Definition:Complementary Error Function

Definition

The complementary error function is the real function $\erfc: \R \to \R$:

\(\ds \map {\erfc} x\)

\(=\)

\(\ds 1 - \map \erf x\)

where $\erf$ denotes the error function

\(\ds \)

\(=\)

\(\ds 1 - \dfrac 2 {\sqrt \pi} \int_0^x \map \exp {-t^2} \rd t\)

where $\exp$ denotes the real exponential function

\(\ds \)

\(=\)

\(\ds \dfrac 2 {\sqrt \pi} \int_x^\infty \map \exp {-t^2} \rd t\)

Also see

• Definition:Error Function

• Results about the complementary error function can be found here.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {IV}$. The Complementary Error function
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Miscellaneous Special Functions: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): complementary error function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error function
• 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 $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$