Error Function Wolfram Alpha
Contents |
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 complementary error function and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands>
Error Function Calculator
MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of inverse error function the Gaussian function). It is an entire function defined 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
Error Function Table
Language as Erf[z]. A two-argument form giving is also implemented as 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 error function matlab be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using 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. 2.8.1
Mathematica Wolfram|Alpha Appliance Enterprise Solutions Corporate Consulting Technical Services Wolfram|Alpha Business Solutions Products for Education Wolfram|Alpha Wolfram|Alpha Pro Problem Generator API Data Drop Mobile Apps Wolfram Cloud App
Error Function Python
Wolfram|Alpha for Mobile Wolfram|Alpha-Powered Apps Services Paid Project Support Training Summer
Erf(1)
Programs All Products & Services » Technologies Wolfram Language Revolutionary knowledge-based programming language. Wolfram Cloud Central equivalent record form infrastructure for Wolfram's cloud products & services. Wolfram Science Technology-enabling science of the computational universe. Computable Document Format Computation-powered interactive documents. Wolfram Engine Software engine implementing the http://mathworld.wolfram.com/Erf.html Wolfram Language. Wolfram Natural Language Understanding System Knowledge-based broadly deployed natural language. Wolfram Data Framework Semantic framework for real-world data. Wolfram Universal Deployment System Instant deployment across cloud, desktop, mobile, and more. Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha. All Technologies » Solutions Engineering, R&D Aerospace & Defense Chemical Engineering Control Systems Electrical Engineering https://reference.wolfram.com/language/ref/Erf.html Image Processing Industrial Engineering Mechanical Engineering Operations Research More... Education All Solutions for Education Web & Software Authoring & Publishing Interface Development Software Engineering Web Development Finance, Statistics & Business Analysis Actuarial Sciences Bioinformatics Data Science Econometrics Financial Risk Management Statistics More... Sciences Astronomy Biology Chemistry More... Trends Internet of Things High-Performance Computing Hackathons All Solutions » Support & Learning Learning Wolfram Language Documentation Fast Introduction for Programmers Training Videos & Screencasts Wolfram Language Introductory Book Virtual Workshops Summer Programs Books Need Help? Support FAQ Wolfram Community Contact Support Premium Support Premier Service Technical Services All Support & Learning » Company About Company Background Wolfram Blog News Events Contact Us Work with Us Careers at Wolfram Internships Other Wolfram Language Jobs Initiatives Wolfram Foundation MathWorld Computer-Based Math A New Kind of Science Wolfram Technology for Hackathons Student Ambassador Program Wolfram for Startups Demonstrations Project Wolfram Innovator Awards Wolfram + Raspberry Pi Summer Programs More... All Company » Search Wolf
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 http://mathworld.wolfram.com/InverseErf.html and Analysis>Special Functions>Erf> Calculus and Analysis>Calculus>Integrals>Definite Integrals> History and Terminology>Wolfram Language Commands> Inverse Erf The inverse erf function is the inverse function of the erf function such that (1) (2) with the http://www.sosmath.com/CBB/viewtopic.php?t=52713 first identity holding for and the second for . It is implemented in the Wolfram Language as InverseErf[x]. It is an odd function since (3) It has the special values (4) (5) error function (6) It is apparently not known if (7) (OEIS A069286) can be written in closed form. It satisfies the equation (8) where is the inverse erfc function. It has the derivative (9) and its integral is (10) (which follows from the method of Parker 1955). Definite integrals are given by (11) (12) (13) (14) (OEIS A087197 and A114864), where is the Euler-Mascheroni constant and is error function wolfram the natural logarithm of 2. The Maclaurin series of is given by (15) (OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1, (16) (OEIS A092676 and A092677). The th coefficient of this series can be computed as (17) where is given by the recurrence equation (18) with initial condition . SEE ALSO: Confidence Interval, Erf, Inverse Erfc, Probable Error RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/InverseErf/, http://functions.wolfram.com/GammaBetaErf/InverseErf2/ REFERENCES: Bergeron, F.; Labelle, G.; and Leroux, P. Ch.5 in Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998. Carlitz, L. "The Inverse of the Error Function." Pacific J. Math. 13, 459-470, 1963. Parker, F.D. "Integrals of Inverse Functions." Amer. Math. Monthly 62, 439-440, 1955. Sloane, N.J.A. Sequences A002067/M4458, A007019/M3126, A069286, A087197, A092676, A092677, A114859, A114860, and A114864 in "The On-Line Encyclopedia of Integer Sequences." CITE THIS AS: Weisstein, Eric W. "Inverse Erf." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InverseErf.html Wolfram Web Resources Mathematica» The #1 tool for creating Demonstrations and anything technical. Wolfram|Alpha» Explore anything with the first computational knowledge engine. Wolfram Demonstrations Project» Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social science
» 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