Rational Chebyshev Approximations For The Inverse Of The Error Function
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Complementary 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
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e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin − 1\operatorname − 0 (x)&={\frac 9{\sqrt {\pi }}}\int _{-x}^ 8e^{-t^ 7}\,\mathrm 6 t\\&={\frac 5{\sqrt {\pi }}}\int _ 4^ 3e^{-t^ 2}\,\mathrm 1 t.\end 0}} In statistics, for nonnegative values of x, http://www.ams.org/mcom/1976-30-136/S0025-5718-1976-0421040-7/ the error function has the following interpretation: for a random variable X that is normally distributed with mean 0 and variance 1 2 {\textstyle {\frac − 7 − 6}} , erf(x) describes the probability of X falling in the range [-x, x]. Contents 1 The name 'error function' 2 Derived and related functions 2.1 Complementary error function 2.2 Imaginary error function https://en.wikipedia.org/wiki/Error_function 2.3 Cumulative distribution function 3 Properties 3.1 Taylor series 3.2 Derivative and integral 3.3 Bürmann series 3.4 Inverse functions 3.5 Asymptotic expansion 3.6 Continued fraction expansion 3.7 Integral of error function with Gaussian density function 4 Approximation with elementary functions 5 Numerical approximations 6 Applications 7 Related functions 7.1 Generalized error functions 7.2 Iterated integrals of the complementary error function 8 Implementations 9 See also 9.1 Related functions 9.2 In probability 10 References 11 Further reading 12 External links The name 'error function'[edit] The error function is used in measurement theory (using probability and statistics), and its use in other branches of mathematics is typically unrelated to the characterization of measurement errors. In statistics, it is common to have a variable Y {\displaystyle Y} and its unbiased estimator Y ^ {\displaystyle {\hat − 3}} . The error is then defined as ε = Y ^ − Y {\displaystyle \varepsilon ={\hat − 1}-Y} . This makes the error a normally distributed random variable with mean 0 (because the estimator is unbiased) and some variance σ 2 {\displaystyle \sigma ^ − 9} ; t
20036983, 2280356863, 49020204823, 65967241200001, 15773461423793767, 655889589032992201, 94020690191035873697, 655782249799531714375489, 44737200694996264619809969 (list; graph; refs; listen; history; text; internal format) OFFSET https://oeis.org/A092676 1,3 COMMENTS Differs from A002067(n) at n = 6, 9, 12, .... Following Blair et al., we use the notation inverf() for the inverse of the error function. LINKS Table of n, a(n) for n=1..15. G. Alkauskas, Algebraic and abelian solutions to the projective translation equation, arXiv preprint arXiv:1506.08028, 2015 J. M. Blair, C. A. error function Edwards and J. H. Johnson, Rational Chebyshev approximations for the inverse of the error function, Math. Comp. 30 (1976), 827-830. L. Carlitz, The inverse of the error function, Pacific J. Math., 13 (1963), 459-470. Eric Weisstein, Mathematica program and first 50 terms of the series Eric Weisstein's World of Mathematics, Inverse Erf Wikipedia, Error inverse error function Function EXAMPLE Inverf(2x/sqrt(Pi)) = x + x^3/3 + 7x^5/30 + 127x^7/630 + 4369x^9/22680 + 34807x^11/178200 + ... The first few coefficients are 1, 1, 7/6, 127/90, 4369/2520, 34807/16200, 20036983/7484400, 2280356863/681080400, ... MAPLE c:=proc(n) option remember; if n <= 0 then 1 else add( c(k)*c(n-k-1)/((k+1)*(2*k+1)), k=0..n-1 ) fi; end; CROSSREFS Cf. A002067, A092677, A052712. For denominators see A132467. Sequence in context: A139291 A274673 A215066 * A002067 A274571 A138523 Adjacent sequences:A092673 A092674 A092675 * A092677 A092678 A092679 KEYWORD nonn,frac AUTHOR Eric W. Weisstein, Mar 02 2004 EXTENSIONS Edited by N. J. A. Sloane, Nov 15 2007 STATUS approved Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages The OEIS Community | Maintained by The OEIS Foundation Inc. License Agreements, Terms of Use, Privacy Policy . Last modified October 25 18:19 EDT 2016. Contains 277129 sequences.
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