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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 $ z$.



"error function" is owned by rspuzio. [ full author list (2) ]
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See Also: area under Gaussian curve

Also defines:  complementary error function
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Cross-references: endpoint, real axis, positive, parallel, entire function, path, Cauchy's theorem, Mittag-Leffler function, hypergeometric functions, confluent, normal random variable, theory, functions
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This is version 4 of error function, born on 2004-10-28, modified 2007-06-12.
Object id is 6429, canonical name is ErrorFunction.
Accessed 11443 times total.

Classification:
AMS MSC33B20 (Special functions :: Elementary classical functions :: Incomplete beta and gamma functions )

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