Power Series Expansion for Complementary Error Function

Theorem

\(\ds \map \erfc x\)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 1 - \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9} {9 \times 4!} - \cdots}\)

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): $\S 35$: Miscellaneous Special Functions: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.4$
• 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 $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $36.4.$