Error function

In mathematics, the error function (also called the Gauss error function), often denoted by ${\displaystyle \mathbf {erf} }$, is the function $${\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,dt.}$$ The integral here is a complex contour integral which is path-independent because ${\displaystyle \exp(-t^{2})}$ is holomorphic on the whole complex plane ${\displaystyle \mathbb {C} }$. In many applications, the function argument is a real number, in which case the function value is also real.

In some older texts, the error function is defined without the factor of ${\displaystyle 2/{\sqrt {\pi }}}$. This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations.

In statistics, for non-negative real values of ${\displaystyle X}$, the error function has the following interpretation: for a real random variable${\displaystyle Y}$ that is normally distributed with mean 0 and standard deviation${\displaystyle 1/{\sqrt {2}}}$, ${\displaystyle \operatorname {erf} (x)}$ is the probability that ${\displaystyle Y}$ falls in the range ${\displaystyle [-x,x]}$.

Two closely related functions are the complementary error function

${\displaystyle \operatorname {erfc} (z)=1-\operatorname {erf} (z)}$

and the imaginary error function

${\displaystyle \operatorname {erfi} (z)=-i\operatorname {erf} (iz),}$

where ${\displaystyle i}$ is the imaginary unit.