Asymptotic Expansion 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 Calculator
complementary error function, error functions Referenced by: §2.11(iv), §7.12(ii), Figure 7.3.6, error function table 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, inverse error function 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
Erf(inf)
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 Karagounis
Error Function Matlab
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 (7.12.6) and
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