mech-math 0.3.3

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

%% Complementary error function of argument

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

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

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

Computes the complementary error function 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`.

The complementary error function is defined as:

$$ erfc(x) = 1 - erf(x)

It is widely used in probability, statistics, and numerical analysis.

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

| Argument | Kind               | Description                           |
|----------|--------------------|---------------------------------------|
| `X`      | `float`, `[float]` | Input value(s), real or complex. |

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

| Argument | Kind               | Description                           |
|----------|--------------------|---------------------------------------|
| `Y`      | matches input      | Complementary error function of the input values. For real `X`, the output is real in the range (0, 2). The shape of `Y` matches the shape of `X`. |

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

(a) Compute the complementary error function of a number

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

(b) Compute the complementary error function for a vector of numbers

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

(c) Compute the complementary error function for a matrix of numbers

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

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

The complementary error function is defined as:

$$ erfc(x) = 1 - erf(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt

For real inputs:

- $$ erfc(0) = 1 $$
- $$ \lim_{x \to \infty} erfc(x) = 0 $$
- $$ \lim_{x \to -\infty} erfc(x) = 2 $$

It is especially useful in probability theory and statistics, particularly in describing tail probabilities of the normal distribution.