Approximation Of 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 x e − t
Approximation Q Function
2 d t = 2 π ∫ 0 x e − t approximation gamma function 2 d t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm
Approximation Normal Distribution
7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc, is defined as erfc ( simple approximations of the error function x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} which error function table 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 Φ 8 (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 Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\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 − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite
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Inverse Error Function Approximation
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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 voted up and rise https://en.wikipedia.org/wiki/Error_function to the top Approximating the error function erf by analytical functions up vote 11 down vote favorite 2 The Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function $f$ which also starts linearly, has $f(0)=0$ and converges against the constant value 1 fast, namely $\tanh{(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}}$. Astoningishly to me, I found that http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions they never differ by more than $|\Delta f|=0.0812$ and converge against each other exponentially fast! I consider $\tanh{(x)}$ to be the somewhat prettyier function, and so I wanted to find an approximation to $\text{erf}$ with "nice functions" by a short expression. I "naturally" tried $f(x)=A\cdot\tanh(k\cdot x^a-d)$ Changing $A=1$ or $d=0$ on it's own makes the approximation go bad and the exponent $a$ is a bit difficult to deal with. However, I found that for $k=\sqrt{\pi}\log{(2)}$ the situation gets "better". I obtained that $k$ value by the requirement that "norm" given by $\int_0^\infty\text{erf}(x)-f(x)dx,$ i.e. the difference of the functions areas, should valish. With this value, the maximal value difference even falls under $|\Delta f| = 0.03$. And however you choose the integration bounds for an interval, the area difference is no more than $0.017$. Numerically speaking and relative to a unit scale, the functions $\text{erf}$ and $\tanh{(\sqrt{\pi}\log{(2)}x)}$ are essentially the same. My question is if I can find, or if there are known, substitutions for this non-elementary function in terms of elementary ones. In the sense above, i.e. the approximation is compact/rememberable while the values are even better, from a numerical point of view. The purpose being for example, that if I see somewhere that for a computation I have to integrate erf, th
institution loginHelpJournalsBooksRegisterJournalsBooksRegisterSign inHelpcloseSign in using your ScienceDirect credentialsUsernamePasswordRemember meForgotten username or password?Sign in via your institutionOpenAthens loginOther institution login Purchase Loading... Export You have selected 1 citation http://www.sciencedirect.com/science/article/pii/0098135480800159 for export. Help Direct export Save to Mendeley Save to RefWorks http://mathworld.wolfram.com/Erf.html Export file Format RIS (for EndNote, ReferenceManager, ProCite) BibTeX Text Content Citation Only Citation and Abstract Export Advanced search Close This document does not have an outline. JavaScript is disabled on your browser. Please enable JavaScript to use all the features on this page. error function Computers & Chemical Engineering Volume 4, Issue 2, 1980, Pages 67-68 A simple approximation of the error function Author links open the overlay panel. Numbers correspond to the affiliation list which can be exposed by using the show more link. Opens overlay H.T. Karlsson ∗, Opens overlay I. Bjerle Division of Chemical Technology, Department error function approximation of Chemical Engineering, Chemical Center, Lund Institute of Technology, P.O.B. 740, S-220 07 Lund 7, Sweden Received 20 February 1979, Available online 30 July 2001 Show more Choose an option to locate/access this article: Check if you have access through your login credentials or your institution. Check access Purchase Sign in using your ScienceDirect credentials Username: Password: Remember me Not Registered? Forgotten username or password? OpenAthens login Login via your institution Other institution login doi:10.1016/0098-1354(80)80015-9 Get rights and content AbstractA very simple approximation formula of the error function, with sufficient accuracy for engineering calculations, is proposed in this investigation. The presented form is compared with some of the less sophisticated approximations available in the literature. Aspects such as mnemonic form, computation time, accuracy and ease of inversion are considered. open in overlay Author to whom correspondence should be directed. Copyright © 1980 Published by Elsevier Ltd. ElsevierAbout ScienceDirectRemote accessShopping cartContact and supportTerms and conditionsPrivacy policyCookies are used by this site. For mor
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 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 (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 , (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 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 c