Inverse Error Function Expansion
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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 erf(2) 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special inverse error function calculator Functions>Erf> Calculus and Analysis>Calculus>Integrals>Definite Integrals> History and Terminology>Wolfram Language Commands> Inverse Erf The inverse erf function is
Inverse Error Function Excel
the inverse function of the erf function such that (1) (2) with the first identity holding for and the second for . It is implemented in the
Inverse Erf
Wolfram Language as InverseErf[x]. It is an odd function since (3) It has the special values (4) (5) (6) It is apparently not known if (7) (OEIS A069286) can be written in closed form. It satisfies the equation (8) where is the inverse erfc function. It has the derivative (9) and its integral is (10) erf function calculator (which follows from the method of Parker 1955). Definite integrals are given by (11) (12) (13) (14) (OEIS A087197 and A114864), where is the Euler-Mascheroni constant and is the natural logarithm of 2. The Maclaurin series of is given by (15) (OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1, (16) (OEIS A092676 and A092677). The th coefficient of this series can be computed as (17) where is given by the recurrence equation (18) with initial condition . SEE ALSO: Confidence Interval, Erf, Inverse Erfc, Probable Error RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/InverseErf/, http://functions.wolfram.com/GammaBetaErf/InverseErf2/ REFERENCES: Bergeron, F.; Labelle, G.; and Leroux, P. Ch.5 in Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998. Carlitz, L. "The Inverse of the Error Function." Pacific J. Math. 13, 459-470, 1963. Parker, F.D. "Integrals of Inverse Functions." Amer. Math. Monthly 62, 439-440, 1955. Sloane, N.J.A. Sequences A002067/M4458, A007019/M3126, A069286, A087197, A092676, A092677, A114859, A114860, and A114864 in "The On-Line Encyclopedia of Inte
Error Function §7.17 Inverse Error Functions Referenced by: §8.12 Permalink: http://dlmf.nist.gov/7.17 See also: info for 7 Contents §7.17(i) Notation §7.17(ii) Power Series §7.17(iii) Asymptotic Expansion of inverfcx
Erf(inf)
for Small x §7.17(i) Notation Keywords: error functions Permalink: http://dlmf.nist.gov/7.17.i error function table See also: info for 7.17 The inverses of the functions x=erfy, x=erfcy, y∈ℝ, are denoted by erf function excel 7.17.1 y =inverfx, y =inverfcx, Defines: inverfcx: inverse complementary error function and inverfx: inverse error function Symbols: x: real variable Permalink: http://dlmf.nist.gov/7.17.E1 Encodings: TeX, TeX, pMML, http://mathworld.wolfram.com/InverseErf.html pMML, png, png See also: info for 7.17(i) respectively. §7.17(ii) Power Series Notes: See Carlitz (1963). Keywords: error functions Permalink: http://dlmf.nist.gov/7.17.ii See also: info for 7.17 With t=12πx, 7.17.2 inverfx=t+13t3+730t5+127630t7+⋯, |x|<1. Symbols: inverfx: inverse error function and x: real variable Permalink: http://dlmf.nist.gov/7.17.E2 Encodings: TeX, pMML, png See also: info for 7.17(ii) For 25S http://dlmf.nist.gov/7.17 values of the first 200 coefficients see Strecok (1968). §7.17(iii) Asymptotic Expansion of inverfcx for Small x Notes: (7.17.3) follows from Blair et al. (1976), after modifications. Keywords: error functions Permalink: http://dlmf.nist.gov/7.17.iii See also: info for 7.17 As x→0 7.17.3 inverfcx∼u-1/2+a2u3/2+a3u5/2+a4u7/2+⋯, Symbols: ∼: Poincaré asymptotic expansion, inverfcx: inverse complementary error function, x: real variable, ai: coefficients and u: expansion variable Referenced by: §7.17(iii) Permalink: http://dlmf.nist.gov/7.17.E3 Encodings: TeX, pMML, png See also: info for 7.17(iii) where 7.17.4 a2 =18v, a3 =-132(v2+6v-6), a4 =1384(4v3+27v2+108v-300), Defines: ai: coefficients (locally) Symbols: v: expansion variable Permalink: http://dlmf.nist.gov/7.17.E4 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: info for 7.17(iii) 7.17.5 u=-2/ln(πx2ln(1/x)), Defines: u: expansion variable (locally) Symbols: lnz: principal branch of logarithm function and x: real variable Permalink: http://dlmf.nist.gov/7.17.E5 Encodings: TeX, pMML, png See also: info for 7.17(iii) and 7.17.6 v=ln(ln(1/x))-2+lnπ. Defines: v: expansion variable (locally) Symbols: lnz: principal branch of logarithm function and x: real variable Permalink: http://dlmf.nist.gov/7.1
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