# 误差函数

$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm dt.$

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

$\operatorname{erfi}(z) = -i\,\,\operatorname{erf}(i\,z).$

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

## 名称由来

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

## 性质

Integrand exp(−z2)
erf(z)

$\operatorname{erf} (-z) = -\operatorname{erf} (z)$

$\operatorname{erf} (\overline{z}) = \overline{\operatorname{erf}(z)}$

### 泰勒级数

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

$\operatorname{erf}(z)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infin\left(z \prod_{k=1}^n {\frac{-(2k-1) z^2}{k (2k+1)}}\right) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^\infin \frac{z}{2n+1} \prod_{k=1}^n \frac{-z^2}{k}$

$\frac{\rm d}{{\rm d}z}\,\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}\,e^{-z^2}.$

$z\,\operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt{\pi}}$

### 逆函数

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

$c_k=\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} = \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\ldots\right\}.$

$\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 ).\$

$\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1}(z).$

### 渐近展开

$\mathrm{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\left [1+\sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdots(2n-1)}{(2x^2)^n}\right ]=\frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n-1)!!}{(2x^2)^n},\,$

$\mathrm{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^{N-1} (-1)^n \frac{(2n-1)!!}{(2x^2)^n}+ R_N(x) \,$

$R_N(x)=O(x^{-2N+1} e^{-x^2})$ as $x\to\infty$.

$R_N(x):= \frac{(-1)^N}{\sqrt{\pi}}2^{-2N+1}\frac{(2N)!}{N!}\int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,$

### 连分式展开

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

## 初等函数近似表达式

$\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+a_3x^3+a_4x^4)^4}$    (最大误差： 5·10−4)

$\operatorname{erf}(x)\approx 1-(a_1t+a_2t^2+a_3t^3)e^{-x^2},\quad t=\frac{1}{1+px}$    (最大误差：2.5·10−5)

$\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+\cdots+a_6x^6)^{16}}$    (最大误差： 3·10−7)

$\operatorname{erf}(x)\approx 1-(a_1t+a_2t^2+\cdots+a_5t^5)e^{-x^2},\quad t=\frac{1}{1+px}$    (maximum error: 1.5·10−7)

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

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

$\operatorname{erf}^{-1}(x)\approx \sgn(x) \sqrt{\sqrt{\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)^2 - \frac{\ln(1-x^2)}{a}} -\left(\frac{2}{\pi a}+\frac{\ln(1-x^2)}{2}\right)}.$

## 数值近似

$\operatorname{erf}(x)=\begin{cases} 1-\tau & \mathrm{for\;}x\ge 0\\ \tau-1 & \mathrm{for\;}x < 0 \end{cases}$

$\begin{array}{rcl} \tau & = & t\cdot\exp\left(-x^{2}-1.26551223+1.00002368\cdot t+0.37409196\cdot t^{2}+0.09678418\cdot t^{3}\right.\\ & & \qquad-0.18628806\cdot t^{4}+0.27886807\cdot t^{5}-1.13520398\cdot t^{6}+1.48851587\cdot t^7\\ & & \qquad\left.-0.82215223\cdot t^{8}+0.17087277\cdot t^{9}\right) \end{array}$
$t=\frac{1}{1+0.5\,|x|}$

## 与其他函数的关系

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

\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}

$\Phi$的逆函数为正态，即概率单位英语Probit函数，

$\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\,\operatorname{erfc}^{-1}(2p).$

$Q(x) =\frac{1}{2} - \frac{1}{2} \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right)=\frac{1}{2}\operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right).$

$\mathrm{erf}(x)= \frac{2x}{\sqrt{\pi}}\,_1F_1\left(\tfrac12,\tfrac32,-x^2\right).$

$\operatorname{erf}(x)=\operatorname{sgn}(x) P\left(\tfrac12, x^2\right)={\operatorname{sgn}(x) \over \sqrt{\pi}}\gamma\left(\tfrac12, x^2\right).$

$\scriptstyle\operatorname{sgn}(x) \$符号函数.

### 广义误差函数

$E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}\,\mathrm dt =\frac{n!}{\sqrt{\pi}}\sum_{p=0}^\infin(-1)^p\frac{x^{np+1}}{(np+1)p!}\,.$

x > 0时，广义误差函数可以用Γ函数和 不完全Γ函数表示，

$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.\$

$\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

$\mathrm i^n \operatorname{erfc}\, (z) = \int_z^\infty \mathrm i^{n-1} \operatorname{erfc}\, (\zeta)\;\mathrm d \zeta.\,$

$\mathrm i^n \operatorname{erfc}\, (z) = \sum_{j=0}^\infty \frac{(-z)^j}{2^{n-j}j! \Gamma \left( 1 + \frac{n-j}{2}\right)}\,,$

$\mathrm i^{2m} \operatorname{erfc} (-z) = - \mathrm i^{2m} \operatorname{erfc}\, (z) + \sum_{q=0}^m \frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!}$

$\mathrm i^{2m+1} \operatorname{erfc} (-z) = \mathrm i^{2m+1} \operatorname{erfc}\, (z) + \sum_{q=0}^m \frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}\,.$

## 函数表

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

## 参考文献

1. ^ Andrews, Larry C.; Special functions of mathematics for engineers
2. ^ 2.0 2.1 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. Handbook of Continued Fractions for Special Functions. Springer-Verlag. 2008. ISBN 978-1-4020-6948-2.
4. ^ Winitzki, Sergei. A handy approximation for the error function and its inverse (PDF). 6 February 2008 [2011-10-03].
5. ^ Numerical Recipes in Fortran 77: The Art of Scientific Computing (ISBN 0-521-43064-X), 1992, page 214, Cambridge University Press.