Introductions to InverseErfc

Transcription

1 Introductions to InverseErfc Introduction to the probability integrals and inverses General The probability integral (error function) erf has a long history beginning with the articles of A. de Moivre (78 733) and P.-S. Laplace (774) where it was expressed through the following integral: t t. Later C. Kramp (799) used this integral for the definition of the complementary error function erfc. P.-S. Laplace (8) derived an asymptotic expansion of the error function. The probability integrals were so named because they are widely applied in the theory of probability, in both normal and limit distributions. To obtain, say, a normal distributed random variable from a uniformly distributed random variable, the inverse of the error function, namely erf is needed. The inverse was systematically investigated in the second half of the twentieth century, especially by J. R. Philip (960) and A. J. Strecok (968). Definitions of probability integrals and inverses The probability integral (error function) erf, the generalied error function erf,, the complementary error function erfc, the imaginary error function erfi, the inverse error function erf, the inverse of the generalied error function erf,, and the inverse complementary error function erfc are defined through the following formulas: erf 0 t t erf, erf erf erfc t t erfi 0 t t erfw ; w erf erf, w ; w erf, erfcw ; w erfc. These seven functions are typically called probability integrals and their inverses.

2 Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series. erf k k k k erfc k k k k erfi k k. k A quick look at the probability integrals and inverses Here is a quick look at the graphics for the probability integrals and inverses along the real axis. erf x erfcx erfix In[0]:= (* set text style options *) Off[General::spell]; Off[General::spell]; SetOptions[#, TextStyle -> {FontFamily -> "Helvetica", FontSlant -> "Italic", FontSie -> 0}]& /@ {Graphics, Graphics3D, Plot3D, ListPlot3D, GraphicsArray, Plot, ListPlot, ContourPlot, DensityPlot, ParametricPlot, ParametricPlot3D, ListContourPlot, ListDensityPlot}; Module[{funs, tab, triggraphic, legend, ε = 0^-8}, funs = {InverseErf, InverseErfc}; tab = Table[ Plot[Evaluate[funs[[k]][x]], {x, x0-4, x0 + }, Frame -> True, Axes -> True, PlotPoints -> 0, DisplayFunction -> Identity, PlotStyle -> {{Hue[0.8 (k - )/Length[funs]], Thickness[0.00]}}, FrameTicks -> {Automatic, Automatic, None, None}, FrameLabel -> {x, f}, AspectRatio -> /, PlotRange -> {All,.0 {-5, 5}}], {k, Length[funs]}, {x0, 0, 0}]; triggraphic = Show[tab]; legend = Graphics[ Table[{Hue[0.8 (k - )/Length[funs]], Line[{{0, -0k + 5}, {, -0k + 5}}], Text[TraditionalForm[funs[[k]][x]], {.3, -0k + 5}]}, {k,, Length[funs]}], Frame -> True, FrameTicks -> None, PlotRange -> {{0,.5}, {0, }}]; Show[GraphicsArray[{trigGraphic, legend}]]];

3 3 erf 0, x erf, x Connections within the group of probability integrals and inverses and with other function groups Representations through more general functions The probability integrals erf, erf,, erfc, and erfi are the particular cases of two more general functions: hypergeometric and Meijer G functions. For example, they can be represented through the confluent hypergeometric functions F and U: erf F ; 3 ; erf U,, erf, F ; 3 ; F ; 3 ; erf, U,, U,, erfc F ; 3 ; erfc U,, erfi F ; 3 ; erfi U,,. Representations of the probability integrals erf, erf,, erfc, and erfi through classical Meijer G functions are rather simple:

4 4 erf G,, 0, erf, G,, G,, 0, 0, erfc G,, 0, erfi G,, 0,. The factor in the last four formulas can be removed by changing the classical Meijer G functions to the generalied one: erf G,,,, 0 erf, G,,,, 0 G,,,, 0 erfc G,0,, 0, erfi G,,,, 0. The probability integrals erf, erf,, erfc, and erfi are the particular cases of the incomplete gamma function, regularied incomplete gamma function, and exponential integral E: erf, erf Q, erf E erf,,, erf, Q, Q,

5 5 erf, E E erfc, erfc Q, erfc E erfi, erfi Q, erfi E. Representations through related equivalent functions The probability integrals erf, erfc, and erfi can be represented through Fresnel integrals by the following formulas: erf C S erfc C S erfi C S. Representations through other probability integrals and inverses The probability integrals and their inverses erf, erf,, erfc, erfi, erf, erf,, and erfc are interconnected by the following formulas: erf erf0, erf erfc erf erfi erferf

6 6 erferf 0, erf, erf, erferfc erfc erf, erfc erf erfc erfi erfcerf erfcerf, erfcerfc erfi erf, 0 erfi erfc erfi erf erfi erf erfi erf 0, erfi erfc erf erf 0, erf erfc erfc erf, erfc erf. The best-known properties and formulas for probability integrals and inverses Real values for real arguments For real values of argument, the values of the probability integrals erf, erf,, erfc, and erfi are real. For real arguments, the values of the inverse error function erf are real; for real arguments,, the values of the inverse of the generalied error function erf, are real; and for real arguments 0, the values of the inverse complementary error function erfc are real. Simple values at ero and one The probability integrals erf, erf,, erfc, and erfi, and their inverses erf, erf,, and erfc have simple values for ero or unit arguments:

