Complementary error function

Remarks

Since $\frac{2}{\sqrt{\pi}} \int_0^\infty e^{-t^2}\, dt = 1$, the definition is equivalent to $$\operatorname{erfc} x = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt.$$

Example: $\operatorname{erfc} 0 = 1$, and $\operatorname{erfc}(-x) = 2 - \operatorname{erfc} x$ because $\operatorname{erf}$ is odd.

Non-example: $x \mapsto 1 + \operatorname{erf} x$ is not $\operatorname{erfc}$; it tends to $2$ rather than $0$ as $x \to \infty$.

For large $x > 0$, $\operatorname{erfc} x \sim \frac{e^{-x^2}}{x\sqrt{\pi}}$. The definition extends to complex $z$ by $\operatorname{erfc} z = 1 - \operatorname{erf} z$, which is an entire function. The scaled function $e^{x^2} \operatorname{erfc} x$ is often used in numerical work.