查看“误差函数”的源代码

[[File:Erf_plot.svg|thumb|right|325px|误差函数]]
在[[数学]]中，误差函数（也称之为'''高斯误差函数'''）是一个[[特殊函数]]（即不是[[初等函数]]），其在[[概率论]]，[[统计学]]以及[[偏微分方程]]中都有广泛的应用。它的定义如下：<ref>Andrews, Larry C.; [http://books.google.co.uk/books?id=2CAqsF-RebgC&pg=PA110#v=onepage&q&f=false ''Special functions of mathematics for engineers'']</ref><ref name="Greene">Greene, William H.; ''Econometric Analysis'' (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11</ref>

:<math>\operatorname{erf}(x) = \frac{1}{\sqrt\pi}\int_{-x}^x e^{-t^2} \,\mathrm{d}t=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm dt.</math>

[[File:Erfc_plot.svg|thumb|right|325px|互补误差函数]]

'''互补误差函数'''，记为 erfc，在误差函数的基础上定义：

:<math>\mbox{erfc}(x) = 1-\mbox{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,\mathrm dt\,.</math>

'''虚误差函数'''，记为 ''erfi''，定义为：
:<math>\operatorname{erfi}(z) = -i\,\,\operatorname{erf}(i\,z).</math>

'''複誤差函數'''，记为''w''(''z'')，也在误差函数的基础上定义：

:<math>w(z) = e^{-z^2}{\textrm{erfc}}(-iz).</math>

==名称由来==
误差函数来自[[测度论]]，后来与测量误差无关的其他领域也用到这一函数，但仍然使用误差函数这一名字。

误差函数与标准[[正态分布]]的积分[[累积分布函数]]<math>\Phi</math>的关系为<ref name="Greene" />

:<math>\Phi (x) = \frac{1}{2}+ \frac{1}{2} \operatorname{erf} \left(\frac{x}{\sqrt{2}}\right).</math>

==性质==
{{multiple image
   | header  =  复平面上的图
   | direction = vertical
   | width     = 250
   | image1    = ComplexEx2.jpg
   | caption1  = Integrand exp(&minus;''z''<sup>2</sup>)
   | image2    = ComplexErf.jpg
   | caption2  = erf(''z'')
  }}

误差函数是[[奇函数与偶函数|奇函数]]：
:<math>\operatorname{erf} (-z) = -\operatorname{erf} (z)</math>

对于任何 [[复数]] ''z'':

:<math>\operatorname{erf} (\overline{z}) = \overline{\operatorname{erf}(z)}  </math>

其中 <math>\overline{z}</math> 表示 ''z''的 [[复共轭]]。

复平面上，函数 ''ƒ''&nbsp;=&nbsp;exp(&minus;''z''<sup>2</sup>) 和 ''ƒ''&nbsp;=&nbsp;erf(''z'') 如图所示。粗绿线表示 Im(''ƒ'')&nbsp;=&nbsp;0，粗红线表示 Im(''ƒ'')&nbsp;<&nbsp;0， 粗蓝线为 Im(''ƒ'')&nbsp;>&nbsp;0。细绿线表示 Im(''ƒ'')&nbsp;=&nbsp;constant，细红线表示 Re(''ƒ'')&nbsp;=&nbsp;constant<0，细蓝线表示 Re(''ƒ'')&nbsp;=&nbsp;constant>0。
<!-- 
 Level of Im(''ƒ'')&nbsp;=&nbsp;0 is shown with a thick green line. Negative integer values of Im(''ƒ'') are shown with thick red lines. Positive integer values of Im(''f'') are shown with thick blue lines. Intermediate levels of Im(''ƒ'')&nbsp;=&nbsp;constant are shown with thin green lines. Intermediate levels of Re(''ƒ'')&nbsp;=&nbsp;constant are shown with thin red lines for negative values and with thin blue lines for positive values.


the relation <math>{\rm erf}(-z)=-{\rm erf}(z)</math> holds.!-->

在实轴上， ''z''&nbsp;→&nbsp;∞时，erf(''z'') 趋于1，''z''&nbsp;→&nbsp;&minus;∞时，erf(''z'') 趋于&minus;1 。在虚轴上， erf(''z'') 趋于 ±i∞。
===泰勒级数===
误差函数是[[整函数]]，没有奇点（无穷远处除外），泰勒展开收敛。

误差函数泰勒级数：

:<math>\operatorname{erf}(z)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\frac{(-1)^n z^{2n+1}}{n! (2n+1)} =\frac{2}{\sqrt{\pi}} \left(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\ \cdots\right)</math>

对每个复数&nbsp;''z''均成立。
上式可以用迭代形式表示：
:<math>\operatorname{erf}(z)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\left(z \prod_{k=1}^n {\frac{-(2k-1) z^2}{k (2k+1)}}\right) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^\infin \frac{z}{2n+1} \prod_{k=1}^n \frac{-z^2}{k}</math>

