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# erf

Error function

## Syntax

``erf(X)``

## Description

example

````erf(X)` represents the error function of `X`. If `X` is a vector or a matrix, `erf(X)` computes the error function of each element of `X`.```

## Examples

### Error Function for Floating-Point and Symbolic Numbers

Depending on its arguments, `erf` can return floating-point or exact symbolic results.

Compute the error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:

`A = [erf(1/2), erf(1.41), erf(sqrt(2))]`
```A = 0.5205 0.9539 0.9545```

Compute the error function for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, `erf` returns unresolved symbolic calls:

`symA = [erf(sym(1/2)), erf(sym(1.41)), erf(sqrt(sym(2)))]`
```symA = [ erf(1/2), erf(141/100), erf(2^(1/2))]```

Use `vpa` to approximate symbolic results with the required number of digits:

```d = digits(10); vpa(symA) digits(d)```
```ans = [ 0.5204998778, 0.9538524394, 0.9544997361]```

### Error Function for Variables and Expressions

For most symbolic variables and expressions, `erf` returns unresolved symbolic calls.

Compute the error function for `x` and ```sin(x) + x*exp(x)```:

```syms x f = sin(x) + x*exp(x); erf(x) erf(f)```
```ans = erf(x) ans = erf(sin(x) + x*exp(x))```

### Error Function for Vectors and Matrices

If the input argument is a vector or a matrix, `erf` returns the error function for each element of that vector or matrix.

Compute the error function for elements of matrix `M` and vector `V`:

```M = sym([0 inf; 1/3 -inf]); V = sym([1; -i*inf]); erf(M) erf(V)```
```ans = [ 0, 1] [ erf(1/3), -1] ans = erf(1) -Inf*1i```

### Special Values of Error Function

`erf` returns special values for particular parameters.

Compute the error function for x = 0, x = ∞, and x = –∞. Use `sym` to convert `0` and infinities to symbolic objects. The error function has special values for these parameters:

`[erf(sym(0)), erf(sym(Inf)), erf(sym(-Inf))]`
```ans = [ 0, 1, -1]```

Compute the error function for complex infinities. Use `sym` to convert complex infinities to symbolic objects:

`[erf(sym(i*Inf)), erf(sym(-i*Inf))]`
```ans = [ Inf*1i, -Inf*1i]```

### Handling Expressions That Contain Error Function

Many functions, such as `diff` and `int`, can handle expressions containing `erf`.

Compute the first and second derivatives of the error function:

```syms x diff(erf(x), x) diff(erf(x), x, 2)```
```ans = (2*exp(-x^2))/pi^(1/2) ans = -(4*x*exp(-x^2))/pi^(1/2)```

Compute the integrals of these expressions:

```int(erf(x), x) int(erf(log(x)), x)```
```ans = exp(-x^2)/pi^(1/2) + x*erf(x) ans = x*erf(log(x)) - int((2*exp(-log(x)^2))/pi^(1/2), x)```

### Plot Error Function

Plot the error function on the interval from -5 to 5. Prior to R2016a, use `ezplot` instead of `fplot`.

```syms x fplot(erf(x),[-5,5]) grid on```

## Input Arguments

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Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

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### Error Function

The following integral defines the error function:

`$erf\left(x\right)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-{t}^{2}}dt$`

## Tips

• Calling `erf` for a number that is not a symbolic object invokes the MATLAB® `erf` function. This function accepts real arguments only. If you want to compute the error function for a complex number, use `sym` to convert that number to a symbolic object, and then call `erf` for that symbolic object.

• For most symbolic (exact) numbers, `erf` returns unresolved symbolic calls. You can approximate such results with floating-point numbers using `vpa`.

## Algorithms

The toolbox can simplify expressions that contain error functions and their inverses. For real values `x`, the toolbox applies these simplification rules:

• ```erfinv(erf(x)) = erfinv(1 - erfc(x)) = erfcinv(1 - erf(x)) = erfcinv(erfc(x)) = x```

• ```erfinv(-erf(x)) = erfinv(erfc(x) - 1) = erfcinv(1 + erf(x)) = erfcinv(2 - erfc(x)) = -x```

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

• `erfcinv(x) = erfinv(1 - x)`

• `erfinv(-x) = -erfinv(x)`

• `erfcinv(2 - x) = -erfcinv(x)`

• `erf(erfinv(x)) = erfc(erfcinv(x)) = x`

• `erf(erfcinv(x)) = erfc(erfinv(x)) = 1 - x`

## References

[1] Gautschi, W. "Error Function and Fresnel Integrals." Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.