在数学中，**误差函数**（英語：）是一个特殊函数，符号{\displaystyle \operatorname {erf} }。误差函数在概率论，统计学以及偏微分方程中都有广泛的应用。它的定义如下：

{\displaystyle \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 {d} t.}
误差函数
互补误差函数

分类
--

**互补误差函数**，记为 erfc，在误差函数的基础上定义：

{\displaystyle {\mbox{erfc}}(x)=1-{\mbox{erf}}(x)={\frac {2}{\sqrt {\pi }}}\int \_{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t\,.}
**虚误差函数**，记为 *erfi*，定义为：

{\displaystyle \operatorname {erfi} (z)=-i\,\,\operatorname {erf} (i\,z).}
**複誤差函數**，记为*w*(*z*)，也在误差函数的基础上定义：

{\displaystyle w(z)=e^{-z^{2}}{\textrm {erfc}}(-iz).}

词源
--

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{\displaystyle \Phi }的关系为

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

性质
--

复平面上的图Integrand exp(−*z*2)erf(*z*)
误差函数是奇函数：

{\displaystyle \operatorname {erf} (-z)=-\operatorname {erf} (z)}
对于任何 复数 *z*:

{\displaystyle \operatorname {erf} ({\overline {z}})={\overline {\operatorname {erf} (z)}}}
其中 {\displaystyle {\overline {z}}} 表示 *z*的 复共轭。

复平面上，函数 *ƒ* = exp(−*z*2) 和 *ƒ* = erf(*z*) 如图所示。粗绿线表示 Im(*ƒ*) = 0，粗红线表示 Im(*ƒ*) < 0， 粗蓝线为 Im(*ƒ*) > 0。细绿线表示 Im(*ƒ*) = constant，细红线表示 Re(*ƒ*) = constant<0，细蓝线表示 Re(*ƒ*) = constant>0。

在实轴上， *z* → ∞时，erf(*z*) 趋于1，*z* → −∞时，erf(*z*) 趋于−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：

{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum \_{n=0}^{\infty }{\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)}
对每个复数 *z*均成立。
上式可以用迭代形式表示：

{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum \_{n=0}^{\infty }\left(z\prod \_{k=1}^{n}{\frac {-(2k-1)z^{2}}{k(2k+1)}}\right)={\frac {2}{\sqrt {\pi }}}\sum \_{n=0}^{\infty }{\frac {z}{2n+1}}\prod \_{k=1}^{n}{\frac {-z^{2}}{k}}}
误差函数的导数：

{\displaystyle {\frac {\rm {d}}{{\rm {d}}z}}\,\mathrm {erf} (z)={\frac {2}{\sqrt {\pi }}}\,e^{-z^{2}}.}
误差函数的 不定积分为：

{\displaystyle z\,\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}}
### 逆函数

逆誤差函數
**逆误差函数** 可由 麦克劳林级数表示：

{\displaystyle \operatorname {erf} ^{-1}(z)=\sum \_{k=0}^{\infty }{\frac {c\_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},\,\!}
其中， *c*0 = 1 ，

{\displaystyle 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\}.}
即：

{\displaystyle \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).\ }
**逆互补误差函数**定义为：

{\displaystyle \operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}(z).}
### 渐近展开

互补误差函数的渐近展开，

  

{\displaystyle \mathrm {erfc} (x)={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left[1+\sum \_{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\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}}},\,}
其中 (2*n* – 1)!! 为 双阶乘，*x*为实数，该级数对有限 *x*发散。对于{\displaystyle N\in \mathbb {N} } ，有

{\displaystyle \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)\,}
其中余项用以 大O符号表示为

{\displaystyle R\_{N}(x)=O(x^{-2N+1}e^{-x^{2}})} as {\displaystyle x\to \infty }.
余项的精确形式为：

{\displaystyle R\_{N}(x):={\frac {(-1)^{N}}{\sqrt {\pi }}}2^{-2N+1}{\frac {(2N)!}{N!}}\int \_{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t,}
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。

### 连分式展开

互补误差函数的连分式展开形式：

{\displaystyle \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.}

初等函数近似表达式
---------

* 

{\displaystyle \operatorname {erf} (x)\approx 1-{\frac {1}{(1+a\_{1}x+a\_{2}x^{2}+a\_{3}x^{3}+a\_{4}x^{4})^{4}}}}    (最大误差： 5·10−4)
其中， *a*1 = 0.278393, *a*2 = 0.230389, *a*3 = 0.000972, *a*4 = 0.078108

* 

{\displaystyle \operatorname {erf} (x)\approx 1-(a\_{1}t+a\_{2}t^{2}+a\_{3}t^{3})e^{-x^{2}},\quad t={\frac {1}{1+px}}}    (最大误差：2.5·10−5)
其中， *p* = 0.47047, *a*1 = 0.3480242, *a*2 = −0.0958798, *a*3 = 0.7478556

* 

{\displaystyle \operatorname {erf} (x)\approx 1-{\frac {1}{(1+a\_{1}x+a\_{2}x^{2}+\cdots +a\_{6}x^{6})^{16}}}}    (最大误差： 3·10−7)
其中， *a*1 = 0.0705230784, *a*2 = 0.0422820123, *a*3 = 0.0092705272, *a*4 = 0.0001520143, *a*5 = 0.0002765672, *a*6 = 0.0000430638

* 

{\displaystyle \operatorname {erf} (x)\approx 1-(a\_{1}t+a\_{2}t^{2}+\cdots +a\_{5}t^{5})e^{-x^{2}},\quad t={\frac {1}{1+px}}}    (最大误差： 1.5·10−7)
其中， *p* = 0.3275911, *a*1 = 0.254829592, *a*2 = −0.284496736, *a*3 = 1.421413741, *a*4 = −1.453152027, *a*5 = 1.061405429

以上所有近似式适用范围是： *x* ≥ 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*) = −erf(−*x*).

