mech-math 0.3.4

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

%% Arc sine of argument in radians

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

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

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

Computes the arc sine (inverse sine) of each element of `X`. The input `X` is interpreted as a numeric value between `-1` and `1` (for real inputs). 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 lie in the range `[-1, 1]`. For complex inputs, values outside this domain are allowed, producing complex results. |

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

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Arc sine of the input values, expressed in radians. For real inputs in `[-1, 1]`, results are in the range `[-π/2, π/2]`. 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 arc sine of a number 

```mech:ex1
y := math/asin(0.5)
```

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

```mech:ex2
x := [0, -0.5, 1]
y := math/asin(x)
```

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

```mech:ex3
x := [0.5, -0.5; 1, 0]
y := math/asin(x)
```

(d) Handling degrees (convert to radians first)

```mech:ex4
x := [30, 45, 90]
y := math/asin(sin(x * 3.14 / 180))
```

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

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

In real analysis, for $$x \in [-1,1]$$:

$$ y = asin(x)

means $$ sin(y) = x $$, where $$y \in [-π/2, π/2]$$.

In the unit circle, $$asin(x)$$ gives the angle in radians corresponding to the given sine value (y-coordinate of a point on the unit circle).

For complex numbers, arc sine can be expressed as:

$$ asin(z) = -i \ln(iz + \sqrt{1 - z^2})

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