误差函数的[[导数]]：
:<math>\frac{\rm d}{{\rm d}z}\,\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}\,e^{-z^2}.</math>

误差函数的 [[不定积分]]为：

:<math>z\,\operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt{\pi}}</math>
===逆函数===
[[File:Inverse Error function.png|thumb|逆誤差函數]]
'''逆误差函数''' 可由 [[麦克劳林级数]]表示：

:<math>\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infin\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1}, \,\!</math>

其中， ''c''<sub>0</sub> = 1 ，

:<math>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},\frac{4369}{2520},\ldots\right\}.</math>
即：
:<math>\operatorname{erf}^{-1}(z)=\tfrac{1}{2}\sqrt{\pi}\left (z+\frac{\pi}{12}z^3+\frac{7\pi^2}{480}z^5+\frac{127\pi^3}{40320}z^7+\frac{4369\pi^4}{5806080}z^9+\frac{34807\pi^5}{182476800}z^{11}+\cdots\right ).\ </math>

'''逆互补误差函数'''定义为：

:<math>\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1}(z).</math>

===渐近展开===
互补误差函数的[[渐近展开]]，


:<math>\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-1)!!}{(2x^2)^n},\,</math>

其中 (2''n''&nbsp;–&nbsp;1)!! 为 [[双阶乘]]，''x''为实数，该级数对有限 ''x''发散。对于<math>N\in\N</math> ，有
:<math>\mathrm{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^{N-1} (-1)^n \frac{(2n-1)!!}{(2x^2)^n}+ R_N(x)  \,</math>
其中余项用以 [[大O符号]]表示为
:<math>R_N(x)=O(x^{-2N+1} e^{-x^2})</math> as <math>x\to\infty</math>.
余项的精确形式为：
:<math>R_N(x):= \frac{(-1)^N}{\sqrt{\pi}}2^{-2N+1}\frac{(2N)!}{N!}\int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,
</math>

对于比较大的 x, 只需渐近展开中开始的几项就可以得到 erfc(''x'')很好的近似值。(对于不太大的 ''x'' ，上文泰勒展开在0处可以快速收敛。)。

===连分式展开===
互补误差函数的连分式展开形式：<ref>{{cite book
  | last1 = Cuyt
  | first1 = Annie A. M.
  | last2 = Petersen
  | first2 = Vigdis B.
  | last3 = Verdonk
  | first3 = Brigitte
  | last4 = Waadeland
  | first4 = Haakon
  | last5 = Jones
  | first5 = William B.
  | title = Handbook of Continued Fractions for Special Functions
  | publisher = [[Springer-Verlag]]
  | year = 2008
  | isbn = 978-1-4020-6948-2
}}</ref>

: <math>\mathrm{erfc}(z) = \frac{z}{\sqrt{\pi}}e^{-z^2} 
\cfrac{a_1}{z^2+
\cfrac{a_2}{1+
\cfrac{a_3}{z^2+
\cfrac{a_4}{1+\dotsb}}}}
\qquad a_1 = 1,\quad a_m = \frac{m-1}{2},\quad m \geq 2.
</math>

==初等函数近似表达式==
*
: <math>\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+a_3x^3+a_4x^4)^4}</math>&nbsp;&nbsp;&nbsp; (最大误差： 5&middot;10<sup>&minus;4</sup>)

其中， ''a''<sub>1</sub>&nbsp;=&nbsp;0.278393, ''a''<sub>2</sub>&nbsp;=&nbsp;0.230389, ''a''<sub>3</sub>&nbsp;=&nbsp;0.000972, ''a''<sub>4</sub>&nbsp;=&nbsp;0.078108
*
: <math>\operatorname{erf}(x)\approx 1-(a_1t+a_2t^2+a_3t^3)e^{-x^2},\quad t=\frac{1}{1+px}</math>&nbsp;&nbsp;&nbsp; (最大误差：2.5&middot;10<sup>&minus;5</sup>)

其中， ''p''&nbsp;=&nbsp;0.47047, ''a''<sub>1</sub>&nbsp;=&nbsp;0.3480242, ''a''<sub>2</sub>&nbsp;=&nbsp;−0.0958798, ''a''<sub>3</sub>&nbsp;=&nbsp;0.7478556
*
: <math>\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+\cdots+a_6x^6)^{16}}</math>&nbsp;&nbsp;&nbsp; (最大误差： 3&middot;10<sup>&minus;7</sup>)