另有近似式：

{\displaystyle \operatorname {erf} (x)\approx \operatorname {sgn}(x){\sqrt {1-\exp \left(-x^{2}{\frac {4/\pi +ax^{2}}{1+ax^{2}}}\right)}}}
其中，

{\displaystyle a={\frac {8(\pi -3)}{3\pi (4-\pi )}}\approx 0.140012.}

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* ≈ 0.147 ，最大误差可减小到0.00012。

逆误差函数近似式:

{\displaystyle \operatorname {erf} ^{-1}(x)\approx \operatorname {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)}}.}

数值近似
----

下式在整个定义域上，最大误差可低至 {\displaystyle 1.2\cdot 10^{-7}}：

{\displaystyle \operatorname {erf} (x)={\begin{cases}1-\tau &\mathrm {for\;} x\geq 0\\\tau -1&\mathrm {for\;} x<0\end{cases}}}
其中，

{\displaystyle {\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}}}
{\displaystyle t={\frac {1}{1+0.5\,|x|}}}

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{\displaystyle \Phi }是等价的，

{\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int \_{-\infty }^{x}e^{\tfrac {-t^{2}}{2}}\,\mathrm {d} t={\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)}
可整理为如下形式：

{\displaystyle {\begin{aligned}\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{aligned}}}
{\displaystyle \Phi }的逆函数为正态分位函数，即概率单位（英语：）函数，

{\displaystyle \operatorname {probit} (p)=\Phi ^{-1}(p)={\sqrt {2}}\,\operatorname {erf} ^{-1}(2p-1)=-{\sqrt {2}}\,\operatorname {erfc} ^{-1}(2p).}
误差函数为标准正态分布的尾概率Q函数（英语：）的关系为，

{\displaystyle 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).}
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，

{\displaystyle \mathrm {erf} (x)={\frac {2x}{\sqrt {\pi }}}\,\_{1}F\_{1}\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}
误差函数用正则Γ函数P和 不完全Γ函数表示为

{\displaystyle \operatorname {erf} (x)=\operatorname {sgn} (x)P\left({\tfrac {1}{2}},x^{2}\right)={\operatorname {sgn} (x) \over {\sqrt {\pi }}}\gamma \left({\tfrac {1}{2}},x^{2}\right).}
{\displaystyle \scriptstyle \operatorname {sgn} (x)\ } 为 符号函数.

### 广义误差函数

广义误差函数图像 *E*n(*x*):  

灰线: *E*1(*x*) = (1 − e−*x*)/{\displaystyle \scriptstyle {\sqrt {\pi }}}  

红线: *E*2(*x*) = erf(*x*)  

绿线: *E*3(*x*)  

蓝线: *E*4(*x*)  

金线: *E*5(*x*).
广义误差函数为：

{\displaystyle E\_{n}(x)={\frac {n!}{\sqrt {\pi }}}\int \_{0}^{x}e^{-t^{n}}\,\mathrm {d} t={\frac {n!}{\sqrt {\pi }}}\sum \_{p=0}^{\infty }(-1)^{p}{\frac {x^{np+1}}{(np+1)p!}}\,.}
其中，*E*0(*x*)为通过原点的直线， {\displaystyle \scriptstyle E\_{0}(x)={\frac {x}{e{\sqrt {\pi }}}}}。*E*2(*x*) 即为误差函数 erf(*x*)。

*x* > 0时，广义误差函数可以用Γ函数和 不完全Γ函数表示，

{\displaystyle 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.\ }
因此，误差函数可以用不完全Γ函数表示为：

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

{\displaystyle \mathrm {i} ^{n}\operatorname {erfc} \,(z)=\int \_{z}^{\infty }\mathrm {i} ^{n-1}\operatorname {erfc} \,(\zeta )\;\mathrm {d} \zeta .\,}
可以展开成幂级数：

{\displaystyle \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)}}\,,}
满足如下对称性质：

{\displaystyle \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)!}}}
和

{\displaystyle \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)!}}\,.}

函数表
---

 

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| 

|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| x | erf(x) | erfc(x) |  | 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 |

  | 

| 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 |

  |

参见
--

* 古德温 - 斯塔顿积分

参考文献
----

1. ↑  Andrews, Larry C.; *Special functions of mathematics for engineers* （页面存档备份，存于）
2. 1 2  Greene, William H.; *Econometric Analysis* (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11
3. ↑  Cuyt, Annie A. M.; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B.. Springer-Verlag. 2008. ISBN 978-1-4020-6948-2.
4. ↑  Winitzki, Sergei.  (PDF). 6 February 2008  [2011-10-03]. [永久]
5. ↑  Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 978-0-521-43064-7), 1992, page 214, Cambridge University Press.

外部链接
----

* MathWorld – Erf（页面存档备份，存于）

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| 规范控制 編輯維基數據鏈接 | 
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* NDL: 00562553

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