Cylindrical Bessel function of the second kind, $Y_\nu(x)$, and its respective derivatives

Computes the cylindrical Bessel function of the second kind ($Y_\nu(z)$), also known as the Neumann function or Weber function, and its derivatives through ycdk().

Usage

yc(l, n)

ycdk(l, n, k)

Arguments

l: Numeric. The order ($\nu$) of the Bessel 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 ($x + iy$ with $x \neq 0$ and $y \neq 0$) are not supported.
k: Non-negative integer. The order of the derivative for ycdk.

Value

A complex vector containing:

yc: $Y_\nu(z)$
ycdk(..., k = 1): $Y'_\nu(z)$ (first derivative)
ycdk(..., k = 2): $Y''_\nu(z)$ (second derivative)
ycdk: $Y_\nu^{(k)}(z)$ (k-th derivative)

Details

The cylindrical Bessel function of the second kind satisfies the same differential equation as $J_\nu(z)$: $$z^2 \frac{d^2 Y_\nu}{dz^2} + z \frac{dY_\nu}{dz} + (z^2 - \nu^2) Y_\nu = 0$$

but represents the linearly independent second solution.

Supported argument types:

Purely real arguments ($z = x$, where $x \in \mathbb{R}$): Fully supported for both positive and negative values.
Purely imaginary arguments ($z = iy$, where $y \in \mathbb{R}$): Computed using the identity $Y_\nu(iy) = i e^{-i\pi\nu/2} I_\nu(y) - \frac{2}{\pi} e^{i\pi\nu/2} K_\nu(y)$ where $I_\nu$ and $K_\nu$ are modified Bessel functions.
General complex arguments ($z = x + iy$, where $x \neq 0$ and $y \neq 0$): Not supported.

Special cases:

$Y_\nu(0) = -\infty$ (singularity at the origin).
For negative real arguments: $Y_\nu(-x) = \cos(\pi\nu) Y_\nu(x) + \sin(\pi\nu) J_\nu(x)$.
For integer order $n$: $Y_n(-x) = (-1)^n Y_n(x)$.
k-th derivative (DLMF 10.6.1): $$ \frac{d^k}{dz^k} Y_\nu(z) = \frac{1}{2^k} \sum_{j=0}^{k} (-1)^j \binom{k}{j} Y_{\nu - k + 2j}(z) $$

Derivative: $$Y'_\nu(z) = Y_{\nu-1}(z) - \frac{\nu}{z} Y_\nu(z)$$

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/

Bessel's equation: https://dlmf.nist.gov/10.2
Negative argument identity ($Y_\nu(-z)$): Eq. 10.4.1 at https://dlmf.nist.gov/10.4
Imaginary argument identity ($Y_\nu(iz)$): Eq. 10.27.8 at https://dlmf.nist.gov/10.27

Examples

# Real argument, integer order
yc(0, 1)
#> [1] 0.08825696+0i
yc(1, 2.5)
#> [1] 0.1459181+0i

# Real argument, fractional order
yc(0.5, 3)
#> [1] 0.4560488+0i

# Negative real argument
yc(2, -1.5)
#> [1] -0.9321938+0i

# Purely imaginary argument
yc(1, 1i)
#> [1] 0.5651591-0.383186i
yc(2, 3i)
#> [1] 0.03915877-2.245212i

# Singularity at origin
yc(0, 0) # Returns -Inf
#> [1] -Inf+0i

# First derivative
ycdk(1, 2, 1)
#> [1] 0.5638919+0i

Cylindrical Bessel function of the second kind, \(Y_\nu(x)\), and its respective derivatives