Asymptotic Error Function
Contents |
that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . asymptotic expansion of complementary error function {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm asymptotic function excel 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc,
Asymptotic Error Constant Newton’s Method
is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle
Error Function Values
{\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}} 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 complementary error function ) = 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 arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 4 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx ( − i z ) . {\displaystyle w(z)=e^{-z^ 2}\operatorname 1 (-iz)=\operatorname 0 (-iz).} Contents 1 The name "error function" 2 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann series 2.
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,
Error Function Calculator
and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire error function table Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" inverse error function encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire function defined by (1) Note that some authors (e.g., Whittaker https://en.wikipedia.org/wiki/Error_function 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, and is a confluent hypergeometric function of the first kind. For , (5) where is http://mathworld.wolfram.com/Erf.html 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 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),
7 September 2010 by John This post will present a couple asymptotic series, explain how to use them, and point to some applications.The most common special function in applications, at least in my experience, is the gamma function Γ(z). It's http://www.johndcook.com/blog/2010/09/07/two-useful-asymptotic-series/ often easier to work with the logarithm of the gamma function than to work with the gamma function directly, so one of the asymptotic series we'll look at is the series for log Γ(z).Another very common special function in applications is the error function erf(z). The error function and its relationship to the normal probability distribution are explained error function here. Even though the gamma function is more common, we'll start with the asymptotic series for the error function because it is a little simpler.Error functionWe actually present the series for the complementary error function erfc(z) = 1 - erf(z). (Why have a separate name for erfc when it's so closely related to erf? Sometimes erfc is easier complementary error function to work with mathematically. Also, sometimes numerical difficulties require separate software for evaluating erf and erfc as explained here.)If you're unfamiliar with the n!! notation, see this explanation of double factorial.Note that the series has a squiggle ~ instead of an equal sign. That is because the partial sums of the right side do not converge to the left side. In fact, the partial sums diverge for any z. Instead, if you take any fixed partial sum you obtain an approximation of the left side that improves as z increases.The series above is valid for any complex value of z as long as |arg(z)| < 3π/4. However, the error term is easier to describe if z is real. In that case, when you truncate the infinite sum at some point, the error is less than the first term that was left out. In fact, the error also has the same sign as the first term left out. So, for example, if you drop the sum entirely and just keep the "1" term o
be down. Please try the request again. Your cache administrator is webmaster. Generated Sat, 01 Oct 2016 13:21:49 GMT by s_hv720 (squid/3.5.20)