= Complementary error function
{title2=$\operatorname{erfc}z=1-\operatorname{erf}z$}
= erfc
{c}
{synonym}
The complementary error function describes the <Gaussian integral> area remaining beyond the endpoint. For a real $x$,
$$
\operatorname{erfc}(x)=1-\operatorname{erf}(x)=\frac{2}{\sqrt\pi}\int_x^\infty e^{-t^2}\,dt.
$$
It is an <entire function> for complex arguments when defined by $1-\operatorname{erf}(z)$. On the <real line> it decreases from two to zero, with $\operatorname{erfc}(0)=1$ and $\operatorname{erfc}(-x)=2-\operatorname{erfc}(x)$.<NIST Digital Library of Mathematical Functions/7.2 Definitions>[§7.2(i), equation 7.2.2 and the entire-function statement]{ref}
For example, $\operatorname{erfc}(1)\approx0.157299$ is a sizeable tail, whereas $\operatorname{erfc}(8)\approx1.12243\times10^{-29}$ is extremely small. In the tested <floating-point arithmetic>, a direct evaluation of $1-\operatorname{erf}(8)$ gives zero because the computed <error function> has already rounded to one. Evaluating the <complementary error function> directly preserves this tail. It is the useful quantity in an upper-tail <normal distribution> <probability> and in the decaying side of an <error-function diffusion front>.
For large positive real $x$, the leading <asymptotic expansion> is
$$
\operatorname{erfc}(x)\sim\frac{e^{-x^2}}{\sqrt\pi x},\qquad x\to+\infty.
$$
The first corrections are $1-1/(2x^2)+3/(4x^4)-\cdots$ multiplying the leading term. This is an <asymptotic expansion>, not an assertion that summing arbitrarily many terms converges at a fixed $x$.<NIST Digital Library of Mathematical Functions/7.12 Asymptotic Expansions>[§7.12(i), equation 7.12.1 and its real-axis remainder statement]{ref} The <complementary error function tail bound> provides finite-$x$ inequalities instead.
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