7 7 erf0 0 erf0, 0 0 erfc0 erfi0 0 erf 0 0 erf erf 0, 0 0 erf 0, erf, 0 erfc 0 erfc 0. Simple values at infinity The probability integrals erf, erfc, and erfi have simple values at infinity: erf erfc 0 erfi. Specific values for specialied arguments In cases when or is equal to 0 or, the generalied error function erf, and its inverse erf, can be expressed through the probability integrals erf, erfc, or their inverses by the following formulas: erf, 0 erf erf0, erf erf, erfc erf, erf erf 0, erf erf, erfc. Analyticity The probability integrals erf, erfc, and erfi, and their inverses erf, and erfc are defined for all complex values of, and they are analytical functions of over the whole complex -plane. The probability integrals erf, erfc, and erfi are entire functions with an essential singular point at, and they do not have branch cuts or branch points.

8 8 The generalied error function erf, is an analytical function of and, which is defined in. For fixed, it is an entire function of. For fixed, it is an entire function of. It does not have branch cuts or branch points. The inverse of the generalied error function erf, is an analytical function of and, which is defined in. Poles and essential singularities The probability integrals erf, erfc, and erfi have only one singular point at. It is an essential singular point. The generalied error function erf, has singular points at and. They are essential singular points. Periodicity The probability integrals erf, erf,, erfc, and erfi, and their inverses erf, erf,, and erfc do not have periodicity. Parity and symmetry The probability integrals erf, erf,, and erfi are odd functions and have mirror symmetry: erf erf erf erf erf, erf, erf, erf, erfi erfi erfi erfi. The generalied error function erf, has permutation symmetry: erf, erf,. The complementary error function erfc has mirror symmetry: erfc erfc. Series representations The probability integrals erf, erf,, erfc, and erfi, and their inverses erf and erfc have the following series expansions: erf ; 0 0 erf k k k k erf, ; 0 0

9 9 erf, k k k k k erfc ; 0 0 erfc k k k k erfi ; 0 0 erfi k k k erf 3 O5 erf k c k k k c m c km ; c 0 c k m m m 0 erf, erf erf erf erf O 3 erf, 4 O 3 erfc 3 O 5 erfc k c k k k c m c km ; c 0 c k m m. m 0 The series for functions erf, erf,, erfc, and erfi converge for all complex values of their arguments. Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through generalied hypergeometric function F, for example: erf F ; F n n k k erf n k k n3 F, n 3 n 3 n ; n, n 5 ; n. Asymptotic series expansions The asymptotic behavior of the probability integrals erf, erfc, and erfi can be described by the following formulas (only the main terms of the asymptotic expansion are given): erf O ;

10 0 erfc O ; erfi O ;. The previous formulas are valid in any direction approaching infinity ( can be simplified to the following relations: ). In particular cases, these formulas erf O ; Re 0 erfc O ; Re 0 erfi O ; Im 0. Integral representations The probability integrals erf, erf,, erfc, and erfi can also be represented through the following equivalent integrals: erfx t sin x t t ; x 0 t erf, t t erfcx t sin x t t ; x 0 t erfix x 0 t sin x tt ; x erfix t x t ; x. t x The symbol in the preceding integral means that the integral evaluates as the Cauchy principal value: b f t a tx t lim xε f t Ε0 a tx t xε Transformations b f t tx t ; a x b. If the arguments of the probability integrals erf, erfc, and erfi contain square roots, the arguments can sometimes be simplified: erf erf

11 erfc erf erfi erfi. Representations of derivatives The derivative of the probability integrals erf, erf,, erfc, and erfi, and their inverses erf, erf,, and erfc have simple representations through elementary functions: erf erf, erf, erfc erfi erf erf erf, erf, erf, erf, erfc erfc. The symbolic n th -order derivatives from the probability integrals erf, erf,, erfc, and erfi have the following simple representations through the regularied generalied hypergeometric function F : n erf n n n F n erf, n n F n, ; n, 3 n ; ; n, ; n, 3 n ; 3 ; n

12 n erf, n n F n n erfc n n n F n erfi n n n F, ; n, 3 n ; 3 ; n, ; n, 3 n ; ; n, ; n, 3 n ; ; n. But the symbolic n th -order derivatives from the inverse probability integrals erf, erf,, and erfc have very complicated structures in which the regularied generalied hypergeometric function F appears in the multidimensional sums, for example: n erf erf n n n erf n n n j 0 n j n 0 n i n i j i,n i j i n n j i i n j i i i erf i erf i j i F, ; i, 3 i ; erf j i ; n. Differential equations The probability integrals erf, erf,, erfc, and erfi satisfy the following second-order linear differential equations: w w 0 ; w erf w0 0 w 0 w w 0 ; w c erf, c w w 0 ; w c erf, c w w 0 ; w erfc w0 w 0 w w 0 ; w erfi w0 0 w 0, where c and c are arbitrary constants. The inverses of the probability integrals erf, erf,, and erfc satisfy the following ordinary secondorder nonlinear differential equations: w w w 0 ; w erf w w w 0 ; w erf, w w w 0 ; w erfc. Applications of probability integrals and inverses

13 3 Applications of probability integrals include solutions of linear partial differential equations, probability theory, Monte Carlo simulations, and the Ewald method for calculating electrostatic lattice constants.

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