Asymptotic Series Of Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − asymptotic expansion of exponential function t 2 d t = 2 π ∫ 0 x e − t
Asymptotic Expansion Of Bessel Function
2 d t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ asymptotic function excel 8}\,\mathrm 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc, is defined as erfc
Taylor Series Of Error Function
( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} erfc asymptotic expansion which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname Φ 8 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid ari
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 function, error functions
Asymptotic Expression
Referenced by: §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:
Error Function Values
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 symbol (or shifted factorial), complementary error function ∼: 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 https://en.wikipedia.org/wiki/Error_function 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 being valid when |phz|≤34π-δ (<34π). When |phz|≤14π the remainder http://dlmf.nist.gov/7.12 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 replacing |τ+1| by its minimum value on the path. Keywords: Fresnel integrals, auxiliary functions for Fresnel integrals Perm
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