mech-math 0.3.3

Math library for the Mech language
Documentation 
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math/exp2
===============================================================================

%% Base-2 Exponential Function

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

```mech:disabled
Y := math/exp2(X)
```

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

Computes the base-2 exponential of each element of `X`. This means it returns 2 raised to the power of `X`. 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, `exp2` returns complex results. |

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

| Argument | Kind                     | Description                           |
|----------|--------------------------|---------------------------------------|
| `Y`      | matches input            | Result of `2^X`. If `X` is real, `Y` is positive real. If `X` is complex, `Y` may have both real and imaginary parts. The shape of `Y` matches the shape of `X`. |

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

(a) Compute 2 raised to a number

```mech:ex1
y := math/exp2(3)
```

(b) Compute base-2 exponential for a vector

```mech:ex2
x := [0, 1, 2, 3]
y := math/exp2(x)
```

(c) Compute base-2 exponential for a matrix

```mech:ex3
x := [0, 1; 2 3]
y := math/exp2(x)
```

(d) Compute fractional exponents

```mech:ex4
x := [0.5, -1.0, 2.5]
y := math/exp2(x)
```

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

The base-2 exponential function is defined as:

$$ exp2(x) = 2^x

This is equivalent to the general exponential function with base `e`:

$$ 2^x = e^{x \cdot ln(2)}

This function is useful in binary systems, computer science, and digital signal processing, where powers of two commonly occur. It can be applied element-wise to vectors and matrices, preserving the shape of the input.

For complex numbers, the definition extends naturally using Euler's formula:

$$ 2^z = e^{z \cdot ln(2)}

where `z` may have both real and imaginary components.