# inverfc

Inverse of the complementary error function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```inverfc(`x`)
```

## Description

`inverfc(x) = inverf(1 - x)` computes the inverse of the complementary error function. This function is defined for all complex arguments `x`.

The inverse function `inverfc` is singular at the points `x = 0` and `x = 2`.

The inverses of the error functions return floating-point values only for floating-point arguments that belong to a particular interval. Thus, the inverse of the complementary error function `inverfc(x)` returns floating-point values for real values `x` from the interval `[0, 2]`. The implemented exact values are: ```inverfc(0) = ∞```, `inverfc(1) = 0`, ```inverfc(2) = -∞```. For all other arguments, the error functions return symbolic function calls.

MuPAD® can simplify expressions that contain error functions and their inverses. For real values `x`, the system applies the following simplification rules:

• ```inverf(erf(x)) = inverf(1 - erfc(x)) = inverfc(1 - erf(x)) = inverfc(erfc(x)) = x```

• ```inverf(-erf(x)) = inverf(erfc(x) - 1) = inverfc(1 + erf(x)) = inverfc(2 - erfc(x)) = -x```

For any value `x`, the system applies the following simplification rules:

• `inverf(-x) = -inverf(x)`

• `inverfc(2 - x) = -inverfc(x)`

• `erf(inverf(x)) = erfc(inverfc(x)) = x`

• `erf(inverfc(x)) = erfc(inverf(x)) = 1 - x`

## Environment Interactions

When called with a floating-point argument, the functions are sensitive to the environment variable `DIGITS`, which determines the numerical working precision.

## Examples

### Example 1

You can call the inverse of the complementary error function with exact and symbolic arguments:

`inverfc(0), inverfc(1), inverfc(2), inverfc(15), inverfc(x/5), inverfc(1/5), inverfc(sqrt(2))`

To approximate exact results with floating-point numbers, use `float`:

`float(inverfc(1/5)), float(inverfc(sqrt(2)))`

Alternatively, use floating-points value as arguments:

`inverfc(0.2), inverfc(sqrt(2.0))`

For floating-point arguments `x` from the interval ```[0, 2]```, `inverfc` returns floating-point values:

`inverfc(0.5), inverfc(1.25)`

For floating-point arguments outside of this interval, `inverfc` returns symbolic function calls:

`inverfc(-1.25), inverfc(2.5)`

### Example 2

`diff`, `float`, `limit`, `rewrite`, `series`, and other functions handle expressions involving the inverse of the complementary error function:

`diff(inverfc(x), x)`

`float(ln(3 + inverfc(sqrt(PI))))`

```limit(1/inverfc(x), x = 1, Right); limit(1/inverfc(x), x = 1, Left)```

`rewrite(inverfc(x), inverf)`

`series(inverfc(x), x = 1)`

## Parameters

 `x` Arithmetical expression

## Return Values

Arithmetical expression