math/jn
===============================================================================
%% Bessel function of the first kind, integer order n
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/jn(N, X)
```
2. Description
-------------------------------------------------------------------------------
Computes the Bessel function of the first kind of integer order `N` for each element of `X`.
The result `Y` has the same shape as the input `X`.
3. Input
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|-------------|
| `N` | `int`, `[int]` | The order of the Bessel function. Must be an integer. |
| `X` | `float`, `[float]` | Input value(s). Can be real. For complex inputs, the function may not be defined. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|---------------|-------------|
| `Y` | matches input | The Bessel function of the first kind of order `N`, evaluated at each element of `X`. The shape of `Y` matches the shape of `X`. |
5. Examples
-------------------------------------------------------------------------------
(a) Compute jn of order 0
```mech:ex1
y := math/jn(0, 1.0)
```
(b) Compute jn of order 2 for a vector of numbers
```mech:ex2
x := [0.0, 1.0, 2.0]
y := math/jn(2, x)
```
(c) Compute jn of order 3 for a matrix of numbers
```mech:ex3
x := [0.0, 1.0; 2.0 3.0]
y := math/jn(3, x)
```
6. Details
-------------------------------------------------------------------------------
The Bessel functions of the first kind, denoted $$J_n(x)$$, are solutions to Bessel's differential equation:
$$ x^2 y'' + x y' + (x^2 - n^2)y = 0
For integer order `n`, the function can be expressed as a power series:
$$ J_n(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!(n+m)!} \left( \frac{x}{2} \right)^{2m+n}
These functions are widely used in applied mathematics, especially in problems involving cylindrical or spherical symmetry, such as waveguides, heat conduction, and vibrations.