NationMaster - Encyclopedia: 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: Image File history File links Erf_plot. ... Image File history File links Erf_plot. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, several functions are important enough to deserve their own name. ... Probability is the likelihood that something is the case or will happen. ... This article is about the field of statistics. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

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

1 Properties
2 Applications
3 Asymptotic expansion
4 Related functions
- 4.1 Generalized error functions
- 4.2 Iterated integrals of the complementary error function
5 Implementation
6 See also
7 References
8 External links

Properties

The error function is odd: In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. ...

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

Also, for any complex number x one has

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

where x * is the complex conjugate of x. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

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: In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by... This article deals with the concept of an integral in calculus. ... As the degree of the Taylor series rises, it approaches the correct function. ...

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

which holds for every real number x, and also throughout the complex plane. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

(This result arises from the Taylor series expansion of $e^{-x^2}$ , which is $sum_{n=0}frac{(-1)^n x^{2n}}{n!}$ , which we then integrate term by term.)

The error function at infinity is exactly 1 (see Gaussian integral). The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

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

$operatorname{erf}^{-1}(x)=sum_{k=0}^infinfrac{c_k}{2k+1}left (frac{sqrt{pi}}{2}xright )^{2k+1}, ,!$

where c₀ = 1 and

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

So we have the series expansion

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

Plot of the complementary error function

The complementary error function, denoted erfc, is defined in terms of the error function: Image File history File links Erfc_plot. ... Image File history File links Erfc_plot. ...

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

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

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

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

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. The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ... In probability and statistics, the standard deviation of a probability distribution, random variable, or population or multiset of values is a measure of the spread of its values. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

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. The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

In digital optical communication system, BER is expressed by: In telecommunication, an error ratio is the ratio of the number of bits, elements, characters, or blocks incorrectly received to the total number of bits, elements, characters, or blocks sent during a specified time interval. ...

$mathrm{BER} = 0.5,operatorname{erfc}left( frac{mu_1 - mu_2}{sqrt{2}left(sigma_1 + sigma_2right)} right).$

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large x is In mathematics an asymptotic expansion, asymptotic series or PoincarÃ© expansion (after Henri PoincarÃ©) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular...

$mathrm{erfc}(x) = frac{e^{-x^2}}{xsqrt{pi}}left [1+sum_{n=1}^infty (-1)^n frac{1cdot3cdot5cdots(2n-1)}{(2x^2)^n}right ]=frac{e^{-x^2}}{xsqrt{pi}}sum_{n=0}^infty (-1)^n frac{(2n)!}{n!(2x)^{2n}}.,$

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. As the degree of the Taylor series rises, it approaches the correct function. ...

Another approximation is given by

$(operatorname{erf}(x))^2approx 1-expleft(-x^2frac{4/pi+ax^2}{1+ax^2}right)$

where

$a = frac{-8}{3pi}frac{pi-3}{pi-4}.$

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, The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. ...

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

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 In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... This article or section does not cite any references or sources. ... In probability theory and statistics the probit function is the inverse cumulative distribution function, or quantile function of the normal distribution. ...

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

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): In mathematics, the Mittag-Leffler function is special function, a complex function which depends on two complex parameters and . ... In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ...

$mathrm{erf}(x)= frac{2x}{sqrt{pi}},_1F_1left(frac{1}{2},frac{3}{2},-x^2right).$

It has a simple expression in terms of the Fresnel integral. In terms of the Regularized Gamma function P and the incomplete gamma function, In mathematics and optics, the two Fresnel integrals, S(x) and C(x), named after Augustin-Jean Fresnel, arise in the description of near field Fresnel diffraction phenomena, and are the integrals defined as follows: Some (including Abramowitz and Stegun, eqs 7. ... In mathematics, the gamma function is defined by a definite integral. ... In mathematics, the gamma function is defined by a definite integral. ...

$operatorname{erf}(x)=operatorname{sgn}(x) Pleft(frac{1}{2}, x^2right)={operatorname{sgn}(x) over sqrt{pi}}gammaleft(frac{1}{2}, x^2right).$

$operatorname{sgn}(x)$ is the sign function. Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...

Generalized error functions

Some authors discuss the more general functions

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

E₂(x) is the error function.

Graph of generalized error functions E_n(x). Grey curve: E₁(x) = 1 − e^−x, red curve: erf(x) = E₂(x), green curve: E₃(x), blue curve: E₄(x), and yellow curve: E₅(x). (The yellow curve is quite close to the y-axis and may not be visible.) After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. The E_n with odd and even n look similar on the positive x side of the graph. Image File history File links Erf. ...

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

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

Implementation

C/C++: It is implemented as the function erf() and erfc() in the header math.h or cmath in the GNU version. This is not part of standard so please check the compiler documentation before use.

References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 7)

Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...

External links

MathWorld - Erf

Categories: Special functions | Special hypergeometric functions

Results from FactBites:

Error function - Wikipedia, the free encyclopedia (420 words)
	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.
	The error function is essentially identical to the standard normal cumulative distribution function, denoted Î¦, as they differ only by scaling and translation.
	The integral defining the error function cannot be evaluated in closed form in terms of elementary functions, but the integrand may be expanded in a power series and integrated term by term.

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