quality:correction, structural:scholar_edition
Factorial series

The inverse factorial series: erfc z = e − z 2 π z ∑ n = 0 ∞ (− 1) n Q n (z 2 + 1) n ¯ = e − z 2 π z {\displaystyle {\begin{aligned}\operatorname {erfc} z&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}Q_{n}}{{\left(z^{2}+1\right)}^{\bar {n}}}}\\&={\frac {e^{-z^{2}}}{{\sqrt {\pi }}\,z}}\left\end{aligned}}} converges for Re(z) > 0. Here Q n = def 1 Γ (1 2) ∫ 0 ∞ τ (τ − 1) ⋯ (τ − n + 1) τ − 1 2 e − τ d τ = ∑ k = 0 n (1 2) k ¯ s (n, k), {\displaystyle {\begin{aligned}Q_{n}&{\overset {\text{def}}{{}={}}}{\frac {1}{\Gamma {\left({\frac {1}{2}}\right)}}}\int _{0}^{\infty }\tau (\tau -1)\cdots (\tau -n+1)\tau ^{-{\frac {1}{2}}}e^{-\tau }\,d\tau \\&=\sum _{k=0}^{n}\left({\frac {1}{2}}\right)^{\bar {k}}s(n,k),\end{aligned}}} z denotes the rising factorial, and s (n, k) denotes a signed Stirling number of the first kind. There also exists a representation by an infinite sum containing the double factorial: erf z = 2 π ∑ n = 0 ∞ (− 2) n (2 n − 1) ! ! (2 n + 1) ! z 2 n + 1 {\displaystyle \operatorname {erf} z={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}}