Error Function Expansion
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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 error function values updated: Tue Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at taylor expansion of error function WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History asymptotic expansion of error function and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian
Error Function Series Expansion
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 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) bessel function expansion 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, (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 comp
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Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's http://mathworld.wolfram.com/Erf.html how it works: Anybody can ask a question Anybody can answer The best answers 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 http://math.stackexchange.com/questions/125328/taylor-expansion-of-error-function $\mathrm{erf}(x):=\frac {2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}dt$. Find its Taylor expansion. I know that the Taylor series of the function $f$ at $a$ is 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+\f
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