error function

The 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 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.

Note. By Cauchy integral 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 $z$ .