Numerical Approximation Error Function
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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 error function integral x e − t 2 d t = 2 π ∫ 0 error function calculator x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi error function table }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted error function matlab erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm
Inverse Error Function
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 / 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^
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How To Calculate Error Function In Casio Calculator
Stack Overflow the company Business Learn more about hiring developers or posting ads with complementary error function table us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people error function excel studying math at any level and professionals in related fields. Join 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 https://en.wikipedia.org/wiki/Error_function voted up and rise to the top efficient and accurate approximation of error function up vote 2 down vote favorite I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$ reference-request special-functions approximation share|cite|improve this question edited Aug 27 '14 at 11:38 Jean-Claude Arbaut 11.7k52354 asked Jun 3 '11 at 2:32 shaikh 493619 Wiki http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function suggests an approximation en.wikipedia.org/wiki/… –user17762 Jun 3 '11 at 2:37 possible duplicate of Definite integral of Normal Distribution –user17762 Jun 3 '11 at 2:43 1 Related: stats.stackexchange.com/questions/7200/… –cardinal Jun 3 '11 at 8:30 possible duplicate of How to accurately calculate erf(x) with a computer? –J. M. Jul 23 '11 at 15:26 You will find implementations in most scientific libraries: cmlib, slatec, nswc, nag, imsl, harwell hsl... Also in gnu gsl, in R, probably octave and Scilab... You can also have a look at ACM TOMS Collected Algorithms. There are plenty of places to look for this. –Jean-Claude Arbaut Aug 27 '14 at 11:40 add a comment| 4 Answers 4 active oldest votes up vote 2 down vote accepted "Efficient and accurate" is probably contradictory... Have you tried the one listed in http://en.wikipedia.org/wiki/Error_function#Approximation_with_elementary_functions ? share|cite|improve this answer answered Jun 3 '11 at 2:39 lhf 106k5120271 yes, I have tried this. Its accuracy is up to 2 decimal places. Do we have more than this? –shaikh Jun 3 '11 at 2:40 @shaikh, C99 has an erf function, which should be quite accurate. –lhf Jun 3 '11 at 2:42 where can I find the derivation of
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: Wed Oct 19 2016 Created, http://mathworld.wolfram.com/Erf.html developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf 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 (1) Note that some authors error function (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 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 , error function table (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 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 conti