Error function

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.

Error function graph

In the theory of special functions, the error function or Gaussian error function is the one defined by the integral

${\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-\tau ^{2}}\,\mathrm {d} \tau }$

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). ${\displaystyle {\tfrac {2}{\sqrt {\pi }}}e^{-\tau ^{2}}}$

For a real argument is a real-valued function ; for generalization to complex arguments see below . ${\displaystyle x}$ ${\displaystyle \operatorname {erf} }$

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 . ${\displaystyle {\textrm {erf}}(x)}$

The complementary (or conjugate ) error function is given by: ${\displaystyle \operatorname {erfc} (x)}$

${\displaystyle \operatorname {erfc} (x)=1-\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-\tau ^{2}}\,\mathrm {d} \tau }$

The generalized error function is given by the integral ${\displaystyle \operatorname {erf} (a,b)}$

${\displaystyle \operatorname {erf} (a,b)={\frac {2}{\sqrt {\pi }}}\int _{a}^{b}e^{-\tau ^{2}}\,\mathrm {d} \tau }$

Are defined.

properties

The following applies:

${\displaystyle \operatorname {erf} (a,b)=\operatorname {erf} (b)-\operatorname {erf} (a)}$

The error function is odd :

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

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 . ${\displaystyle (-1,1)}$ ${\displaystyle [0,1]}$

It applies to the standard normal distribution

${\displaystyle \Phi (x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}$

or for the distribution function of any normal distribution with standard deviation and expected value ${\displaystyle F}$ ${\displaystyle \sigma }$ ${\displaystyle \mu }$

${\displaystyle F(x)={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right).}$

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 ). ${\displaystyle \sigma }$ ${\displaystyle \textstyle \operatorname {erf} \left({\frac {a}{\sigma {\sqrt {2}}}}\right)}$ ${\displaystyle -a}$ ${\displaystyle +a}$ ${\displaystyle a}$

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

${\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)n!}}={\frac {2}{\sqrt {\pi }}}\left(x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}-{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}-\dotsb \right),}$

for large real values ​​with the continued fraction expansion

${\displaystyle \operatorname {erf} (x)=1-{\frac {1}{\sqrt {\pi }}}\cdot {\frac {e^{-x^{2}}}{x+{\frac {1}{2x+{\frac {2}{x+{\frac {3}{2x+{\frac {4}{x+\dotsb }}}}}}}}}}.}$

The following approximation is available for the entire range of values ​​with a maximum error of : ${\displaystyle 1{,}2\cdot 10^{-7}}$

${\displaystyle \operatorname {erf} (x)\approx {\begin{cases}1-\tau (x){\text{,}}&{\text{falls }}x\geq 0{\text{,}}\\\tau (-x)-1&{\text{sonst,}}\end{cases}}}$

With

${\displaystyle {\begin{array}{rcl}\tau (x)&=&t\cdot \exp {\bigl (}-x^{2}-1{,}26551223+1{,}00002368\cdot t+0{,}37409196\cdot t^{2}+0{,}09678418\cdot t^{3}\\&&\qquad -0{,}18628806\cdot t^{4}+0{,}27886807\cdot t^{5}-1{,}13520398\cdot t^{6}+1{,}48851587\cdot t^{7}\\&&\qquad -0{,}82215223\cdot t^{8}+0{,}17087277\cdot t^{9}{\bigr )}\end{array}}}$

and

${\displaystyle t={\frac {1}{1+0{,}5\,|x|}}.}$

A rapidly converging expansion for all real values ​​of can be obtained using Heinrich H. Bürmann's theorem: ${\displaystyle x}$

${\displaystyle {\begin{aligned}\operatorname {erf} (x)&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left(1-{\frac {1}{12}}\left(1-e^{-x^{2}}\right)-{\frac {7}{480}}\left(1-e^{-x^{2}}\right)^{2}-{\frac {5}{896}}\left(1-e^{-x^{2}}\right)^{3}-{\frac {787}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\ \cdots \right)\\&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-k\,x^{2}}\right)\end{aligned}}}$

A suitable choice of and results in an approximation whose largest relative error is less than : ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$ ${\displaystyle \textstyle x=\pm 1{,}3796}$ ${\displaystyle \textstyle 3{,}6127\cdot 10^{-3}}$

${\displaystyle \operatorname {erf} (x)\approx {\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x){\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+{\frac {31}{200}}\,e^{-x^{2}}-{\frac {341}{8000}}\,e^{-2\,x^{2}}\right)}$

Table of values

${\displaystyle x}$	${\displaystyle \operatorname {erf} (x)}$	${\displaystyle \operatorname {erfc} (x)}$		${\displaystyle x}$	${\displaystyle \operatorname {erf} (x)}$	${\displaystyle \operatorname {erfc} (x)}$
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 complex error function in the area and . The hue indicates the angle, the lightness the absolute value of the complex number. ${\displaystyle \operatorname {erf} (z)}$ ${\displaystyle -3<\operatorname {Im} (z)<3}$ ${\displaystyle -3<\operatorname {Re} (z)<3}$

The definition equation of the error function can be extended to complex arguments : ${\displaystyle z}$

${\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-\tau ^{2}}\,\mathrm {d} \tau }$

In this case is a complex valued function . Under complex conjugation applies ${\displaystyle \operatorname {erf} }$

${\displaystyle \operatorname {erf} (z^{*})=\operatorname {erf} (z)^{*}}$.

The imaginary error function is given by ${\displaystyle \operatorname {erfi} (x)}$

${\displaystyle \operatorname {erfi} (x)={\frac {\operatorname {erf} (\mathrm {i} x)}{\mathrm {i} }}}$

with the series development

${\displaystyle \operatorname {erfi} (x)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{n!(2n+1)}}={\frac {2}{\sqrt {\pi }}}\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{10}}+{\frac {x^{7}}{42}}+{\frac {x^{9}}{216}}+\dotsb \right)}$.

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 . ${\displaystyle \operatorname {erf,erfi,erfc} }$ ${\displaystyle w(z)}$

literature

• Milton Abramowitz , Irene A. Stegun (Eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Dover, New York 1972, Chapter 7.
• William H. Press, Saul A. Teukolsky, William T. Vetter Ling, Brian P. Flannery: Numerical Recipes in C . 2nd Edition. Cambridge 1992, p. 220 ff. (PDF; 76 kB)

Individual evidence

1. ↑ 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. 
2. ↑ Numerical Recipes in Fortran 77: The Art of Scientific Computing . Cambridge University Press, 1992, ISBN 0-521-43064-X , p. 214.
3. ↑ 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 .
4. ↑ Moritz Cantor:  Bürmann, Henry . In: Allgemeine Deutsche Biographie (ADB). Volume 47, Duncker & Humblot, Leipzig 1903, pp. 392-394.
5. ↑ EW Weis Stone: Bürmann's Theorem . mathworld
6. ↑ Steven G. Johnson, Joachim Wuttke: libcerf.