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 $z$.