math/j0
===============================================================================
%% Bessel function of the first kind, order 0
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/j0(X)
```
2. Description
-------------------------------------------------------------------------------
Computes the Bessel function of the first kind of order 0, `J₀(x)`, for each element of `X`.
This special function frequently appears in solutions to differential equations with cylindrical symmetry, such as heat conduction, wave propagation, and signal processing.
The function is defined as:
$$ J_0(x) = \sum_{m=0}^{\infty} \frac{(-1)^m}{(m!)^2} \left( \frac{x}{2} \right)^{2m}
3. Input
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|-------------------------------------|
| `X` | `float`, `[float]` | Input value(s), real. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|------------------------------------------------|
| `Y` | matches input | Values of the Bessel function of order 0 at each input point. |
5. Examples
-------------------------------------------------------------------------------
(a) Compute the Bessel J0 function of a number
```mech:ex1
y := math/j0(0.0)
```
(b) Compute the J0 function for a vector of numbers
```mech:ex2
x := [0.0, 1.0, 2.0, 3.0]
y := math/j0(x)
```
(c) Compute the J0 function for a matrix
```mech:ex3
x := [0.0, 2.0; 4.0, 6.0]
y := math/j0(x)
```
6. Details
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- $$J0(0) = 1$$
- $$J0(x)$$ oscillates and decays in amplitude as $$x → ∞$$.
- Useful in physics and engineering applications involving cylindrical symmetry.
Special cases:
- $$j0(±∞) = 0$$
- $$j0(NaN) = NaN$$