mech-math 0.3.3

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

%% Arctangent of argument in radians

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

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

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

Computes the arctangent (inverse tangent) of each element of `X`. The input `X` is interpreted as a real or complex number. 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. If `X` is complex, atan returns complex results. |

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

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Arctangent of the input values. If `X` is real, `Y` is real in the range `(-π/2, π/2)`. If `X` is complex, `Y` may have both real and imaginary parts. The shape of `Y` matches the shape of `X`. |

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

(a) Find the arctangent of a number 

```mech:ex1
y := math/atan(1)
```

(b) Find the arctangent for a vector of numbers

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

(c) Find the arctangent for a matrix of numbers

```mech:ex3
x := [0, 1; -1, 2]
y := math/atan(x)
```

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

The arctangent function is the inverse of the tangent function. For a real number `x`, it returns the angle `y` in radians such that:

$$ tan(y) = x, \quad y \in (-\pi/2, \pi/2)

In trigonometry, atan is used to determine an angle given the ratio of the opposite side to the adjacent side of a right triangle.

For complex numbers, the arctangent is defined using logarithms:

$$ atan(z) = \frac{i}{2} \left( \ln(1 - iz) - \ln(1 + iz) \right)

This definition extends the arctangent function to complex inputs.