math/erf FIX 6
===============================================================================
%% Error Function
1. Usage
-------------------------------------------------------------------------------
```mech:disabled
Y := math/erf(X)
```
2. Description
-------------------------------------------------------------------------------
Computes the error function of each element of `X`. The error function is a special function widely used in probability, statistics, and partial differential equations. 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, `erf` returns complex results. |
4. Output
-------------------------------------------------------------------------------
| Argument | Kind | Description |
|----------|--------------------------|---------------------------------------|
| `Y` | matches input | Error function evaluated at the input values. The shape of `Y` matches the shape of `X`. |
5. Examples
-------------------------------------------------------------------------------
(a) Error function of zero
```mech:ex1
y := math/erf(0)
```
(b) Error function of a positive number
```mech:ex2
y := math/erf(1)
```
(c) Error function for a vector
```mech:ex3
x := [0, 0.5, 1.0]
y := math/erf(x)
```
(d) Error function for a matrix
```mech:ex4
x := [0, -1; 1, 2]
y := math/erf(x)
```
6. Details
-------------------------------------------------------------------------------
The error function is defined as:
$$ erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt
It is an odd function, meaning $$erf(-x) = -erf(x)$$.
For real inputs:
- $$ erf(0) = 0 $$
- $$ \lim_{x \to \infty} erf(x) = 1 $$
- $$ \lim_{x \to -\infty} erf(x) = -1 $$
The error function is closely related to the cumulative distribution function (CDF) of the normal distribution, and appears frequently in statistics and probability theory.
Properties:
- $$erf(0) = 0$$
- $$erf(∞) = 1$$
- $$erf(-∞) = -1$$
- The function is odd: $$erf(-x) = -erf(x)$$
The error function arises naturally in probability theory as it relates to the cumulative distribution function (CDF) of the normal distribution:
$$ \Phi(x) = \frac{1}{2} [1 + erf(x / \sqrt{2})]
For complex arguments, the error function can be extended into the complex plane, where it is related to the Faddeeva function.