Erfc is the complementary error function, commonly denoted , is an entire function
defined by
It is implemented in the Wolfram Language
as Erfc[z].
Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of
.
For ,
|
(3)
|
where is the incomplete
gamma function.
The derivative is given by
|
(4)
|
and the indefinite integral by
|
(5)
|
It has the special values
It satisfies the identity
|
(9)
|
It has definite integrals
For , is bounded by
|
(13)
|
Erfc can also be extended to the complex plane, as illustrated above.
A generalization is obtained from the erfc
differential equation
|
(14)
|
(Abramowitz and Stegun 1972, p. 299; Zwillinger 1997, p. 122). The general solution is then
|
(15)
|
where is the repeated erfc integral.
For integer ,
(Abramowitz and Stegun 1972, p. 299), where is a confluent
hypergeometric function of the first kind and is a gamma function.
The first few values, extended by the definition for and 0, are given by
See also
Erf,
Erfc Differential Equation,
Erfi,
Hh
Function,
Inverse Erfc
Related Wolfram sites
http://functions.wolfram.com/GammaBetaErf/Erfc/
Explore with Wolfram|Alpha
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 and Its Complement " and "The and
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
Subject classifications