Asymptotic Expansion Of The Complementary Error Function
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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, error function integral complementary error function, error functions Referenced by: §2.11(iv), §7.12(ii), Figure 7.3.6,
Error Function Calculator
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)
Error Function Table
∼2-e-z2π∑m=0∞(-1)m(12)mz2m+1, 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
Erf(inf)
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, 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 error function matlab 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 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
where it was expressed through the following integral: Later C. Kramp (1799) used this integral for the definition of the complementary error function . P.‐S. Laplace (1812) inverse error function derived an asymptotic expansion of the error function. The probability integrals were so erfc named because they are widely applied in the theory of probability, in both normal and limit distributions. To error function excel obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely is needed. The inverse was systematically investigated in the second half http://dlmf.nist.gov/7.12 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) , 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 http://functions.wolfram.com/GammaBetaErf/Erfi/introductions/ProbabilityIntegrals/ShowAll.html 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 functions and : Representations of the probability integrals , , , and through classical Meijer G functions are rather simple: The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one: The probability integrals , , , and are the particular cases of the incomplete gamma function, regularized incomplete gamma function, and exponential integral : Representations through related equivalent funbe down. Please try the request again. Your cache administrator is webmaster. Generated Sat, 01 Oct 2016 06:29:45 GMT by s_hv997 (squid/3.5.20)
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