mech-math 0.3.4

Math library for the Mech language
Documentation 
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math/acsc
===============================================================================

%% Arc cosecant of argument in radians

1. Usage
-------------------------------------------------------------------------------

```mech:disabled
Y := math/acsc(X)
```

2. Description
-------------------------------------------------------------------------------

Computes the arc cosecant (inverse cosecant) of each element of `X`. The input `X` is interpreted as a numeric value. The result `Y` has the same shape as the input `X`.

3. Input
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `X`      | `float`, `[float]`       | Input value(s). Can be real or complex. For real inputs, values must satisfy `|X| >= 1`. For complex inputs, results are computed accordingly. |

4. Output
-------------------------------------------------------------------------------

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Arc cosecant of the input values, expressed in radians. For real inputs, results are in the range `[-π/2, π/2]` excluding `0`. The shape of `Y` matches the shape of `X`. |

5. Examples  
-------------------------------------------------------------------------------

(a) Find the arc cosecant of a number 

```mech:ex1
y := math/acsc(2.0)
```

(b) Find the arc cosecant for a vector of numbers

```mech:ex2
x := [1, -2, 10]
y := math/acsc(x)
```

(c) Find the arc cosecant for a matrix of numbers

```mech:ex3
x := [2, -3; 4, -5]
y := math/acsc(x)
```

(d) Relationship with asin

```mech:ex4
x := [2, 3, 4]
y := math/acsc(x)  # equivalent to asin(1/x)
```

6. Details
-------------------------------------------------------------------------------

The arc cosecant (inverse cosecant) function returns the angle whose cosecant is the specified value.

For real numbers $$x$$ with $$|x| \geq 1$$:

$$ y = acsc(x)

means $$ csc(y) = x $$, with $$ y \in [-\pi/2, \pi/2] $$ and $$ y \neq 0 $$.

It can be expressed in terms of arcsine:

$$ acsc(x) = asin(1/x)

for $$x \neq 0$$.

For complex numbers, arc cosecant can be expressed as:

$$ acsc(z) = -i \ln \left( \frac{i}{z} + \sqrt{1 - \frac{1}{z^2}} \right)

This definition extends the function beyond real numbers, allowing it to handle complex inputs.