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Erf

erf(z) is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by

erf(z)=2/(sqrt(pi))int_0^ze^(-t^2)dt. (1)

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

Erf is implemented in the Wolfram Language as Erf[_z_]. A two-argument form giving erf(z_1)-erf(z_0) is also implemented as Erf[z0,_z1_].

Erf satisfies the identities

where erfc(z) is erfc, the complementary error function, and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind. For z>0,

erf(z)=pi^(-1/2)gamma(1/2,z^2), (5)

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

Erf can also be defined as a Maclaurin series

(OEIS A007680). Similarly,

erf^2(z)=4/pi(z^2-2/3z^4+(14)/(45)z^6-4/(35)z^8+(166)/(4725)z^(10)+...) (8)

(OEIS A103979 and A103980).

For x<<1, erf(x) may be computed from

(OEIS A000079 and A001147; Acton 1990).

For x>>1,

Using integration by parts gives

so

erf(x)=1-(e^(-x^2))/(sqrt(pi)x)(1-1/(2x^2)-...) (17)

and continuing the procedure gives the asymptotic series

(OEIS A001147 and A000079).

Erf has the values

It is an odd function

erf(-z)=-erf(z), (23)

and satisfies

erf(z)+erfc(z)=1. (24)

Erf may be expressed in terms of a confluent hypergeometric function of the first kind M as

Its derivative is

(d^n)/(dz^n)erf(z)=(-1)^(n-1)2/(sqrt(pi))H_(n-1)(z)e^(-z^2), (27)

where H_n is a Hermite polynomial. The first derivative is

d/(dz)erf(z)=2/(sqrt(pi))e^(-z^2), (28)

and the integral is

interf(z)dz=zerf(z)+(e^(-z^2))/(sqrt(pi)). (29)

ErfReIm

ErfContours

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

A simple integral involving erf that Wolfram Language cannot do is given by

int_0^pe^(-x^2)erf(p-x)dx=1/2sqrt(pi)[erf(1/2sqrt(2)p)]^2 (30)

(M. R. D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include

int_0^infty(e^(-(p+x)y))/(pi(p+x))sin(asqrt(x))dx=-sinh(asqrt(p)) +(e^(-asqrt(p)))/2erf(a/(2sqrt(y))-sqrt(py))+(e^(asqrt(p)))/2erf(a/(2sqrt(y))+sqrt(py)) int_0^infty(sqrt(x)e^(-(p+x)y))/(pi(p+x))cos(asqrt(x))dx=(e^(-[py+a^2/(4y)]))/(sqrt(piy))+sqrt(p)[-cosh(asqrt(p))-(e^(-asqrt(p)))/2erf(a/(2sqrt(y))-sqrt(py))+(e^(asqrt(p)))/2erf(a/(2sqrt(y))+sqrt(py))] (31)

(M. R. D'Orsogna, pers. comm., Dec. 15, 2005).

Erf has the continued fraction

(Wall 1948, p. 357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p. 139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp. 8-9).

Definite integrals involving erf(x) include Definite integrals involving erf(x) include

The first two of these appear in Prudnikov et al. (1990, p. 123, eqns. 2.8.19.8 and 2.8.19.11), with R[p]>0,|arg(a)|,|argb|,|argc|<pi/4.

A complex generalization of erf(x) is defined as

Integral representations valid only in the upper half-plane I[z]>0 are given by

See also

Dawson's Integral, Erfc, Erfi, Fresnel Integrals,Gaussian Function, Gaussian Integral, Inverse Erf, Normal Distribution Function, Owen T-Function, Probability Integral Explore this topic in the MathWorld classroom

http://functions.wolfram.com/GammaBetaErf/Erf/,http://functions.wolfram.com/GammaBetaErf/Erf2/

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Error Function and Fresnel Integrals." Ch. 7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 297-309, 1972.Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 16, 1990.Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 568-569, 1985.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 105, 2003.Olds, C. D. Continued Fractions. New York: Random House, 1963.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 2: Special Functions. New York: Gordon and Breach, 1990.Sloane, N. J. A. Sequences A000079/M1129,A001147/M3002, A007680/M2861,A103979, A103980 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Error Function erf(x) and Its Complement erfc(x)." Ch. 40 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 385-393, 1987.Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.Watson, G. N. "Theorems Stated by Ramanujan (IV): Theorems on Approximate Integration and Summation of Series." J. London Math. Soc. 3, 282-289, 1928.Whittaker, E. T. and Robinson, G. "The Error Function." ยง92 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 179-182, 1967.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

Referenced on Wolfram|Alpha

Erf

Cite this as:

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

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