Inverse Error Function Wikipedia
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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 . complementary error function {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm error function calculator − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted
Error Function Table
erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) ,
Erf(inf)
{\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 ( x | x ≥ 0 error function matlab ) = 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 Properties 2.1 Taylor series 2.2 Derivative and integral 2.3 Bürmann s
For other uses, see Bell curve (disambiguation). Normal distribution Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation N ( μ , σ
Erf(1)
2 ) {\displaystyle {\mathcal σ 4}(\mu ,\,\sigma ^ σ 3)} Parameters μ inverse error function calculator ∈ R — mean (location) σ2 > 0 — variance (squared scale) Support x ∈ R PDF 1 inverse error function excel 2 σ 2 π e − ( x − μ ) 2 2 σ 2 {\displaystyle {\frac σ 0{\sqrt − 9\pi }}}\,e^{-{\frac {(x-\mu )^ − 8} − 7}}}} CDF https://en.wikipedia.org/wiki/Error_function 1 2 [ 1 + erf ( x − μ σ 2 ) ] {\displaystyle {\frac − 2 − 1}\left[1+\operatorname − 0 \left({\frac 9{\sigma {\sqrt 8}}}\right)\right]} Quantile μ + σ 2 erf − 1 ( 2 F − 1 ) {\displaystyle \mu +\sigma {\sqrt 2}\operatorname 1 ^{-1}(2F-1)} Mean μ Median μ Mode μ Variance https://en.wikipedia.org/wiki/Normal_distribution σ 2 {\displaystyle \sigma ^ − 8\,} Skewness 0 Ex. kurtosis 0 Entropy 1 2 ln ( 2 σ 2 π e ) {\displaystyle {\tfrac − 6 − 5}\ln(2\sigma ^ − 4\pi \,e\,)} MGF exp { μ t + 1 2 σ 2 t 2 } {\displaystyle \exp\{\mu t+{\frac − 0 σ 9}\sigma ^ σ 8t^ σ 7\}} CF exp { i μ t − 1 2 σ 2 t 2 } {\displaystyle \exp\ σ 2 σ 1}\sigma ^ σ 0t^ μ 9\}} Fisher information ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) {\displaystyle {\begin μ 41/\sigma ^ μ 3&0\\0&1/(2\sigma ^ μ 2)\end μ 1}} In probability theory, the normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.[1][2] The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (w
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 updated: Tue Sep 27 2016 Created, http://mathworld.wolfram.com/Erfc.html developed, and nurturedbyEricWeisstein at WolframResearch Calculus and Analysis>Special Functions>Erf> Calculus and Analysis>Complex https://www.scribd.com/document/50267654/Error-function-Wikipedia-the-free-encyclopedia Analysis>Entire Functions> Calculus and Analysis>Calculus>Integrals>Definite Integrals> More... Interactive Entries>webMathematica Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire function defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some error function authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . For , (3) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) It has the special values (6) (7) (8) It satisfies the identity (9) It has definite integrals (10) (11) (12) For , is bounded inverse error function by (13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above. A generalization is obtained from the erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122). The general solution is then (15) where is the repeated erfc integral. For integer , (16) (17) (18) (19) (Abramowitz and Stegun 1972, p.299), where is a confluent hypergeometric function of the first kind and is a gamma function. The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). "Repeated Integrals of the Error Function." §7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.299-300, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.568-569, 1985. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; and Vetterling, W.T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in
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