# Error function

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In mathematics, the error function is a function associated with the cumulative distribution function of the normal distribution.

The definition is

${\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}\exp(-t^{2})dt.\,}$

The complementary error function is defined as

${\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x).\,}$

The probability that a normally distributed random variable X with mean μ and variance σ2 exceeds x is

${\displaystyle F(x;\mu ,\sigma )={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right].}$