This article deals with the Gaussian error function, the error function in
approximation theory , which describes the
difference between a function and its best approximation, see there.
In the theory of special functions, the error function or Gaussian error function is the one defined by the integral
defined function. Thus the error function is an antiderivative of , namely the only odd one (even functions with an antiderivative have exactly one odd one).
For a real argument is a real-valued function ; for generalization to complex arguments see below .
The error function is a sigmoid function , is used in statistics and in the theory of partial differential equations and is closely related to the error integral .
Designations
The name comes from he ror f unction .
The complementary (or conjugate ) error function is given by:
The generalized error function is given by the integral
Are defined.
properties
The following applies:
The error function is odd :
use
Relationship to the normal distribution
The error function is somewhat similar to the distribution function of the normal distribution . However, it has a target amount of , while a distribution function must necessarily assume values from the range .
It applies to the standard normal distribution
or for the distribution function of any normal distribution with standard deviation and expected value
If the deviations of the individual results of a measurement series from the common mean can be described by a normal distribution with standard deviation and expected value 0, then the probability with which the measurement error of a single measurement is between and (for positive ).
The error function can be used to generate normally distributed pseudorandom numbers using the inversion method.
Thermal equation
The error function and the complementary error function occur, for example, in solutions to the heat conduction equation if boundary value conditions are given by the Heaviside function .
Numerical calculation
Like the distribution function of the normal distribution, the error function cannot be represented by a closed function and must be determined numerically .
For small real values, the calculation is carried out with the series expansion
for large real values with the continued fraction expansion
The following approximation is available for the entire range of values with a maximum error of :
With
and
A rapidly converging expansion for all real values of can be obtained using Heinrich H. Bürmann's theorem:
A suitable choice of and results in an approximation whose largest relative error is less than :
Table of values
|
|
|
|
|
|
|
0.00
|
0.0000000
|
1.0000000
|
1.30
|
0.9340079
|
0.0659921
|
0.05
|
0.0563720
|
0.9436280
|
1.40
|
0.9522851
|
0.0477149
|
0.10
|
0.1124629
|
0.8875371
|
1.50
|
0.9661051
|
0.0338949
|
0.15
|
0.1679960
|
0.8320040
|
1.60
|
0.9763484
|
0.0236516
|
0.20
|
0.2227026
|
0.7772974
|
1.70
|
0.9837905
|
0.0162095
|
0.25
|
0.2763264
|
0.7236736
|
1.80
|
0.9890905
|
0.0109095
|
0.30
|
0.3286268
|
0.6713732
|
1.90
|
0.9927904
|
0.0072096
|
0.35
|
0.3793821
|
0.6206179
|
2.00
|
0.9953223
|
0.0046777
|
0.40
|
0.4283924
|
0.5716076
|
2.10
|
0.9970205
|
0.0029795
|
0.45
|
0.4754817
|
0.5245183
|
2.20
|
0.9981372
|
0.0018628
|
0.50
|
0.5204999
|
0.4795001
|
2.30
|
0.9988568
|
0.0011432
|
0.55
|
0.5633234
|
0.4366766
|
2.40
|
0.9993115
|
0.0006885
|
0.60
|
0.6038561
|
0.3961439
|
2.50
|
0.9995930
|
0.0004070
|
0.65
|
0.6420293
|
0.3579707
|
2.60
|
0.9997640
|
0.0002360
|
0.70
|
0.6778012
|
0.3221988
|
2.70
|
0.9998657
|
0.0001343
|
0.75
|
0.7111556
|
0.2888444
|
2.80
|
0.9999250
|
0.0000750
|
0.80
|
0.7421010
|
0.2578990
|
2.90
|
0.9999589
|
0.0000411
|
0.85
|
0.7706681
|
0.2293319
|
3.00
|
0.9999779
|
0.0000221
|
0.90
|
0.7969082
|
0.2030918
|
3.10
|
0.9999884
|
0.0000116
|
0.95
|
0.8208908
|
0.1791092
|
3.20
|
0.9999940
|
0.0000060
|
1.00
|
0.8427008
|
0.1572992
|
3.30
|
0.9999969
|
0.0000031
|
1.10
|
0.8802051
|
0.1197949
|
3.40
|
0.9999985
|
0.0000015
|
1.20
|
0.9103140
|
0.0896860
|
3.50
|
0.9999993
|
0.0000007
|
Complex error function
The definition equation of the error function can be extended to complex arguments :
In this case is a complex valued function . Under complex conjugation applies
-
.
The imaginary error function is given by
with the series development
-
.
For calculation, and other related functions can also be expressed by the Faddeeva function . The Faddeeva function is a scaled complex complementary error function and is also known as a relativistic plasma dispersion function. It is related to the Dawson integrals and the Voigt profile . A numerical implementation by Steven G. Johnson is available as the C library libcerf .
literature
Individual evidence
-
↑ For a concrete implementation see z. B. Peter John Acklam: An algorithm for computing the inverse normal cumulative distribution function . ( Memento of the original from May 5, 2007 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / home.online.no
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↑ Numerical Recipes in Fortran 77: The Art of Scientific Computing . Cambridge University Press, 1992, ISBN 0-521-43064-X , p. 214.
-
↑ HM Schöpf, PH Supancic: On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion . In: The Mathematica Journal , 2014. doi: 10.3888 / tmj.16-11 .
-
↑ Moritz Cantor: Bürmann, Henry . In: Allgemeine Deutsche Biographie (ADB). Volume 47, Duncker & Humblot, Leipzig 1903, pp. 392-394.
-
↑ EW Weis Stone: Bürmann's Theorem . mathworld
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↑ Steven G. Johnson, Joachim Wuttke: libcerf.