Error Function Series Approximation
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Taylor Series Approximation Error
Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is taylor series approximation error bound 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 the leading factor of . Erf gamma function approximation 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 as a Maclaurin series (6) (7) (OEIS A007680). Similarly,
Approximation Q Function
(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 rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). Definite integrals involving include Definite integrals involving include (34) (35) (36)
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Error Function Values
people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers http://mathworld.wolfram.com/Erf.html are voted up and rise to the top Taylor Expansion of Error Function up vote 4 down vote favorite 3 I am working on a question that involves finding the Taylor expansion of the error function. The question is stated as follows The error function is defined by $\mathrm{erf}(x):=\frac {2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}dt$. Find its Taylor expansion. I know that the Taylor series of the function $f$ at $a$ is http://math.stackexchange.com/questions/125328/taylor-expansion-of-error-function given by $$f(x)=\sum_{n=0}^{\infty}\frac {f^{(n)}(a)}{n!}(x-a)^{n}.$$ However, the question doesn't give a point $a$ with which to center the Taylor series. How should I interpret this? May I use a Maclaurin series, with $a=0$? This appears to be what was done on the Wikipedia page here: http://en.wikipedia.org/wiki/Error_function Any explanations and advice would be appreciated. calculus special-functions taylor-expansion share|cite|improve this question edited Apr 28 '12 at 13:06 J. M. 52.8k5118254 asked Mar 28 '12 at 5:08 fitzgeraldo 14127 6 $a=0$ seems OK for me. I would expand $e^{-t^2}$ in a power series and integrate term by term. –marty cohen Mar 28 '12 at 5:38 add a comment| 1 Answer 1 active oldest votes up vote 2 down vote Elaborating a little on Marty's comment gives the following: $f^{(n)}(a)$ can be written in terms of Hermite polynomials $H_n$: $$ H_0(x)=1,\, H_1(x)=2x,\, H_2(x)=4x^2-2,\, H_3(x)=8x^3-12x,\, H_4(x)=16x^4-48x^2+12,\, H_5(x)=32x^5-160x^3+120x,\, H_6(x)=64x^6-480x^4+720x^2-120,\dots\, $$ You recognize that $H_{2n-1}(0)=0$, which gives the power series for $e^{-x^2}$ at $a=0$: $$ e^{-x^2} = 1 - \frac{2}{2!}x^2+\frac{12}{4!}x^4-\frac{120}{6!}x^6+\cdots $$ (see here). After multiplying by $2/\sqrt{\pi}$, this integrates to $$ \operatorname{erf}(z) =\frac{2}{\sqrt{\pi}} \left(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\ \cdots\right) . $$ EDIT: Since $\displaystyle \frac{d^n}{dx^n}e^{-x^2}= (-1)^n e^{-x^2} H_n(x), $ one can do a Taylor Series for every $a$: $$ \text{erf}_a(x)= e^{-a^2} \sum_{n=0}^\infty (-1)^n \frac{H_n(a)}{n!}(x-a)^n $$ s
series methods allow to write down this indefinite integral: Since its indefinite integral, called the error function, has the Taylor expansion Both series http://www.sosmath.com/calculus/tayser/tayser06/tayser06.html have as their radius of convergence. In the following picture, is depicted in blue, while its integral is shown in red. The constant of integration has been chosen so that the red graph reflects the area under the blue graph. The simple pendulum The angular motion of a typical undamped simple pendulum can be described by the differential equation Here x(t) denotes error function the angle (in radian measure) at time t between the pendulum and the resting position of the pendulum. Let's suppose the pendulum starts at time t=0 in its resting position ,i.e., x(0)=0, with a certain initial angular velocity, say x'(0)=A. We want to describe the angular motion x(t) of the pendulum over time. Suppose also that x(t) has a Taylor series taylor series approximation with center : Plugging in t=0 yields the information that . Let's take the first derivative: Plugging in t=0 yields the information that . Let's take the second derivative: Now the differential equation tells us that . Plugging in t=0 yields the information that . One more time. Let's take the third derivative: Now the differential equation tells us that by the chain rule. Plugging in t=0 yields the information that , so . Thus the Taylor series for the pendulum motion starts out with Actually computing a few more terms yields Here are the graphs of the pendulum movements for various values of the initial velocity A. Try it yourself! For each function in the picture, move your arm to show the way the pendulum is swinging! What is the significance of the two thin red lines? Helmut Knaust Tue Jul 16 16:53:21 MDT 1996 This module consists of 6 HTML pages. Copyright © 1999-2016 MathMedics, LLC. All rights reserved. Contact us Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA users online during the last hour