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Error function

In mathematics, the error function (also called

the Gauss error function) is a special function (non-elementary) ofsigmoid shape which occurs in

probability, statistics andpartial differential equations. Itis defined as:

(When x is negative, the integral is interpreted as the negative ofthe integral from x to zero.)

The complementary error function, denoted erfc, is defined as

The imaginary error function, denoted erfi, is defined as

The complex error function, denoted w(x) and also known as

the Faddeeva function, is defined as

The name "error function"

The error function is used in measurement theory (using

probability and statistics), and although its use in other

branches of mathematics has nothing to do with the

characterization of measurement errors, the name hasstuck.

http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Special_functionhttp://en.wikipedia.org/wiki/Elementary_functionhttp://en.wikipedia.org/wiki/Sigmoid_functionhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Special_functionhttp://en.wikipedia.org/wiki/Elementary_functionhttp://en.wikipedia.org/wiki/Sigmoid_functionhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Statisticshttp://en.wikipedia.org/wiki/Partial_differential_equationhttp://en.wikipedia.org/wiki/Mathematics

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The error function is related to the cumulative

distribution , the integral of the standard normal

distribution (the "bell curve"), by

The error function, evaluated at for positive x values,

gives the probability that a measurement, under theinfluence of normally distributed errors with standard

deviation , has a distance less than x from the mean

value. This function is used in statistics to predictbehavior of any sample with respect to the population

mean. This usage is similar to the Q-function, which in

fact can be written in terms of the error function.

http://en.wikipedia.org/wiki/Standard_normal_distributionhttp://en.wikipedia.org/wiki/Standard_normal_distributionhttp://en.wikipedia.org/wiki/Q-functionhttp://en.wikipedia.org/wiki/Standard_normal_distributionhttp://en.wikipedia.org/wiki/Standard_normal_distributionhttp://en.wikipedia.org/wiki/Q-function

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Properties

The property means that the error

function is an odd function.

For any complex numberz:

Where is the complex conjugate ofz.

The integrand = exp(z2) and = erf(z) are shown in

the complex z-plane in figures 2 and 3. Level of Im() = 0 is shown with a thick green line. Negative integer

values of Im () are shown with thick red lines. Positive

integer values of are shown with thick blue lines.

Intermediate levels of Im () = constant are shown with

thin green lines. Intermediate levels of Re () = constant

are shown with thin red lines for negative values and

with thin blue lines for positive values.

At the real axis, erf (z) approaches unity at z + and

1 at z . At the imaginary axis, it tends to i.

Taylor series

The error function is an entire function; it has nosingularities (except that at infinity) and its Taylor

expansion always converges.

The defining integral cannot be evaluated in closed form

in terms ofelementary functions, but by expanding the

integrandez2 into its Taylor series and integrating term

by term, one obtains the error function's Taylor series as:

http://en.wikipedia.org/wiki/Even_and_odd_functionshttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Entire_functionhttp://en.wikipedia.org/wiki/Taylor_expansionhttp://en.wikipedia.org/wiki/Taylor_expansionhttp://en.wikipedia.org/wiki/Closed-form_expressionhttp://en.wikipedia.org/wiki/Elementary_function_(differential_algebra)http://en.wikipedia.org/wiki/Integrandhttp://en.wikipedia.org/wiki/Even_and_odd_functionshttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_conjugatehttp://en.wikipedia.org/wiki/Entire_functionhttp://en.wikipedia.org/wiki/Taylor_expansionhttp://en.wikipedia.org/wiki/Taylor_expansionhttp://en.wikipedia.org/wiki/Closed-form_expressionhttp://en.wikipedia.org/wiki/Elementary_function_(differential_algebra)http://en.wikipedia.org/wiki/Integrand

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Which holds for every complex numberz. Thedenominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the

following alternative formulation may be useful:

Because expresses the multiplier to turn the kth

term into the (k+ 1)th term (considering zas the first

term).

The error function at + is exactly 1 (see Gaussianintegral).

