error functionErrorFunction2004-10-28 22:59:452009-01-14 11:27:34Definitioncomplementary error function% this is the default PlanetMath preamble. as your knowledge
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% define commands hereThe \emph{error function} ${\rm erf} \colon \mathbb{C} \to \mathbb{C}$ is defined as follows:
$${\rm erf} (z) = {2 \over \sqrt{\pi}} \int_0^z e^{-t^2} \, dt$$
The \emph{complementary error function} ${\rm erfc} \colon \mathbb{C} \to \mathbb{C}$ is defined as
$${\rm erfc} (z) = {2 \over \sqrt{\pi}} \int_z^\infty e^{-t^2} \, dt$$
The name ``error function'' comes from the role that these functions play in the theory of the normal random variable. It is also worth noting that the error function is a special case of the confluent hypergeometric functions and of the Mittag-Leffler function.
\textbf{Note.}\, By \PMlinkname{Cauchy integral theorem}{SecondFormOfCauchyIntegralTheorem}, the choice path of integration in the definition of ${\rm erf}$ is irrelevant since the integrand is an entire function. In the definition of ${\rm erfc}$, the path may be taken to be a half-line parallel to the positive real axis with endpoint $z$.