Asymptotic Expansion Of Gauss Error Function
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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 Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at derivative of error function WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica erf function calculator Examples> More... History and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the
Error Function Table
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 and Watson 1990, p.341) define without
Inverse Error Function
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 the incomplete gamma function. Erf can also be defined erf(inf) 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), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and red
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 often easier to work with the logarithm of the gamma function than
Error Function Matlab
to work with the gamma function directly, so one of the asymptotic series erf(1) 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 excel error function and its relationship to the normal probability distribution are explained here. Even though the gamma function is more common, we'll start with the asymptotic series for the error function because it is a http://mathworld.wolfram.com/Erf.html 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 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 http://www.johndcook.com/blog/2010/09/07/two-useful-asymptotic-series/ 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 on the right side, the error is negative and the absolute value of the error is less than 1/2 z2.One way this series is used in practice is to bound the tails of the normal distribution function. A slight more involved application can be found here.Log gammaThe next series is the asymptotic series for log Γ(z).If you throw out all the terms involving powers of 1/z you get Stirling's app
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