Power Series Expansion for Complementary Error Function

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Theorem

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

where:

$\erfc$ is the complementary error function
$x$ is a real number.


Proof

\(\ds \map \erfc x\) \(=\) \(\ds 1 - \map \erf x\) Definition of Complementary Error Function
\(\ds \) \(=\) \(\ds 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }\) Power Series Expansion for Error Function

$\blacksquare$


Sources