# Asymptotic Expansion for Complementary Error Function

## Theorem

$\ds \map \erfc x \sim \frac {e^{-x^2} } {\sqrt \pi x} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n - 1}!!} {\paren {2 x^2}^n}$

where:

$\erfc$ denotes the complementary error function
$n!!$ denotes the double factorial of $n$
$\sim$ denotes asymptotic equivalence as $x \to \infty$.