PlanetMath: error function

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error function

(Definition)

The error function ${\rm Erf} \colon \mathbb{C} \to \mathbb{C}$ is defined as follows:

$\displaystyle {\rm Erf} (z) = {2 \over \sqrt{\pi}} \int_0^z e^{-t^2} \, dt$

The complementary error function ${\rm Erfc} \colon \mathbb{C} \to \mathbb{C}$ is defined as

$\displaystyle {\rm Erfc} (z) = {2 \over \sqrt{\pi}} \int_z^\infty e^{-t^2} \, dt$

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.

Note: By Cauchy's theorem, 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 .

"error function" is owned by rspuzio. [ full author list (2) ]

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