Error Function Mathworld
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Error Function Matlab
at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... error function mathematica Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function defined by
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(1) (2) 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 gaussian mathworld the special values (6) (7) (8) It satisfies the identity (9) It has definite 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 Table
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
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2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf>
Beta Distribution Mathworld
Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Erfi Min Max Min complementary error function Max Re Im The "imaginary error function" is an entire function defined by (1) where is the erf function. It is implemented in the Wolfram Language as http://mathworld.wolfram.com/Erfc.html Erfi[z]. has derivative (2) and integral (3) It has series about given by (4) (where the terms are OEIS A084253), and series about infinity given by (5) (OEIS A001147 and A000079). SEE ALSO: Dawson's Integral, Erf, Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfi/ REFERENCES: Sloane, N.J.A. Sequences A000079/M1129, A001147/M3002, and A084253 in "The On-Line Encyclopedia of Integer Sequences." http://mathworld.wolfram.com/Erfi.html Referenced on Wolfram|Alpha: Erfi CITE THIS AS: Weisstein, Eric W. "Erfi." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Erfi.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 sciences, and more. Computerbasedmath.org» Join the initiative for modernizing math education. Online Integral Calculator» Solve integrals with Wolfram|Alpha. Step-by-step Solutions» Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Wolfram Problem Generator» Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Wolfram Education Portal» Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Wolfram Language» Knowledge-based programming for everyone. Contact the MathWorld Team © 1999-2016 Wolfram Research, Inc. | Terms of Use THINGS TO TRY: erf erfi x erfi (0)
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 http://mathworld.wolfram.com/NormalDistributionFunction.html Last updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Probability and Statistics>Statistical Distributions>Continuous Distributions> Interactive Entries>Interactive Demonstrations> Normal Distribution Function A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range , (1) It is related to the error function probability integral (2) by (3) Let so . Then (4) Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range is therefore given by (5) Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and error function mathworld root extractions, and so must be either computed numerically or otherwise approximated. Note that a function different from is sometimes defined as "the" normal distribution function (6) (7) (8) (9) (Feller 1968; Beyer 1987, p.551), although this function is less widely encountered than the usual . The notation is due to Feller (1971). The value of for which falls within the interval with a given probability is a related quantity called the confidence interval. For small values , a good approximation to is obtained from the Maclaurin series for erf, (10) (OEIS A014481). For large values , a good approximation is obtained from the asymptotic series for erf, (11) (OEIS A001147). The value of for intermediate can be computed using the continued fraction identity (12) A simple approximation of which is good to two decimal places is given by (14) The plots below show the differences between and the two appro