Introductions to InverseErf

Introduction to the probability integrals and inverses

General

The probability integral (error function) erf(z) has a long history beginning with the articles of A. de Moivre (1718–1733) and  P.-S. Laplace (1774) where it was expressed through the following integral:

Later C. Kramp (1799) used this integral for the definition of the complementary error function erfc(z). P.-S. Laplace (1812) 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^(-1))(z) is needed. The inverse was systematically investigated in the second half of the twentieth century, especially by J. R. Philip (1960) and A. J. Strecok (1968).

Definitions of probability integrals and inverses

The probability integral (error function) erf(z), the generalized error function  erf((Subscript[z, 1]),(Subscript[z, 2])), the complementary error function erfc(z), the imaginary error function erfi(z), the inverse error function (erf^(-1))(z), the inverse of the generalized error function (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and the inverse complementary error function (erfc^(-1))(z) are defined through the following formulas:

These seven functions are typically called probability integrals and their inverses.

Instead of using definite integrals, the three univariate error functions can be defined through the following infinite series.

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.

(* set text style options *)
Off[General::spell]; Off[General::spell1];
SetOptions[#, TextStyle -> {FontFamily -> "Helvetica", 
                            FontSlant -> "Italic", FontSize -> 10}]& /@ 
     {Graphics, Graphics3D, Plot3D, ListPlot3D, GraphicsArray, 
      Plot, ListPlot, ContourPlot, DensityPlot, ParametricPlot, ParametricPlot3D, 
      ListContourPlot, ListDensityPlot};
      
Module[{funs, tab, trigGraphic, legend1, ε = 10^-8},
funs = {InverseErf, InverseErfc};
tab = Table[
Plot[Evaluate[funs[[k]][x]], {x, x0 - 41, x0 + 1}, 
           Frame -> True, Axes -> True, PlotPoints -> 120,
     DisplayFunction -> Identity,
     PlotStyle -> {{Hue[0.8 (k - 1)/Length[funs]], Thickness[0.002]}},
     FrameTicks -> {Automatic,  Automatic, None, None}, 
                       FrameLabel -> {x, f}, 
     AspectRatio -> 1/2, PlotRange -> {All, 1.01 {-5, 5}}], 
                    {k, Length[funs]}, {x0, 0, 0}];
trigGraphic = Show[tab];
legend1 = Graphics[
Table[{Hue[0.8 (k - 1)/Length[funs]], Line[{{0, -10k + 5}, {1, -10k + 5}}],
      Text[TraditionalForm[funs[[k]][x]], {1.3, -10k + 5}]},
      {k, 1, Length[funs]}], 
      Frame -> True, FrameTicks -> None, 
      PlotRange -> {{0, 1.5},  {10, -10 4 -10}}];
Show[GraphicsArray[{trigGraphic, legend1}]]];

Connections within the group of probability integrals and inverses and with other function groups

Representations through more general functions

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) 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 (Subscript[, 1])(Subscript[F, 1]) and U:

Representations of the probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) through classical Meijer G functions are rather simple:

The factor z in the last four formulas can be removed by changing the classical Meijer G functions to the generalized one:

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) are the particular cases of the incomplete gamma function, regularized incomplete gamma function, and exponential integral E:

Representations through related equivalent functions

The probability integrals erf(z), erfc(z), and erfi(z) can be represented through Fresnel integrals by the following formulas:

Representations through other probability integrals and inverses

The probability integrals and their inverses erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), erfi(z), (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) are interconnected by the following formulas:

The best-known properties and formulas for probability integrals and inverses

Real values for real arguments

For real values of argument z, the values of the probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) are real. For real arguments -1<z<1, the values of the inverse error function (erf^(-1))(z) are real; for real arguments -1<(Subscript[z, 1])<1, -1<(Subscript[z, 2])<1, the values of the inverse of the generalized error function (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])) are real; and for real arguments 0<z<2, the values of the inverse complementary error function (erfc^(-1))(z) are real.

