Error Function Asymptotic Approximation
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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 derivative of error function Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and erf function calculator Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> error function table 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
Inverse Error Function
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 as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, error function matlab 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) 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 d
Permalink: http://dlmf.nist.gov/7.12 See also: info for 7 Contents §7.12(i) Complementary Error Function §7.12(ii) Fresnel Integrals §7.12(iii) Goodwin–Staton Integral §7.12(i) Complementary Error Function Keywords: Stokes phenomenon, complementary error error function excel function, error functions Referenced by: §2.11(iv), §7.12(ii), Figure 7.3.6, Figure 7.3.6, 7.3.6 Permalink:
Error Function Python
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, Symbols: (a)n: Pochhammer’s
Complementary Error Function Table
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) http://mathworld.wolfram.com/Erf.html Referenced by: §3.5(ix), Other Changes Permalink: http://dlmf.nist.gov/7.12.E1 Encodings: TeX, 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 http://dlmf.nist.gov/7.12 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 leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv) and use (7.11.3). (Note that some of these re-expansions themselves involve the complementary error function.) §7.12(ii) Fresnel Integrals Notes: (7.12.2) and (7.12.3) follow from (7.7.10) and (7.7.11) by applying Watson’s lemma in its extended form (§2.4(i)). (7.12.4)–(7.12.7) follow from (7.7.10), (7.7.11), and the identity (t2+1)-1=∑m=0n-1(-1)mt2m+(-1)nt2n(t2+1)-1. The error bounds are obtained by setting t=τ in (7.12.6) and (7.12.7), rotating the integration path in the τ-plane through an angle -4phz, and then replac
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