Handy Approximation Error Function Its Inverse
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 error function integral . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^
Error Function Calculator
− 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error error function table function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( error function matlab x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} 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 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc (
Inverse Error Function
x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\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 Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} 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 8 (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^ 6}\operatorname 5 (-iz)=\operatorname 4 (-iz).} Contents 1 The name "error function" 2 Prop
the article in the profile.DoneDuplicate citationsThe following articles are merged in complementary error function table Scholar. Their combined citations are counted only error function excel for the first article.DoneMerge duplicatesCitations per yearScholarFollowEmailFollow new articlesFollow new citationsCreate alertCancelSergei
Inverse Error Function Excel
WinitzkiUnknown affiliationPhysics, Cosmology, Mathematics, Computer ScienceVerified email at cosmos.phy.tufts.edu - HomepageScholarGet my own profileGoogle ScholarCitation indicesAllSince 2011Citations1627783h-index2114i10-index2919200820092010201120122013201420152016137158138175131107125126119Co-authorsView all…László https://en.wikipedia.org/wiki/Error_function Árpád GergelyTitle1–20Cited byYearIntroduction to quantum effects in gravityV Mukhanov, S WinitzkiCambridge University Press, 20073132007Probabilities in the inflationary multiverseJ Garriga, D Schwartz-Perlov, A Vilenkin, S WinitzkiJournal of Cosmology and Astroparticle Physics 2006 (01), 017, 20061722006Predictability crisis in inflationary cosmology https://scholar.google.com/citations?user=Q9U40gUAAAAJ and its resolutionV Vanchurin, A Vilenkin, S WinitzkiPhysical Review D 61 (8), 083507, 20001022000Minkowski functional description of microwave background GaussianityS Winitzki, A KosowskyNew Astronomy 3 (2), 75-99, 1998931998Predictions in eternal inflationS WinitzkiInflationary Cosmology, 157-191, 2008692008Uniform approximations for transcendental functionsS WinitzkiInternational Conference on Computational Science and Its Applications, 780-789, 2003642003Signatures of kinetic and magnetic helicity in the cosmic microwave background radiationL Pogosian, T Vachaspati, S WinitzkiPhysical Review D 65 (8), 083502, 2002612002A handy approximation for the error function and its inverseS WinitzkiA lecture note obtained through private communication, 2008602008Attractor scenarios and superluminal signals in k-essence cosmologyJU Kang, V Vanchurin, S WinitzkiPhysical Review D 76 (8), 083511, 2007592007Probability distribution for Ω in open-universe infla
here for a quick overview of the site Help Center Detailed answers to any questions you might http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions have Meta Discuss the workings and 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 error function 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 are voted up and rise to the top Approximating the error function erf by analytical functions up vote 12 down vote favorite 2 The error function table Error function $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt$ shows up in many contexts, but can't be represented using elementary functions. I compared it with another function $f$ which also starts linearly, has $f(0)=0$ and converges against the constant value 1 fast, namely $\tanh{(x)} = \frac {e^x - e^{-x}} {e^x + e^{-x}}$. Astoningishly to me, I found that they never differ by more than $|\Delta f|=0.0812$ and converge against each other exponentially fast! I consider $\tanh{(x)}$ to be the somewhat prettyier function, and so I wanted to find an approximation to $\text{erf}$ with "nice functions" by a short expression. I "naturally" tried $f(x)=A\cdot\tanh(k\cdot x^a-d)$ Changing $A=1$ or $d=0$ on it's own makes the approximation go bad and the exponent $a$ is a bit difficult to deal with. However, I found that for $k=\sqrt{\pi}\log{(2)}$ the situation gets "better". I obtained that $k$ value by the requirement that "norm" given by $\int_0^\infty\text{erf}(x)-f(x)dx,$ i.e. the difference of the functions areas, should valish. With this value, the maximal value difference
be down. Please try the request again. Your cache administrator is webmaster. Generated Mon, 17 Oct 2016 12:07:11 GMT by s_ac15 (squid/3.5.20)