Error Function Erf Wiki
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Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive excel error function erf Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error erf error function ti-89 function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined q function erf by (1) Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as
Normal Distribution Erf
Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and is a confluent hypergeometric function of the first kind. For , (5) where is the incomplete gamma function. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using gaussian erf integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im 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 (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include (31) (M.R.D'Orsogna, pers. comm., Dec.15, 2005). Erf has the continued fraction (32) (33) (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 include Definite integrals involving include (34) (35) (36) (37) (38) The first two of these appear in Prudnikov et al. (1990, p.123, eqns. 2.8.19.8 and 2.8.19.11), with , . A complex generalization of is defined as (39) (40) Integral representations valid only in the upper half-plane are given by (41) (42) SEE ALSO: Dawson's Integral, Erfc, E
Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources» 13,594 entries Last updated: Tue Sep 27 mathematica erf 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf>
Erf Erfc
Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> History and Terminology>Wolfram
Erf Definition
Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function defined by (1) (2) It is implemented in the Wolfram Language http://mathworld.wolfram.com/Erf.html 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 (6) (7) (8) It satisfies the identity (9) It has definite http://mathworld.wolfram.com/Erfc.html integrals (10) (11) (12) For , is bounded by (13) Min Max Re Im 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 , (16) (17) (18) (19) (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 (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ 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.
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» Probability and Statistics All times are UTC [ DST ] What is erf(x)? Moderators: mak, helmut, Shadow, outermeasure, Ilaggoodly Page 1 of 1 [ 3 posts ] Print view Previous topic | Next topic Author Message yoyobarn Post subject: What is erf(x)?Posted: Fri, 7 Jan 2011 14:57:00 UTC Member Joined: Fri, 12 Nov 2010 07:00:12 UTCPosts: 48 What is erf(x)? (in layman terms) The official wolframalpha definition is here: http://www.wolframalpha.com/input/?i=erf%28x%29 And, can we calculate erf(x) from the Normal Distribution Table? Top Ilaggoodly Post subject: Posted: Fri, 7 Jan 2011 18:16:02 UTC Site Admin Joined: Thu, 15 Feb 2007 06:35:15 UTCPosts: 755 its the Error Function, a function in probability theory that is related to the integral cumulative density function of the normal distribution, (the pdf doesn't yield a pretty integral) And because of this, it can be approximated with a normal distribution table, read the wiki page http://en.wikipedia.org/wiki/Normal_distribution Top aswoods Post subject: Posted: Sat, 8 Jan 2011 08:11:28 UTC Member of the 'S.O.S. Math' Hall of Fame Joined: Mon, 23 Feb 2009 23:20:33 UTCPosts: 1049Location: Adelaide, Australia If the same measurement is taken hundreds of times, then in many situations the results tend to look like a bell curve, with most of the reported measurements clustered around the correct value. The standard normal distribution is an idealized version of this curve, centred on 0. If a new measurement is taken and mapped onto this idealized version (as m), and you know that it is over the correct value of "0", then erf(x) tells you the probability that your measurement is less than x. Given 0