Asymptotic Expansion for Complementary Error Function

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Theorem

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

where:

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


Proof


Sources