math/acosh
===============================================================================
%% Inverse hyperbolic cosine of argument
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/acosh(X)
```
2. Description
-------------------------------------------------------------------------------
Computes the inverse hyperbolic cosine (area hyperbolic cosine) 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). For real inputs, values must be greater than or equal to `1`. For complex inputs, values outside this domain are allowed, producing complex results. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------------|---------------------------------------|
| `Y` | matches input | Inverse hyperbolic cosine of the input values. For real inputs `x >= 1`, results are real and nonnegative. For complex inputs, results may include both real and imaginary parts. The shape of `Y` matches the shape of `X`. |
5. Examples
-------------------------------------------------------------------------------
(a) Find the inverse hyperbolic cosine of a number
```mech:ex1
y := math/acosh(2.0)
```
(b) Find the inverse hyperbolic cosine for a vector of numbers
```mech:ex2
x := [1, 2, 10]
y := math/acosh(x)
```
(c) Find the inverse hyperbolic cosine for a matrix of numbers
```mech:ex3
x := [1, 2; 3, 4]
y := math/acosh(x)
```
(d) Complex input example
```mech:ex4
x := [-1, 0, 0.5]
y := math/acosh(x)
```
6. Details
-------------------------------------------------------------------------------
The inverse hyperbolic cosine function returns the value whose hyperbolic cosine is the specified input.
For real numbers $$x \geq 1$$:
$$ y = acosh(x)
means $$ cosh(y) = x $$, where $$y \geq 0$$.
The definition in terms of natural logarithms is:
$$ acosh(x) = \ln(x + \sqrt{x^2 - 1})
This form extends the function to complex inputs as well, allowing it to handle values outside the real domain.