Error function - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and This article discusses the concept of elementary functions in differential algebra Probability is the likelihood or chance that something is the case or will happen Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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:

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

1 Properties
2 Applications
3 Asymptotic expansion
4 Related functions
- 4.1 Generalised 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:

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

Also, for any complex number x one has

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

where $x *$ is the complex conjugate of x. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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:

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

which holds for every real number x, and also throughout the complex plane. This article discusses the concept of elementary functions in differential algebra The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

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 i^th 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 Gaussian integral, or probability integral, is the Improper integral of the Gaussian function e^ over the entire real line

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

Note that error function's value at 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, for positive 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 is a measure of the dispersion of a collection of values

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 distribution of Heat (or variation in temperature in a given region over time In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints The Heaviside step function, H, also called the unit step function, is a discontinuous function whose value is zero for negative

In digital optical communication system, BER is expressed by:

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

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. In Telecommunication, an error ratio is the Ratio of the number of Bits elements, characters, or blocks incorrectly received In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which 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. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

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

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

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. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields 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. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B See also Quantile. In Probability theory, a quantile function of a Probability distribution is the inverse In Probability theory and Statistics, the probit function is the inverse Cumulative distribution function (CDF or Quantile function

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 Mathematics, the Mittag-Leffler function E &alpha&beta is Special function, a complex function which depends on two In Mathematics, a confluent hypergeometric functions is a solution of a confluent hypergeometric equation, which is a degenerate form of a Hypergeometric differential Fresnel integrals, S ( x) and C ( x) are two Transcendental functions named after Augustin-Jean Fresnel that are used 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. In Mathematics, the Gamma function is defined by a definite integral. In Mathematics, the Gamma function is defined by a definite integral.

Generalised error functions

Graph of generalised error functions E_n(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 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!. 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:

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

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 provided by C99 as the functions double erf(double x) and double erfc(double x) in the header math. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function 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). gfortran is the name of the GNU Fortran Compiler, which is part of the GNU Compiler Collection (GCC

References

Milton Abramowitz and Irene A. In Mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form f(x = a e^{- { (x-b^2 \over 2 Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Abramowitz and Stegun is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U New York: Dover, 1972. (See Chapter 7)

External links

MathWorld - Erf

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