漸近展開
誤差関数
erfc  ( x ) = 2 π ∫ x ∞ e − t 2 d t {\displaystyle \operatorname {erfc} (x)={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}dt}
は、以下の様な漸近展開を持つ。
erfc  ( x ) ∼ e − x 2 π x ( 1 − 1 2 x 2 + 1 ⋅ 3 2 2 x 4 − 1 ⋅ 3 ⋅ 5 2 3 x 6 + ⋯ ) ( x → ∞ ) {\displaystyle \operatorname {erfc} (x)\sim {\frac {e^{-x^{2}}}{{\sqrt {\pi }}x}}\left(\ 1-{\frac {1}{2x^{2}}}+{\frac {1\cdot 3}{2^{2}x^{4}}}-{\frac {1\cdot 3\cdot 5}{2^{3}x^{6}}}+\cdots \right)\ \ \ (x\to \infty )}