The derivative of the error function follows immediately

from its definition:

An ant derivative of the error function is

http://en.wikipedia.org/wiki/Complex_numberhttp://oeis.org/A007680http://en.wikipedia.org/wiki/OEIShttp://en.wikipedia.org/wiki/Gaussian_integralhttp://en.wikipedia.org/wiki/Gaussian_integralhttp://en.wikipedia.org/wiki/Antiderivativehttp://en.wikipedia.org/wiki/Complex_numberhttp://oeis.org/A007680http://en.wikipedia.org/wiki/OEIShttp://en.wikipedia.org/wiki/Gaussian_integralhttp://en.wikipedia.org/wiki/Gaussian_integralhttp://en.wikipedia.org/wiki/Antiderivative

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Inverse functions

The inverse error function can be defined in terms ofthe Maclaurin series

Where c0 = 1 and

So we have the series expansion (note that common

factors have been canceled from numerators and

denominators):

(After cancellation the numerator/denominator fractions

are entries A092676/A132467 in the OEIS; without

cancellation the numerator terms are given in entry

A002067.) Note that the error function's value at is

equal to 1.

The inverse complementary error function is definedas

http://en.wikipedia.org/wiki/OEIShttp://en.wikipedia.org/wiki/OEIS

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Asymptotic expansion

A useful asymptotic expansion of the complementaryerror function (and therefore also of the error function)

for large x is

Where (2n1)!! Is the double factorial: the product of all

odd numbers up to (2n1). This series diverges for everyfinite x, and its meaning as asymptotic expansion is that,

for any one has

Where the remainder, in Landau notation, is

as .

Indeed, the exact value of the remainder is

Which follows easily by induction, writing

and integrating by parts.

For large enough values of x, only the first few terms of

this asymptotic expansion are needed to obtain a good

approximation of erfc(x)

http://en.wikipedia.org/wiki/Asymptotic_expansionhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Landau_notationhttp://en.wikipedia.org/wiki/Asymptotic_expansionhttp://en.wikipedia.org/wiki/Double_factorialhttp://en.wikipedia.org/wiki/Landau_notation

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Continued fractions expansion

A continued fractions expansion of the complementary

error function is:

Approximation with elementary functions

Abramowitz and Stegun give several approximations of

varying accuracy (equations 7.1.25-28). This allows one

to choose the fastest approximation suitable for a given

application. In order of increasing accuracy, they are:

(Maximum error: 5104)

Where a1=0.278393, a2=0.230389, a3=0.000972,

a4=0.078108

(Maximum error: 2.5105)

Where p=0.47047, a1=0.3480242, a2=-0.0958798,

a3=0.7478556

(Maximum

error: 3107)

http://en.wikipedia.org/wiki/Abramowitz_and_Stegunhttp://en.wikipedia.org/wiki/Abramowitz_and_Stegun

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Where a1=0.0705230784, a2=0.0422820123,

a3=0.0092705272, a4=0.0001520143, a5=0.0002765672,

a6=0.0000430638

(Maximum error: 1.5107)

Where p=0.3275911, a1=0.254829592,

a2=0.284496736, a3=1.421413741, a4=1.453152027,

a5=1.061405429

All of these approximations are valid for x0. To use

these approximations for negative x, use the fact that

erf(x) is an odd function, so erf(x)=erf(x).

Another approximation is given by

Where

This is designed to be very accurate in a neighborhood of

0 and a neighborhood of infinity, and the error is less

than 0.00035 for all x. Using the alternate value

a 0.147 reduces the maximum error to about 0.00012.

This approximation can also be inverted to calculate the

inverse error function:

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Applications

When the results of a series of measurements are

described by a normal distribution with standard

deviation and expected value 0, then is the

probability that the error of a single measurement lies

between a and +a, for positive a. This is useful, forexample, in determining thebit error rate of a digital

communication system.

The error and complementary error functions occur, for

example, in solutions of the heat equation whenboundary conditions are given by the Heaviside step

function.

