Error function

In mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ⁡ ( z ) = 2 π ∫ 0 z e − t 2 d t . {\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 exp ⁡ ( − t 2 ) {\displaystyle \exp(-t^{2})} is holomorphic on the whole complex plane C {\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 2 / π {\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 X {\displaystyle X} , the error function has the following interpretation: for a real random variable Y {\displaystyle Y} that is normally distributed with mean 0 and standard deviation 1 / 2 {\displaystyle 1/{\sqrt {2}}} , erf ⁡ ( x ) {\displaystyle \operatorname {erf} (x)} is the probability that Y {\displaystyle Y} falls in the range [ − x , x ] {\displaystyle [-x,x]} . Two closely related functions are the complementary error function erfc ⁡ ( z ) = 1 − erf ⁡ ( z ) {\displaystyle \operatorname {erfc} (z)=1-\operatorname {erf} (z)} and the imaginary error function erfi ⁡ ( z ) = − i erf ⁡ ( i z ) , {\displaystyle \operatorname {erfi} (z)=-i\operatorname {erf} (iz),} where i {\displaystyle i} is the imaginary unit.