math/j1
===============================================================================
%% Bessel function of the first kind, order 1
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/j1(X)
```
2. Description
-------------------------------------------------------------------------------
Computes the Bessel function of the first kind of order 1, `J₁(x)`, for each element of `X`.
This special function arises in problems with cylindrical symmetry, such as electromagnetic waves, vibrations, and heat conduction.
The function is defined as:
$$ J_1(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{m!(m+1)!} \left( \frac{x}{2} \right)^{2m+1}
3. Input
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|-------------------------------------|
| `X` | `float`, `[float]` | Input value(s), real. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|------------------------------------------------|
| `Y` | matches input | Values of the Bessel function of order 1 at each input point. |
5. Examples
-------------------------------------------------------------------------------
(a) Compute the Bessel J1 function of a number
```mech:ex1
y := math/j1(0.0)
```
(b) Compute the J1 function for a vector of numbers
```mech:ex2
x := [0.0, 1.0, 2.0, 3.0]
y := math/j1(x)
```
(c) Compute the J1 function for a matrix
```mech:ex3
x := [0.0, 2.0; 4.0, 6.0]
y := math/j1(x)
```
6. Details
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- $$J1(0) = 0$$
- $$J1(x)$$ oscillates and decays in amplitude as $$x → ∞$$.
- Appears in many physics and engineering applications involving cylindrical symmetry.
Special cases:
- $$j1(±∞) = 0$$
- $$j1(NaN) = NaN$$