Error function

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Plot of the error function
Plot of the error function

In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. It is defined as:

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt.

The complementary error function, denoted erfc, is defined in terms of the error function:

\mbox{erfc}(x) = 1-\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt.

The complex error function, denoted w(x), (also known as the Faddeeva function) is also defined in terms of the error function:

w(x) = e^{-x^2}{\textrm{erfc}}(-ix).\,\!

Contents

[edit] Properties

The error function is odd:

\operatorname{erf} (-x) = -\operatorname{erf} (x).

Also, for any complex number x one has

\operatorname{erf} (x^{*}) = \operatorname{erf}(x)^{*}

where x * is the complex conjugate of x.

The integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand in a Taylor series, one obtains the Taylor series for the error function as follows:

\operatorname{erf}(x)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\frac{(-1)^n x^{2n+1}}{n! (2n+1)} =\frac{2}{\sqrt{\pi}} \left(x-\frac{x^3}{3}+\frac{x^5}{10}-\frac{x^7}{42}+\frac{x^9}{216}-\ \cdots\right)

which holds for every real number x, and also throughout the complex plane. This result arises from the Taylor series expansion of e^{-x^2} which is \sum_{n=0}^\infin\frac{(-1)^n x^{2n}}{n!} and is then integrated term by term. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternate formulation may be useful:

\operatorname{erf}(x)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\left(x \prod_{i=1}^n{\frac{-(2i-1) x^2}{i (2i+1)}}\right) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^\infin \frac{x}{2n+1} \prod_{i=1}^n \frac{-x^2}{i}

because \frac{-(2i-1) x^2}{i (2i+1)} expresses the multiplier to turn the ith term into the (i+1)th term (assuming we number the "x" as the first term).

The error function at infinity is exactly 1 (see Gaussian integral).

The derivative of the error function follows immediately from its definition:

\frac{d}{dx}\,\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\,e^{-x^2}.

The inverse error function has series

\operatorname{erf}^{-1}(x)=\sum_{k=0}^\infin\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}x\right )^{2k+1}, \,\!

where c0 = 1 and

c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\ldots\right\}.

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

\operatorname{erf}^{-1}(x)=\frac{1}{2}\sqrt{\pi}\left (x+\frac{\pi}{12}x^3+\frac{7\pi^2}{480}x^5+\frac{127\pi^3}{40320}x^7+\frac{4369\pi^4}{5806080}x^9+\frac{34807\pi^5}{182476800}x^{11}+\cdots\right ). \,\![1]

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.)

Plot of the complementary error function
Plot of the complementary error function

Note that error function's value at plus/minus infinity is equal to plus/minus 1.

[edit] Applications

When the results of a series of measurements are described by a normal distribution with standard deviation σ and expected value 0, then \operatorname{erf}\,\left(\,\frac{a}{\sigma \sqrt{2}}\,\right) is the probability that the error of a single measurement lies between −a and +a, for positive a.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

In digital optical communication system, BER is expressed by:

\mathrm{BER} = 0.5\,\operatorname{erfc}\left( \frac{\mu_1 - \mu_2}{\sqrt{2}\left(\sigma_1 + \sigma_2\right)} \right).

[edit] Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large x is

\mathrm{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\left [1+\sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdots(2n-1)}{(2x^2)^n}\right ]=\frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}}.\,

This series diverges for every finite x. However, in practice only the first few terms of this expansion are needed to obtain a good approximation of erfc(x), whereas the Taylor series given above converges very slowly.

Another approximation is given by

\operatorname{erf}^2(x)\approx 1-\exp\left(-x^2\frac{4/\pi+ax^2}{1+ax^2}\right)

where

a = -\frac{8(\pi-3)}{3\pi(\pi-4)}.

[edit] Related functions

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, as they differ only by scaling and translation. Indeed,

\Phi(x) = \frac{1}{2}\left[1+\mbox{erf}\left(\frac{x}{\sqrt{2}}\right)\right]=\frac{1}{2}\,\mbox{erfc}\left(-\frac{x}{\sqrt{2}}\right).

The inverse of \Phi\, is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\,\operatorname{erfc}^{-1}(2p).

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function):

\mathrm{erf}(x)= \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\frac{1}{2},\frac{3}{2},-x^2\right).

It has a simple expression in terms of the Fresnel integral. In terms of the Regularized Gamma function P and the incomplete gamma function,

\operatorname{erf}(x)=\operatorname{sgn}(x) P\left(\frac{1}{2}, x^2\right)={\operatorname{sgn}(x) \over \sqrt{\pi}}\gamma\left(\frac{1}{2}, x^2\right).

\operatorname{sgn}(x) \ is the sign function.

[edit] Generalised error functions

Graph of generalised error functions En(x): grey curve: E1(x) = (1 − e −x)/ red curve: E2(x) = erf(x) green curve: E3(x) blue curve: E4(x) gold curve: E5(x).
Graph of generalised error functions En(x):
grey curve: E1(x) = (1 − e −x)/\sqrt{\pi}
red curve: E2(x) = erf(x)
green curve: E3(x)
blue curve: E4(x)
gold curve: E5(x).

Some authors discuss the more general functions

E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}\,dt =\frac{n!}{\sqrt{\pi}}\sum_{p=0}^\infin(-1)^p\frac{x^{np+1}}{(np+1)p!}\,.

Notable cases are:

  • E0(x) is a straight line through the origin: E_0(x)=\frac{x}{e \sqrt{\pi}}
  • E2(x) is the error function, erf(x).

After division by n!, all the En for odd n look similar (but not identical) to each other. Similarly, the En for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n>0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x>0 using the Gamma function:

E_n(x) = \frac{x\left(x^n\right)^{-1/n}\Gamma(n)\left(\Gamma\left(\frac{1}{n}\right)-\Gamma\left(\frac{1}{n},x^n\right)\right)}{\sqrt\pi}, \quad \quad x>0

Therefore, we can define the error function in terms of the Gamma function:

\operatorname{erf}(x) = 1 - \frac{\Gamma\left(\frac{1}{2},x^2\right)}{\sqrt\pi}

[edit] Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

\mathrm i^n \operatorname{erfc}\, (z) = \int_z^\infty \mathrm i^{n-1} \operatorname{erfc}\, (\zeta)\;\mathrm d \zeta.\,

They have the power series

\mathrm i^n \operatorname{erfc}\, (z) = \sum_{j=0}^\infty \frac{(-z)^j}{2^{n-j}j! \Gamma \left( 1 + \frac{n-j}{2}\right)}\,,

from which follow the symmetry properties

\mathrm i^{2m} \operatorname{erfc} (-z) = - \mathrm i^{2m} \operatorname{erfc}\, (z) + \sum_{q=0}^m \frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!}

and

\mathrm i^{2m+1} \operatorname{erfc} (-z) = \mathrm i^{2m+1} \operatorname{erfc}\, (z) + \sum_{q=0}^m \frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}\,.

[edit] Implementation

C/C++: It is provided by C99 as the functions double erf(double x) and double erfc(double x) in the header math.h or cmath. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. GCC makes these functions available in C++ too.

Fortran: E.g. gfortran provides the intrinsic real function ERF(X) and the double precision function DERF(X).

[edit] See also

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