Inverse Error Function Asymptotic Expansion
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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 complementary error function 7 Contents §7.17(i) Notation §7.17(ii) Power Series §7.17(iii) Asymptotic error function calculator Expansion of inverfcx for Small x §7.17(i) Notation Keywords: error functions Permalink: error function table http://dlmf.nist.gov/7.17.i See also: info for 7.17 The inverses of the functions x=erfy, x=erfcy, y∈ℝ, are denoted by 7.17.1 y =inverfx, y
Erf(inf)
=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, pMML, png, png See also: info for 7.17(i) respectively. §7.17(ii) Power Series Notes: See Carlitz (1963). Keywords: error functions error function matlab 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 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: expans
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 http://mathworld.wolfram.com/Erf.html Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the http://arxiv.org/abs/math/0607230 "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 error function 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) inverse error function 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 generalization of is defined as (39) (40) Integral representations valid only in the upper half-plane are given by (41) (42) SEE ALSO: Dawson's Inte
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