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

Computes the cylindrical Bessel function of the first kind ($J_\nu(z)$) and its k-th derivatives.

Usage

jc(l, n)

jcdk(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 jcdk.

Value

A complex vector containing:

jc: $J_\nu(z)$
jcdk(..., k = 1): $J'_\nu(z)$ (first derivative)
jcdk(..., k = 2): $J''_\nu(z)$ (second derivative)
jcdk: $J_\nu^{(k)}(z)$ (k-th derivative)

Details

The cylindrical Bessel function of the first kind satisfies Bessel's differential equation:

$$z^2 \frac{d^2 J_\nu}{dz^2} + z \frac{dJ_\nu}{dz} + (z^2 - \nu^2) J _\nu = 0$$

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 $J_\nu(iy) = e^{i\pi\nu/2} I_\nu(y)$ where $I_\nu$ is the modified Bessel function of the first kind.
General complex arguments ($z = x + iy$, where $x \neq 0$ and $y \neq 0$): Not supported.

Special cases:

$J_\nu(0) = 1$ if $\nu = 0$, otherwise $J_\nu(0) = 0$.
For negative real arguments with integer order $n$: $J_n(-x) = (-1)^n J_n(x)$.
For negative real arguments with non-integer order $\nu$: $J_\nu(-x) = e^{i\pi\nu} J_\nu(x)$ (complex result).

Derivatives:

First derivative: $J'_\nu(z) = J_{\nu-1}(z) - \frac{\nu}{z} J_\nu(z)$
Second derivative: $J''_\nu(z) = \frac{1}{4}\left[J_{\nu-2}(z) - 2J_\nu(z) + J_{\nu+2}(z)\right]$
k-th derivative (DLMF 10.6.1): $$ \frac{d^k}{dz^k} J_\nu(z) = \frac{1}{2^k} \sum_{j=0}^{k} (-1)^j \binom{k}{j} J_{\nu - k + 2j}(z) $$

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/

Bessel's equation: https://dlmf.nist.gov/10.2
Negative argument identity ($J_\nu(-z)$): Eq. 10.4.1 at https://dlmf.nist.gov/10.4
Imaginary argument identity ($J_\nu(iz)$): Eq. 10.27.6 at https://dlmf.nist.gov/10.27

Examples

# Real argument, integer order
jc(0, 1)
#> [1] 0.7651977+0i
jc(1, 2.5)
#> [1] 0.4970941+0i

# Real argument, fractional order
jc(0.5, 3)
#> [1] 0.06500818+0i

# Negative real argument (integer order gives real result)
jc(2, -1.5)
#> [1] 0.2320877+0i

# Negative real argument (fractional order gives complex result)
jc(0.5, -2)
#> [1] 3.141318e-17+0.5130161i

# Purely imaginary argument
jc(1, 1i)
#> [1] 0+0.5651591i
jc(2, 3i)
#> [1] -2.245212+0i

# First derivative
jcdk(1, 2, 1)
#> [1] -0.06447162+0i

# Second derivative
jcdk(1, 2, 2)
#> [1] -0.4003078+0i

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