derivative: $\frac{d^2}{dz^2}H^{(1)}_\nu(z) = H^{(1)}_{\nu-2}(z) - \frac{2\nu-1}{z} H^{(1)}_{\nu-1}(z) + \frac{\nu^2+\nu}{z^2} H^{(1)}_\nu(z)$
k-th derivative (DLMF 10.6.1): $\frac{d^k}{dz^k}H^{(1)}_\nu(z) = \frac{1}{2^k} \sum_{j=0}^{k} (-1)^j \binom{k}{j} H^{(1)}_{\nu-k+2j}(z)$

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 9.

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

Hankel function definition: https://dlmf.nist.gov/10.2
k-th derivative formula: Eq. 10.6.1 at https://dlmf.nist.gov/10.6

Examples

# Hankel function
hc(0, 1)
#> [1] 0.7651977+0.08825696i
hc(1, 2.5)
#> [1] 0.4970941+0.1459181i

# Fractional order
hc(0.5, 3)
#> [1] 0.06500818+0.4560488i

# Purely imaginary argument
hc(1, 1i)
#> [1] 0.383186+1.130318i

# First derivative
hcdk(1, 2, 1)
#> [1] -0.06447162+0.5638919i

# Second derivative
hcdk(1, 2, 2)
#> [1] -0.4003078-0.2016716i

# k-th derivative
hcdk(1, 2, 3) # Third derivative
#> [1] 0.08820851-0.154352i

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