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Erfc

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Erfc

Erfc is the complementary error function, commonly denoted erfc(z), is an entire function defined by

erfc(z)=1-erf(z)
(1)
=2/(sqrt(pi))int_z^inftye^(-t^2)dt.
(2)

It is implemented in the Wolfram Language as Erfc[z].

Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define erfc(z) without the leading factor of 2/sqrt(pi).

For z>0,

erfc(z)=(Gamma(1/2,z^2))/(sqrt(pi)),
(3)

where Gamma(a,x) is the incomplete gamma function.

The derivative is given by

d/(dz)erfc(z)=-(2e^(-z^2))/(sqrt(pi)),
(4)

and the indefinite integral by

interfc(z)dz=zerfc(z)-(e^(-z^2))/(sqrt(pi))+C.
(5)

It has the special values

erfc(-infty)=2
(6)
erfc(0)=1
(7)
erfc(infty)=0.
(8)

It satisfies the identity

erfc(-x)=2-erfc(x).
(9)

It has definite integrals

int_0^inftyerfc(x)dx=1/(sqrt(pi))
(10)
int_0^inftyerfc^2(x)dx=(2-sqrt(2))/(sqrt(pi))
(11)
int_0^inftysin(x^2)erfc(x)dx=(pi-2sinh^(-1)1)/(4sqrt(2pi)).
(12)
ErfcBounds

For x>0, erfc(x) is bounded by

2/(sqrt(pi))(e^(-x^2))/(x+sqrt(x^2+2))<erfc(x)<=2/(sqrt(pi))(e^(-x^2))/(x+sqrt(x^2+4/pi)).
(13)
ErfcReImAbs
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Erfc can also be extended to the complex plane, as illustrated above.

Erfci

A generalization is obtained from the erfc differential equation

(d^2y)/(dz^2)+2z(dy)/(dz)-2ny=0
(14)

(Abramowitz and Stegun 1972, p. 299; Zwillinger 1997, p. 122). The general solution is then

y=Aerfc_n(z)+Berfc_n(-z),
(15)

where erfc_n(z) is the repeated erfc integral. For integer n>=1,

erfc_n(z)=int_z^infty...int_z^infty_()_(n)erfc(z)dz
(16)
=-2/(sqrt(pi))int_z^infty((t-z)^n)/(n!)e^(-t^2)dt
(17)
=(e^(-z^2))/(sqrt(pi)n!)[Gamma(1/2(n+1))_1F_1(1/2(n+1);1/2;z^2)-nz_1F_1(1+1/2n;3/2;z^2)]
(18)
=2^(-n)e^(-z^2)[(_1F_1(1/2(n+1);1/2;z^2))/(Gamma(1+1/2n))-(2z_1F_1(1+1/2n;3/2;z^2))/(Gamma(1/2(n+1)))]
(19)

(Abramowitz and Stegun 1972, p. 299), where _1F_1(a;b;z) is a confluent hypergeometric function of the first kind and Gamma(z) is a gamma function. The first few values, extended by the definition for n=-1 and 0, are given by

erfc_0(z)=erfc(z)
(20)
erfc_1(z)=(e^(-z^2))/(sqrt(pi))-zerfc(z)
(21)
erfc_2(z)=1/4[(1+2z^2)erfc(z)-(2ze^(-z^2))/(sqrt(pi))].
(22)

See also

Erf, Erfc Differential Equation, Erfi, Hh Function, Inverse Erfc

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/Erfc/

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References

Abramowitz, M. and Stegun, I. 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.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568-569, 1985.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 erf(x) and Its Complement erfc(x)" and "The exp(x) and erfc(sqrt(x)) and Related Functions." Chs. 40 and 41 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393 and 395-403, 1987.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.

Referenced on Wolfram|Alpha

Erfc

Cite this as:

Weisstein, Eric W. "Erfc." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erfc.html

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