Simple values at zero and one

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z), and their inverses (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) have simple values for zero or unit arguments:

Simple values at infinity

The probability integrals erf(z), erfc(z), and erfi(z) have simple values at infinity:

Specific values for specialized arguments

In cases when (Subscript[z, 1]) or (Subscript[z, 2]) is equal to 0 or ∞, the generalized error function erf((Subscript[z, 1]),(Subscript[z, 2])) and its inverse (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])) can be expressed through the probability integrals erf(z), erfc(z), or their inverses by the following formulas:

Analyticity

The probability integrals erf(z), erfc(z), and erfi(z), and their inverses (erf^(-1))(z), and (erfc^(-1))(z) are defined for all complex values of z, and they are analytical functions of z over the whole complex z-plane. The probability integrals erf(z), erfc(z), and erfi(z) are entire functions with an essential singular point at z = (OverTilde[∞]), and they do not have branch cuts or branch points.

The generalized error function erf((Subscript[z, 1]),(Subscript[z, 2])) is an analytical function of (Subscript[z, 1]) and (Subscript[z, 2]), which is defined in (ℂ^2). For fixed (Subscript[z, 1]), it is an entire function of (Subscript[z, 2]). For fixed (Subscript[z, 2]), it is an entire function of (Subscript[z, 1]). It does not have branch cuts or branch points. 
The inverse of the generalized error function (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])) is an analytical function of (Subscript[z, 1]) and (Subscript[z, 2]), which is defined in (ℂ^2).

Poles and essential singularities

The probability integrals erf(z), erfc(z), and erfi(z) have only one singular point at z = (OverTilde[∞]). It is an essential singular point.

The generalized error function erf((Subscript[z, 1]),(Subscript[z, 2])) has singular points at (Subscript[z, 1])= (OverTilde[∞]) and (Subscript[z, 2])==(OverTilde[∞]). They are essential singular points.

Periodicity

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z), and their inverses (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) do not have periodicity.

Parity and symmetry

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), and erfi(z) are odd functions and have mirror symmetry:

The generalized error function erf((Subscript[z, 1]),(Subscript[z, 2])) has permutation symmetry:

The complementary error function  erfc(z) has mirror symmetry:

Series representations

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z), and their inverses (erf^(-1))(z) and (erfc^(-1))(z) have the following series expansions:

The series for functions erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) 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 generalized hypergeometric function (Subscript[, 2])(Subscript[F, 2]), for example:

Asymptotic series expansions

The asymptotic behavior of the probability integrals erf(z), erfc(z), and erfi(z) can be described by the following formulas (only the main terms of the asymptotic expansion are given):

The previous formulas are valid in any direction approaching infinity (|z|->∞). In particular cases, these formulas can be simplified to the following relations:

Integral representations

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) can also be represented through the following equivalent integrals:

The symbol  𝒫 in the preceding  integral means that the integral evaluates as the Cauchy principal value: 𝒫(∫_a^b)((f(t))/(t-x))dt==(lim_(ϵ->0)) ((∫_a^(x-ϵ))((f(t))/(t-x))dt+(∫_(x+ϵ)^b)((f(t))/(t-x))dt)/;a<x<b.

Transformations

If the arguments of the probability integrals erf(z), erfc(z), and erfi(z) contain square roots, the arguments can sometimes be simplified:

Representations of derivatives

The derivative of the probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z), and their inverses (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) have simple representations through elementary functions:

The symbolic (n^th)-order derivatives from the probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) have the following simple representations through the regularized generalized hypergeometric function (Subscript[, 2])(Subscript[(OverTilde[F]), 2]):

But the symbolic (n^th)-order derivatives from the inverse probability integrals (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) have very complicated structures in which the regularized generalized hypergeometric function (Subscript[, 2])(Subscript[(OverTilde[F]), 2]) appears in the multidimensional sums, for example:

Differential equations

The probability integrals erf(z), erf((Subscript[z, 1]),(Subscript[z, 2])), erfc(z), and erfi(z) satisfy the following second-order linear differential equations:

where (Subscript[c, 1]) and (Subscript[c, 2]) are arbitrary constants.

The inverses of the probability integrals (erf^(-1))(z), (erf^(-1))((Subscript[z, 1]),(Subscript[z, 2])), and (erfc^(-1))(z) satisfy the following ordinary second-order nonlinear differential equations:

Applications of probability integrals and inverses

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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