A Handy Approximation For The Error Function
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Approximation Q Function
the first article.DoneMerge duplicatesCitations per yearScholarFollowEmailFollow new articlesFollow new citationsCreate alertCancelSergei WinitzkiUnknown approximation gamma function affiliationPhysics, Cosmology, Mathematics, Computer ScienceVerified email at cosmos.phy.tufts.edu - HomepageScholarGet my own profileGoogle ScholarCitation indicesAllSince 2011Citations1624780h-index2114i10-index2919200820092010201120122013201420152016137158138175131107125126116Co-authorsView all…László Árpád GergelyTitle1–20Cited approximation normal distribution 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 and its
Gaussian Approximation
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, 2002602002A handy approximation for the error function and its inverseS WinitzkiA lecture note obtained through private communication, 2008592008Attractor 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 inflationA Vilenkin, S WinitzkiPhysical Review
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Error Function Integral
site for 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 https://scholar.google.com/citations?user=Q9U40gUAAAAJ The best answers are voted up and rise to the top Approximating the error function erf by analytical functions up vote 11 down vote favorite 2 The 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 $\tanh{(x)} = http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions \frac {e^x - e^{-x}} {e^x + e^{-x}}$. 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 even falls under $|\Delta f| = 0.03$. And however you choose the integration bounds for an interval, the area difference is no more than $0.017$. Numerically speaking and relative to a unit scale, the functions $\text{erf}$ and $\tanh{(\sqrt{\pi}\log{(2)}x)}$ are essentially the same. My question is if I can find, or if there are known, substitutions for this non-elementary function in terms of elementary ones. In the sense above, i.e. the approximation is compact/rememberable while the values are even better, f
the error function and its inverse". TODO: mctad.inverseErf = function (x) { a handy approximation var a = 0.147; return mctad.sign(x) * Math.sqrt( Math.sqrt( Math.pow((2 / (mctad.π * a) + (Math.log(1 - Math.pow(x, 2)) / 2)), 2) - Math.log(1 - Math.pow(x, 2)) / a ) - ((2 / (mctad.π * a)) + Math.log(1 - Math.pow(x, 2)) / 2) ); };
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