In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as:

${\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.}$

Some authors define ${\displaystyle \operatorname {erf} }$ without the factor of ${\displaystyle 2/{\sqrt {\pi }}}$. This integral is a special (non-elementary) sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.

In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2, erf x is the probability that Y falls in the range [−x, x].

Two closely related functions are the complementary error function (erfc) defined as

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

and the imaginary error function (erfi) defined as

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

where i is the imaginary unit.