Inverse Error Function Taylor Series
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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» inverse error function excel 13,594 entries Last updated: Tue Sep 27 2016 Created, inverse error function calculator developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Calculus>Integrals>Definite Integrals> History and erf(2) Terminology>Wolfram Language Commands> Inverse Erf The inverse erf function is the inverse function of the erf function such that (1) (2) with the first
Inverse Erf
identity holding for and the second for . It is implemented in the Wolfram Language as InverseErf[x]. It is an odd function since (3) It has the special values (4) (5) (6) It is apparently not known if (7) (OEIS A069286) can be written in closed form. It satisfies erf function calculator the equation (8) where is the inverse erfc function. It has the derivative (9) and its integral is (10) (which follows from the method of Parker 1955). Definite integrals are given by (11) (12) (13) (14) (OEIS A087197 and A114864), where is the Euler-Mascheroni constant and is the natural logarithm of 2. The Maclaurin series of is given by (15) (OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1, (16) (OEIS A092676 and A092677). The th coefficient of this series can be computed as (17) where is given by the recurrence equation (18) with initial condition . SEE ALSO: Confidence Interval, Erf, Inverse Erfc, Probable Error RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/InverseErf/, http://functions.wolfram.com/GammaBetaErf/InverseErf2/ REFERENCES: Bergeron, F.; Labelle, G.; and Leroux, P. Ch.5 in Combinatorial Species and Tree-Like Structures. Cambridge, England: Cambridge University Press, 1998. Carlitz, L. "The Inverse of the Error
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here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and http://math.stackexchange.com/questions/1313831/inverse-complementary-error-function-values-near-0 policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ 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 error function a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top inverse complementary error function values near 0 up vote 2 down vote favorite $\newcommand{\erf}{\operatorname{erf}}\newcommand{\erfc}{\operatorname{erfc}}$Let's define for each $x>0$ $$\erf(x)=\frac {2}{\sqrt{\pi}}\int_0^xe^{-t^2} dt$$ and $$\erfc(x)=\frac {2}{\sqrt{\pi}}\int_x^\infty e^{-t^2} dt=1-\erf(x)$$ I want to approximate the inverse functions inverse error function $\erf^{-1}$ and $\erfc^{-1}$ for very small values of $x$. I know that $\erf\sim f(x)=\frac{2x}{\sqrt{\pi}}$ now since the inverse function of $f$ is close to $0$ when $x$ is close to $0$ I can use that as an approximation for $\erf^{-1}$, in fact $\erf(f(x))\sim x$ when $x<<1$. Now this doesn't work with erfc since the inverse function of $1-f(x)$ is not small for small values of $x$ Though I know that $\erfc(x) \sim g(x)=\frac{1}{\pi}\frac{e^{-x^2}}x$ for $x\gg1$. Since $g^{-1}$ is an increasing function is should be true that $g^{-1}(x)\gg1$ when $x\gg1$ but is there any simple way to compute the inverse of this function? Thank you! real-analysis approximation error-function gaussian-integral share|cite|improve this question edited Jun 6 '15 at 4:06 Michael Hardy 158k15145350 asked Jun 5 '15 at 22:04 giulio 344110 add a comment| 2 Answers 2 active oldest votes up vote 1 down vote accepted This is only a partial answer. For the inverse error function, for small arguments, Taylor series seem to be quite good $$\text{erf}^{-1}(x)=\frac{\sqrt{\pi } }{2}x\Big(1+\frac{\pi }{12}x^2+\frac{7 \pi ^2 }{480}x^4+\frac{127 \pi ^3 }{40320}x^6+O\left(x^8\right)\Big)$$ Pade approximants $$\text{erf}^{-1