Inverse Error Function Wolfram
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Inverse Complementary Error Function
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Series representations (8 formulas) http://functions.wolfram.com/GammaBetaErf/InverseErf/ Differential equations (1 formula) Differentiation (5 formulas) Integration (1 formula) Representations through more general functions (1 formula) Representations through equivalent functions (2 formulas) Zeros (1 formula) History (0 formulas) InverseErf[z1,z2]
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 https://en.wikipedia.org/wiki/Error_function − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm https://stat.ethz.ch/pipermail/r-help/2006-June/108153.html 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 error function d t = e − x 2 erfcx ( 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 inverse error function 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( 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 (
[ author ] Nathan Dabney wrote: > Do any of the R libraries have an implementation of the Inverse Error > Function (Inverse ERF)? > > ref: > http://mathworld.wolfram.com/InverseErf.html > http://functions.wolfram.com/GammaBetaErf/InverseErf/ > > Thanks, > Nathan > > [[alternative HTML version deleted]] > > ______________________________________________ > R-help at stat.math.ethz.ch mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Don't know of a built-in function, but you can try this: ## if you want the so-called 'error function' ## from ?pnorm erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1 erf.inv <- function(x) qnorm((x + 1)/2)/sqrt(2) erf.inv(1) erf.inv(0) erf.inv(-1) erf.inv(erf(.25)) erf(erf.inv(.25)) erf.inv(.5) HTH, --sundar Previous message: [R] Inverse Error Function Next message: [R] Inverse Error Function Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] More information about the R-help mailing list