Related functions

The error function is essentially identical to the standard

normal cumulative distribution function, denoted , also

named norm(x) by software languages, as they differonly by scaling and translation. Indeed,

Or rearranged for erf and erfc:

Consequently, the error function is also closely related to

the Q-function, which is the tail probability of the

standard normal distribution. The Q-function can be

expressed in terms of the error function as

http://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Bit_error_ratehttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Boundary_conditionhttp://en.wikipedia.org/wiki/Heaviside_step_functionhttp://en.wikipedia.org/wiki/Heaviside_step_functionhttp://en.wikipedia.org/wiki/Normal_cumulative_distribution_functionhttp://en.wikipedia.org/wiki/Q-functionhttp://en.wikipedia.org/wiki/Normal_distributionhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Standard_deviationhttp://en.wikipedia.org/wiki/Expected_valuehttp://en.wikipedia.org/wiki/Bit_error_ratehttp://en.wikipedia.org/wiki/Heat_equationhttp://en.wikipedia.org/wiki/Boundary_conditionhttp://en.wikipedia.org/wiki/Heaviside_step_functionhttp://en.wikipedia.org/wiki/Heaviside_step_functionhttp://en.wikipedia.org/wiki/Normal_cumulative_distribution_functionhttp://en.wikipedia.org/wiki/Q-function

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The inverse of is known as the normal quantile function,

orprobit function and may be expressed in terms of theinverse error function as

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):

It has a simple expression in terms of the Fresnelintegral. In terms of the Regularized Gamma function P

and the incomplete gamma function,

Is the sign function.

Generalized error functions

Some authors discuss the more general functions

http://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Quantile_functionhttp://en.wikipedia.org/wiki/Probithttp://en.wikipedia.org/wiki/Mittag-Leffler_functionhttp://en.wikipedia.org/wiki/Mittag-Leffler_functionhttp://en.wikipedia.org/wiki/Confluent_hypergeometric_functionhttp://en.wikipedia.org/wiki/Confluent_hypergeometric_functionhttp://en.wikipedia.org/wiki/Fresnel_integralhttp://en.wikipedia.org/wiki/Fresnel_integralhttp://en.wikipedia.org/wiki/Incomplete_Gamma_function#Regularized_Gamma_functionshttp://en.wikipedia.org/wiki/Incomplete_gamma_functionhttp://en.wikipedia.org/wiki/Sign_functionhttp://en.wikipedia.org/wiki/Inverse_functionhttp://en.wikipedia.org/wiki/Quantile_functionhttp://en.wikipedia.org/wiki/Probithttp://en.wikipedia.org/wiki/Mittag-Leffler_functionhttp://en.wikipedia.org/wiki/Mittag-Leffler_functionhttp://en.wikipedia.org/wiki/Confluent_hypergeometric_functionhttp://en.wikipedia.org/wiki/Confluent_hypergeometric_functionhttp://en.wikipedia.org/wiki/Fresnel_integralhttp://en.wikipedia.org/wiki/Fresnel_integralhttp://en.wikipedia.org/wiki/Incomplete_Gamma_function#Regularized_Gamma_functionshttp://en.wikipedia.org/wiki/Incomplete_gamma_functionhttp://en.wikipedia.org/wiki/Sign_function

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Notable cases are:

E0(x) is a straight line through the origin:

E2(x) is the error function, erf(x).

After division by n!, all the En for odd n look similar (butnot 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 generalized error functions for

n > 0 look similar on the positive x side of the graph.

These generalized functions can equivalently be

expressed forx > 0 using the Gamma function and

incomplete Gamma function:

Therefore, we can define the error function in terms of

the incomplete Gamma function:

http://en.wikipedia.org/wiki/Gamma_functionhttp://en.wikipedia.org/wiki/Incomplete_Gamma_functionhttp://en.wikipedia.org/wiki/Gamma_functionhttp://en.wikipedia.org/wiki/Incomplete_Gamma_function

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Iterated integrals of the complementary errorfunction

The iterated integrals of the complementary error

function are defined by

They have the power series

From which follow the symmetry properties

And