Erfc

The ``complementary error function''

$$\mathop{\rm erfc}\nolimits (x) \equiv {2\over \sqrt{\pi }} \int^\infty_x e^{-t^2}\,dt$$ (1)

$$= 1-\mathop{\rm erf}\nolimits (x)$$ (2)

$$= \sqrt{\pi}\, \gamma({\textstyle{1\over 2}}, z^2),$$ (3)

where $\gamma$ is the incomplete Gamma Function. It has the values

$$\mathop{\rm erfc}\nolimits (0) = 1$$ (4)

$$\mathop{\rm erfc}\nolimits (\infty) = 0$$ (5)

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

$$\int_0^\infty \mathop{\rm erfc}\nolimits (x)\,dx = {1\over \sqrt{\pi}}$$ (7)

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

A generalization is obtained from the differential equation

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

The general solution is then

$$y=A\mathop{\rm erfci}\nolimits _n(z)+B\mathop{\rm erfci}\nolimits _n(-z),$$ (10)

where erfci${}_n(z)$ is the erfc integral. For integral $n \geq 1$,

$$\mathop{\rm erfci}\nolimits _n(z) = \underbrace{\int\cdots\int}_n \mathop{\rm erfc}\nolimits (z)\,dz$$ (11)

$$= {2\over\sqrt{\pi}} \int_z^\infty {(t-z)^n\over n!} e^{-t^2}\,dt.$$ (12)

The definition can be extended to $n=-1$ and 0 using

$$\mathop{\rm erfci}\nolimits _{-1}(z) = {2\over \sqrt{\pi}} e^{-z^2}$$ (13)

$$\mathop{\rm erfci}\nolimits _0(z) = \mathop{\rm erfc}\nolimits (z).$$ (14)

See also Erf, Erfi

References

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.