math/y1
===============================================================================
%% Bessel function of the second kind (order 1)
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/y1(X)
```
2. Description
-------------------------------------------------------------------------------
Computes the Bessel function of the second kind of order 1, denoted as `Y₁(x)`, for each element of `X`.
The result `Y` has the same shape as the input `X`.
3. Input
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------|-----------------------------------------------------------------------------|
| `X` | `float`, `[float]` | Input value(s). Must be real. Values should be positive since `Y₁(x)` is undefined at zero or negative values. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|---------------|-----------------------------------------------------------------------------|
| `Y` | matches input | Bessel function of the second kind of order 1 evaluated at the input values. The shape of `Y` matches the shape of `X`. |
5. Examples
-------------------------------------------------------------------------------
(a) Compute Y₁ of a number
```mech:ex1
y := math/y1(1.0)
```
(b) Compute Y₁ for a vector of numbers
```mech:ex2
x := [0.5, 1.0, 2.0]
y := math/y1(x)
```
(c) Compute Y₁ for a matrix of numbers
```mech:ex3
x := [0.5, 1.0; 2.0, 3.0]
y := math/y1(x)
```
6. Details
-------------------------------------------------------------------------------
The Bessel functions of the second kind, `Y₁(x)`, also known as **Neumann functions** or **Weber functions**, are solutions to Bessel's differential equation:
$$x^2 y'' + x y' + (x^2 - u^2)y = 0
for order `ν = 1`.
Unlike Bessel functions of the first kind `J₁(x)`, which are finite at the origin, `Y₁(x)` diverges as `x → 0`.
These functions are important in solving problems with cylindrical symmetry, such as wave propagation and static potentials.