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The ``complementary error function''

$\displaystyle \mathop{\rm erfc}\nolimits (x)$ $\textstyle \equiv$ $\displaystyle {2\over \sqrt{\pi }} \int^\infty_x e^{-t^2}\,dt$ (1)
  $\textstyle =$ $\displaystyle 1-\mathop{\rm erf}\nolimits (x)$ (2)
  $\textstyle =$ $\displaystyle \sqrt{\pi}\, \gamma({\textstyle{1\over 2}}, z^2),$ (3)

where $\gamma$ is the incomplete Gamma Function. It has the values
$\displaystyle \mathop{\rm erfc}\nolimits (0)$ $\textstyle =$ $\displaystyle 1$ (4)
$\displaystyle \mathop{\rm erfc}\nolimits (\infty)$ $\textstyle =$ $\displaystyle 0$ (5)

\mathop{\rm erfc}\nolimits (-x)=2-\mathop{\rm erfc}\nolimits (x)
\end{displaymath} (6)

$\displaystyle \int_0^\infty \mathop{\rm erfc}\nolimits (x)\,dx$ $\textstyle =$ $\displaystyle {1\over \sqrt{\pi}}$ (7)
$\displaystyle \int_0^\infty \mathop{\rm erfc}\nolimits ^2(x)\,dx$ $\textstyle =$ $\displaystyle {2-\sqrt{2}\over \sqrt{\pi}}.$ (8)

A generalization is obtained from the differential equation

{d^2y\over dz^2}+2z{dy\over dz}-2ny=0.
\end{displaymath} (9)

The general solution is then
y=A\mathop{\rm erfci}\nolimits _n(z)+B\mathop{\rm erfci}\nolimits _n(-z),
\end{displaymath} (10)

where erfci${}_n(z)$ is the erfc integral. For integral $n \geq 1$,
$\displaystyle \mathop{\rm erfci}\nolimits _n(z)$ $\textstyle =$ $\displaystyle \underbrace{\int\cdots\int}_n \mathop{\rm erfc}\nolimits (z)\,dz$ (11)
  $\textstyle =$ $\displaystyle {2\over\sqrt{\pi}} \int_z^\infty {(t-z)^n\over n!} e^{-t^2}\,dt.$ (12)

The definition can be extended to $n=-1$ and 0 using
$\displaystyle \mathop{\rm erfci}\nolimits _{-1}(z)$ $\textstyle =$ $\displaystyle {2\over \sqrt{\pi}} e^{-z^2}$ (13)
$\displaystyle \mathop{\rm erfci}\nolimits _0(z)$ $\textstyle =$ $\displaystyle \mathop{\rm erfc}\nolimits (z).$ (14)

See also Erf, Erfi


Abramowitz, M. and Stegun, C. A. (Eds.). ``Repeated Integrals of the Error Function.'' §7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 299-300, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.

Spanier, J. and Oldham, K. B. ``The Error Function $\mathop{\rm erf}\nolimits (x)$ and Its Complement $\mathop{\rm erfc}\nolimits (x)$'' and ``The $\mathop{\rm exp}\nolimits (x)$ and $\mathop{\rm erfc}\nolimits (\sqrt{x}\,)$ and Related Functions.'' Chs. 40 and 41 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393 and 395-403, 1987.

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© 1996-9 Eric W. Weisstein