Asymptotic Expression Error Function
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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 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex asymptotic expansion of exponential function Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... asymptotic expansion of bessel function Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It asymptotic expansion of modified bessel function 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
Asymptotic Approach
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 error function values 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 generalization of is defined as (39) (40) Integ
Permalink: http://dlmf.nist.gov/7.12 See also: info for 7 Contents §7.12(i) Complementary Error Function §7.12(ii) derivative of error function Fresnel Integrals §7.12(iii) Goodwin–Staton Integral §7.12(i) Complementary
Erf Function Calculator
Error Function Keywords: Stokes phenomenon, complementary error function, error functions Referenced by:
Error Function Table
§2.11(iv), §7.12(ii), Figure 7.3.6, Figure 7.3.6, 7.3.6 Permalink: http://dlmf.nist.gov/7.12.i See also: info for 7.12 As z→∞ 7.12.1 erfcz ∼e-z2π∑m=0∞(-1)m(12)mz2m+1, erfc(-z) ∼2-e-z2π∑m=0∞(-1)m(12)mz2m+1, http://mathworld.wolfram.com/Erf.html Symbols: (a)n: Pochhammer’s symbol (or shifted factorial), ∼: Poincaré asymptotic expansion, erfcz: complementary error function, e: base of exponential function and z: complex variable A&S Ref: 7.1.23 (in different form) Referenced by: §3.5(ix), Other Changes Permalink: http://dlmf.nist.gov/7.12.E1 Encodings: TeX, http://dlmf.nist.gov/7.12 TeX, pMML, pMML, png, png Notational Change (effective with 1.0.9): Previously the RHS of these equations were written as e-z2πz∑m=0∞(-1)m1⋅3⋅5⋯(2m-1)(2z2)m and 2-e-z2πz∑m=0∞(-1)m1⋅3⋅5⋯(2m-1)(2z2)m. We have rewritten these sums more concisely using Pochhammer’s symbol. Reported 2014-03-13 by Giorgos Karagounis See also: info for 7.12(i) both expansions being valid when |phz|≤34π-δ (<34π). When |phz|≤14π the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when phz=0. When 14π≤|phz|<12π the remainder terms are bounded in magnitude by csc(2|phz|) times the first neglected terms. For these and other error bounds see Olver (1997b, pp. 109–112), with α=12 and z replaced by z2; compare (7.11.2). For re-expansions of the remainder terms leadin
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