Error Function Formula
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin −
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6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − error function table 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc inverse error function ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1
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(x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π /
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2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 6}\operatorname 5 (-iz)=\operatorname 4 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.4 Inverse functions 2.5 Asymptotic expansion 2.6 Continued fraction expansion 2.7 Integral of error function with Gaussian
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 error function python at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Calculus and
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Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error erf(1) function, commonly denoted , is an entire function defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some authors (e.g., Whittaker and Watson 1990, p.341) https://en.wikipedia.org/wiki/Error_function 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 integrals (10) (11) (12) For , is bounded by (13) Min Max Re Im Erfc can also http://mathworld.wolfram.com/Erfc.html 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.; 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. Sp
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the http://math.stackexchange.com/questions/97/how-to-accurately-calculate-the-error-function-erfx-with-a-computer workings and policies of this site About Us Learn more about https://support.office.com/en-us/article/IFERROR-function-c526fd07-caeb-47b8-8bb6-63f3e417f611 Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join error function them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top How to accurately calculate the error function erf(x) with a computer? up vote 9 down vote favorite 2 I am looking for an accurate algorithm to calculate error function formula the error function I have tried using [this formula] (http://stackoverflow.com/a/457805) (Handbook of Mathematical Functions, formula 7.1.26), but the results are not accurate enough for the application. statistics algorithms numerical-methods special-functions share|cite|improve this question edited Jan 10 '14 at 4:47 pnuts 1056 asked Jul 20 '10 at 20:20 badp 6741225 You may want to take a look at python's code.google.com/p/mpmath or other libraries that advertise a "multiple precision" feature. Also, this may be a better question for stack overflow instead, since it's more of a computer science thing. –Jon Bringhurst Jul 20 '10 at 20:26 @Jon: Nope, I'm not interested in a library, there is no such library for the language I'm writing in (yet). I need the mathematical algorithm. –badp Jul 20 '10 at 20:49 Have you tried numerical integration? Gaussian Quadrature is an accurate technique –Digital Gal Aug 28 '10 at 1:25 GQ is nice, but with (a number of) efficient methods for computing $\mathrm{erf}$ already known, I don't see the point. &
Applies To: Excel 2016, Excel 2013, Excel 2010, Excel 2007, Excel 2016 for Mac, Excel for Mac 2011, Excel Online, Excel for iPad, Excel for iPhone, Excel for Android tablets, Excel Starter, Excel Mobile, Excel for Android phones, Less Applies To: Excel 2016 , Excel 2013 , Excel 2010 , Excel 2007 , Excel 2016 for Mac , Excel for Mac 2011 , Excel Online , Excel for iPad , Excel for iPhone , Excel for Android tablets , Excel Starter , Excel Mobile , Excel for Android phones , More... Which version do I have? More... This article describes the formula syntax and usage of the IFERROR function in Microsoft Excel. Description Returns a value you specify if a formula evaluates to an error; otherwise, returns the result of the formula. Use the IFERROR function to trap and handle errors in a formula. Syntax IFERROR(value, value_if_error) The IFERROR function syntax has the following arguments: Value Required. The argument that is checked for an error. Value_if_error Required. The value to return if the formula evaluates to an error. The following error types are evaluated: #N/A, #VALUE!, #REF!, #DIV/0!, #NUM!, #NAME?, or #NULL!. Remarks If Value or Value_if_error is an empty cell, IFERROR treats it as an empty string value (""). If Value is an array formula, IFERROR returns an array of results for each cell in the range specified in value. See the second example below. Examples Copy the example data in the following table, and paste it in cell A1 of a new Excel worksheet. For formulas to show results, select them, press F2, and then press Enter. If you need to, you can adjust the column widths to see all the data. Quota Units Sold 210 35 55 0 23 Formula Description Result =IFERROR(A2/B2, "Error in calculation") Checks for an error in the formula in the first argument (divide 210 by 35), finds no error, and then returns the resul