Asymptotic Expansion for Complementary Error Function

Theorem

\(\ds \map \erfc x\)

\(\sim\)

\(\ds \frac {e^{-x^2} } {\sqrt \pi x} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n \paren {2 n - 1}!!} {\paren {2 x^2}^n}\)

\(\ds \)

\(\sim\)

\(\ds \frac {e^{-x^2} } {\sqrt \pi x} \paren {1 - \dfrac 1 {2 x^2} + \dfrac {1 \cdot 3} {\paren {2 x^2}^2} - \dfrac {1 \cdot 3 \cdot 5} {\paren {2 x^2}^3} + \cdots}\)

where:

$\erfc$ denotes the complementary error function

$n!!$ denotes the double factorial of $n$

$\sim$ denotes asymptotic equivalence as $x \to \infty$.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.5$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 36$: Miscellaneous and Riemann Zeta Functions: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $36.5.$