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Erfc

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

			(1)
			(2)

It is implemented in theWolfram LanguageasErfc[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 ,

(3)

where Gamma(a,x) is theincomplete gamma function.

The derivative is given by

(4)

and the indefinite integral by

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

(5)

It has the special values

			(6)
			(7)
			(8)

It satisfies the identity

(9)

It has definite integrals

			(10)
			(11)
			(12)

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)

Erfc can also be extended to the complex plane, as illustrated above.

A generalization is obtained from theerfcdifferential equation

(14)

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

(15)

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

			(16)
			(17)
			(18)
			(19)

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

			(20)
			(21)
			(22)

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Repeated Integrals of the Error Function." §7.2 inHandbook 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 inNumerical 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

and Related Functions." Chs. 40 and 41 inAn 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.

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Erfc

Cite this as:

Weisstein, Eric W. "Erfc." FromMathWorld--AWolfram Resource.https://mathworld.wolfram.com/Erfc.html

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