### 7.3.11 Error function

The error function erf is defined by:

erf(x)=

2

√

π

∫

x
0

e−t2d t,

where the constant 2/√π is chosen so that erf(+∞)=1 and erf(−∞)=−1, since

∫

+∞
0

e−t2d t=

√

π

2

.

The erf command computes the error function.

- erf takes a, a number.
- erf(a) returns the value of erf(a).

#### Examples

erf(1)

erf | ⎛ ⎝ | 1 | ⎞ ⎠

erf(1.0)

0.84270079295 |  |  |  |  |  |  |  |  |  |

erf(1/(sqrt(2)))*1/2+0.5

0.841344746069 |  |  |  |  |  |  |  |  |  |

##### Remark.

The relation between erf and normal_cdf (see Section 20.4.7) is:

normal_cdf(x)=

1
2

+

1
2

erf

⎛
⎜
⎜
⎜
⎜
⎝

x

√

2

⎞
⎟
⎟
⎟
⎟
⎠

Indeed, making the change of variable t=u√2 in normal_cdf(x)=1/2+1/√2π∫0xe−t2/2d t yields:

normal_cdf(x)=

1
2

+

1

√

π

∫

x

√

2

0

e−u2d u=

1
2

+

1
2

erf

⎛
⎜
⎜
⎜
⎜
⎝

x

√

2

⎞
⎟
⎟
⎟
⎟
⎠

.