Error Function

The error function (sometimes called Gaussian error function) is denoted $ \operatorname {erf} $ and is defined by the formula $$ \operatorname{erf}(x)= \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, \mathrm{d}t $$

The error function returns a result between -1 and 1.

Example: $ \operatorname{erf}(0) = 0 $, $ \operatorname{erf}(-\infty) = -1 $, $ \operatorname{erf}(+\infty) = 1 $

The erf function is an odd function

The complementary error function is defined by $$ \operatorname{erfc}(x) = 1- \operatorname{erf}(x) $$

Example: $ \operatorname{erf}(x) + \operatorname{erfc}(x) = 1 $

The expansion in integer series of 'erf ()' allow a fast computation (with approximations) by the formula: $$ \operatorname{erf}(x)= \frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{n!(2n+1)} = \frac{2}{\sqrt{\pi}} \left( x - \frac{z^3}{3} + \frac{z^5}{10} - \frac{z^7}{42} + \frac{z^9}{216} + \cdots \right) $$

The image of a negative value $ -z $ is the opposite of the image of the positive value: $ \operatorname{erf}(-z) = -\operatorname{erf}(z) $

The derivative of the error function is $$ \frac{d}{dz} \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}e^{-z^2} $$

The integral of the error function is $$ \int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi }}+x \operatorname{erf}(x) + c $$