其中， ''a''<sub>1</sub>&nbsp;=&nbsp;0.0705230784, ''a''<sub>2</sub>&nbsp;=&nbsp;0.0422820123, ''a''<sub>3</sub>&nbsp;=&nbsp;0.0092705272, ''a''<sub>4</sub>&nbsp;=&nbsp;0.0001520143, ''a''<sub>5</sub>&nbsp;=&nbsp;0.0002765672, ''a''<sub>6</sub>&nbsp;=&nbsp;0.0000430638
*
: <math>\operatorname{erf}(x)\approx 1-(a_1t+a_2t^2+\cdots+a_5t^5)e^{-x^2},\quad t=\frac{1}{1+px}</math>&nbsp;&nbsp;&nbsp; (maximum error: 1.5&middot;10<sup>&minus;7</sup>)

其中， ''p''&nbsp;=&nbsp;0.3275911, ''a''<sub>1</sub>&nbsp;=&nbsp;0.254829592, ''a''<sub>2</sub>&nbsp;=&nbsp;&minus;0.284496736, ''a''<sub>3</sub>&nbsp;=&nbsp;1.421413741, ''a''<sub>4</sub>&nbsp;=&nbsp;&minus;1.453152027, ''a''<sub>5</sub>&nbsp;=&nbsp;1.061405429

以上所有近似式适用范围是： ''x''&nbsp;≥&nbsp;0.  对于负的 ''x'', 误差函数是奇函数这一性质得到误差函数的值， erf(''x'')&nbsp;=&nbsp;&minus;erf(&minus;''x'').

另有近似式：

: <math>\operatorname{erf}(x)\approx \sgn(x) \sqrt{1-\exp\left(-x^2\frac{4/\pi+ax^2}{1+ax^2}\right)}</math>

其中，

: <math>a = \frac{8(\pi-3)}{3\pi(4-\pi)} \approx 0.140012.</math>
<!--
The range of approximation and the precision are not reported; the fitting may take place in vicinity of the real axis. -->
该近似式在0或无穷的邻域非常准确，''x''整个定义域上，近似式最大误差小于0.00035，取 ''a''&nbsp;≈&nbsp;0.147 ，最大误差可减小到0.00012。<ref>{{Cite web |last=Winitzki |first=Sergei |date=6 February 2008 |title=A handy approximation for the error function and its inverse |url=http://sites.google.com/site/winitzki/sergei-winitzkis-files/erf-approx.pdf |format=PDF |accessdate=2011-10-03 }}{{Dead link|date=2019年5月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><!-- Note: a=0.14784 gives a maximum error of ~.000104, better than a = 0.147 -->

逆误差函数近似式:

:<math>\operatorname{erf}^{-1}(x)\approx \sgn(x) \sqrt{\sqrt{\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)^2 - \frac{\ln(1-x^2)}{a}}
-\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)}.</math>
== 数值近似 ==
下式在整个定义域上，最大误差可低至 <math>1.2\cdot10^{-7}</math>：<ref>Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 978-0-521-43064-7), 1992, page 214, Cambridge University Press.</ref>
:<math>\operatorname{erf}(x)=\begin{cases}
1-\tau & \mathrm{for\;}x\ge 0\\
\tau-1 & \mathrm{for\;}x < 0
\end{cases}</math>
其中，
:<math>\begin{array}{rcl}
\tau & = & t\cdot\exp\left(-x^{2}-1.26551223+1.00002368\cdot t+0.37409196\cdot t^{2}+0.09678418\cdot t^{3}\right.\\
 &  & \qquad-0.18628806\cdot t^{4}+0.27886807\cdot t^{5}-1.13520398\cdot t^{6}+1.48851587\cdot t^7\\
 &  & \qquad\left.-0.82215223\cdot t^{8}+0.17087277\cdot t^{9}\right)
\end{array}</math>

:<math>t=\frac{1}{1+0.5\,|x|}</math>

==与其他函数的关系==
误差函数本质上与标准正态[[累积分布函数]]<math>\Phi</math>是等价的，

: <math>\Phi(x) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^\tfrac{-t^2}{2}\,\mathrm dt = \frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\right]=\frac{1}{2}\,\operatorname{erfc}\left(-\frac{x}{\sqrt{2}}\right)</math>

可整理为如下形式：

:<math>\begin{align}
\mathrm{erf}(x)  &= 2 \Phi \left ( x \sqrt{2} \right ) - 1 \\
\mathrm{erfc}(x) &= 2 \Phi \left ( - x \sqrt{2} \right )=2\left(1-\Phi \left ( x \sqrt{2} \right)\right).
\end{align}</math>

<math>\Phi</math>的逆函数为正态{{link-en|分位函数|Quantile function}}，即{{link-en|概率单位|Probit}}函数，
:<math>
\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\,\operatorname{erfc}^{-1}(2p).
</math>

误差函数为标准正态分布的尾概率{{link-en|Q函数|Q-function}}的关系为，
:<math>
Q(x) =\frac{1}{2} - \frac{1}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right)=\frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right).
</math>

