error function

The error function $\mathrm{erf}:ℂ\to ℂ$ is defined as follows:

$$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi }}{\int }_0^z{e}^{-t^2}𝑑t$$

The complementary error function $\mathrm{erfc}:ℂ\to ℂ$ is defined as

$$\mathrm{erfc}(z)=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{e}^{-t^2}𝑑t$$

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 integral theorem (http://planetmath.org/SecondFormOfCauchyIntegralTheorem), the choice path of integration in the definition of $\mathrm{erf}$ is irrelevant since the integrand is an entire function. In the definition of $\mathrm{erfc}$, the path may be taken to be a half-line parallel to the positive real axis with endpoint $z$.

Title	error function
Canonical name	ErrorFunction
Date of creation	2013-03-22 14:46:51
Last modified on	2013-03-22 14:46:51
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	10
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 33B20
Related topic	AreaUnderGaussianCurve
Related topic	ListOfImproperIntegrals
Related topic	UsingConvolutionToFindLaplaceTransform
Defines	complementary error function