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.

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

The inverse error function is defined by the inverse (reciprocal) function $$ \operatorname{erf}^{-1}(\operatorname{erf}(x)) = x $$

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 inverse function has a series expansion: $$ \operatorname{erf}^{-1}(x) = \frac{1}{2} \sqrt{\pi} \left( x + \frac{\pi}{12}x^3 + \frac{7\pi^2}{480} x^5 + \frac{127\pi^3}{40320} x^7 + \frac{4369\pi^4}{5806080} x^9 + O(x^{10}) \right) $$

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

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 $$ \int \operatorname{erf}(x)\, dx = \frac{e^{-x^2}}{\sqrt{\pi}}+x \operatorname{erf}(x) + c $$

The error function is bounded between -1 and 1, that is, $ -1 \leq \operatorname{erf}(x) \leq 1 $ for all real $ x $.