误差函数是[[米塔-列夫勒函数]]的特例，可以表示为[[合流超几何函数]]，

:<math>\mathrm{erf}(x)=
\frac{2x}{\sqrt{\pi}}\,_1F_1\left(\tfrac12,\tfrac32,-x^2\right).</math>

误差函数用正则[[Γ函数]]P和 [[不完全Γ函数]]表示为

:<math>\operatorname{erf}(x)=\operatorname{sgn}(x) P\left(\tfrac12, x^2\right)={\operatorname{sgn}(x) \over \sqrt{\pi}}\gamma\left(\tfrac12, x^2\right).</math>

<math>\scriptstyle\operatorname{sgn}(x) \ </math> 为 [[符号函数]].

===广义误差函数===
[[File:Error Function Generalised.svg|right|thumb|400px|广义误差函数图像 ''E''<sub>n</sub>(''x''):<br />
灰线: ''E''<sub>1</sub>(''x'') = (1&nbsp;&minus;&nbsp;e<sup>&nbsp;&minus;''x''</sup>)/<math>\scriptstyle\sqrt{\pi}</math><br />
红线: ''E''<sub>2</sub>(''x'') = erf(''x'')<br />
绿线: ''E''<sub>3</sub>(''x'')<br />
蓝线: ''E''<sub>4</sub>(''x'')<br />
金线: ''E''<sub>5</sub>(''x'').]]

广义误差函数为：
:<math>E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}\,\mathrm dt
=\frac{n!}{\sqrt{\pi}}\sum_{p=0}^\infin(-1)^p\frac{x^{np+1}}{(np+1)p!}\,.</math>

其中，''E''<sub>0</sub>(''x'')为通过原点的直线，  <math>\scriptstyle E_0(x)=\frac{x}{e \sqrt{\pi}}</math>。''E''<sub>2</sub>(''x'') 即为误差函数 erf(''x'')。

''x''&nbsp;>&nbsp;0时，广义误差函数可以用Γ函数和 不完全Γ函数表示，

:<math>E_n(x) = \frac{\Gamma(n)\left(\Gamma\left(\frac{1}{n}\right)-\Gamma\left(\frac{1}{n},x^n\right)\right)}{\sqrt\pi},
\quad \quad
x>0.\ </math>

因此，误差函数可以用不完全Γ函数表示为：

:<math>\operatorname{erf}(x) = 1 - \frac{\Gamma\left(\frac{1}{2},x^2\right)}{\sqrt\pi}.\ </math>
===互补误差函数的迭代积分===
互补误差函数的迭代积分定义为：

:<math>
\mathrm i^n \operatorname{erfc}\, (z) = \int_z^\infty \mathrm i^{n-1} \operatorname{erfc}\, (\zeta)\;\mathrm d \zeta.\,
</math>

可以展开成幂级数：

:<math>
\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)}\,,
</math>
满足如下对称性质：
:<math>
\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)!}
</math>

和

:<math>
\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)!}\,.
</math>

==函数表==

{{Col-begin}}
{{Col-2}}
:{| class="wikitable"
|--class="hintergrundfarbe6"
!x
!erf(x)
!erfc(x)
|rowspan=24 style="width:2px;background-color:#000000;padding:0px;"|
!x
!erf(x)
!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
|}
{{Col-2}}
:{| class="wikitable"
|--class="hintergrundfarbe6"
!x
!erfc(x)/2
|-
|1
|7.86496e−2
|-
|2
|2.33887e−3
|-
|3
|1.10452e−5
|-
|4
|7.70863e−9
|-
|5
|7.6873e−13
|-
|6
|1.07599e−17
|-
|7
|2.09191e−23
|-
|8
|5.61215e−30
|-
|9
|2.06852e−37
|-
|10
|1.04424e−45
|-
|11
|7.20433e−55
|-
|12
|6.78131e−65
|-
|13
|8.69779e−76
|-
|14
|1.51861e−87
|-
|15
|3.6065e−100
|-
|16
|1.16424e−113
|-
|17
|5.10614e−128
|-
|18
|3.04118e−143
|-
|19
|2.45886e−159
|-
|20
|2.69793e−176
|-
|21
|4.01623e−194
|-
|22
|8.10953e−213
|-
|23
|2.22063e−232
|-
|24
|8.24491e−253
|-
|25
|4.15009e−274
|-
|26
|2.8316e−296
|-
|27
|2.61855e−319
|}

{{Col-end}}

==参见==
*[[古德温 - 斯塔顿积分]]

==参考文献==
{{reflist}}

==外部链接==
* [http://mathworld.wolfram.com/Erf.html MathWorld – Erf]

{{Authority control}}
[[Category:特殊函数|E]]
[[Category:特殊超几何函数|E]]