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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Complementary Error Function $\ds\map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.4$