Asymptotic Expansion Inverse Error Function
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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 inverse complementary error function Expansion of inverfcx for Small x §7.17(i) Notation Keywords: error
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functions Permalink: http://dlmf.nist.gov/7.17.i See also: info for 7.17 The inverses of the functions x=erfy, x=erfcy, asymptotic expansion of bessel function y∈ℝ, are denoted by 7.17.1 y =inverfx, y =inverfcx, Defines: inverfcx: inverse complementary error function and inverfx: inverse error function Symbols: x: real variable error function integral Permalink: http://dlmf.nist.gov/7.17.E1 Encodings: TeX, TeX, pMML, 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,
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pMML, png See also: info for 7.17(ii) For 25S 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+ln
(1774) where it was expressed through the following integral: Later C. Kramp (1799) used this integral for erf(inf) the definition of the complementary error function . P.‐S. Laplace error function table (1812) derived an asymptotic expansion of the error function. The probability integrals were so named
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because they are widely applied in the theory of probability, in both normal and limit distributions. To obtain, say, a normal distributed random variable from http://dlmf.nist.gov/7.17 a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in the second half of the twentieth century, especially by J. R. Philip (1960) and A. J. Strecok (1968).
Definitions of probability integrals and inverses The probability integral (error function) http://functions.wolfram.com/GammaBetaErf/InverseErfc/introductions/ProbabilityIntegrals/ShowAll.html , the generalized error function , the complementary error function , the imaginary error function , the inverse error function , the inverse of the generalized error function , and the inverse complementary error function are defined through the following formulas: These seven functions are typically called probability integrals and their inverses. Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series. A quick look at the probability integrals and inversesHere is a quick look at the graphics for the probability integrals and inverses along the real axis. Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals , , , and are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric funRandom 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 http://mathworld.wolfram.com/Erf.html 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 error function 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 asymptotic expansion of (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 complex
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