# 误差函数

（重定向自誤差函數

${\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,\mathrm {d} t.}$

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

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

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

## 名称由来

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

## 性质

Integrand exp(−z2)
erf(z)

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

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

### 泰勒级数

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

${\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }\left(z\prod _{k=1}^{n}{\frac {-(2k-1)z^{2}}{k(2k+1)}}\right)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {z}{2n+1}}\prod _{k=1}^{n}{\frac {-z^{2}}{k}}}$

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

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

### 逆函数

Inverse error function

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

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

${\displaystyle \operatorname {erf} ^{-1}(z)={\tfrac {1}{2}}{\sqrt {\pi }}\left(z+{\frac {\pi }{12}}z^{3}+{\frac {7\pi ^{2}}{480}}z^{5}+{\frac {127\pi ^{3}}{40320}}z^{7}+{\frac {4369\pi ^{4}}{5806080}}z^{9}+{\frac {34807\pi ^{5}}{182476800}}z^{11}+\cdots \right).\ }$

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

### 渐近展开

${\displaystyle \mathrm {erfc} (x)={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left[1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\cdots (2n-1)}{(2x^{2})^{n}}}\right]={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\sum _{n=0}^{\infty }(-1)^{n}{\frac {(2n-1)!!}{(2x^{2})^{n}}},\,}$

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

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

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

### 连分式展开

${\displaystyle \mathrm {erfc} (z)={\frac {z}{\sqrt {\pi }}}e^{-z^{2}}{\cfrac {a_{1}}{z^{2}+{\cfrac {a_{2}}{1+{\cfrac {a_{3}}{z^{2}+{\cfrac {a_{4}}{1+\dotsb }}}}}}}}\qquad a_{1}=1,\quad a_{m}={\frac {m-1}{2}},\quad m\geq 2.}$

## 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)

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

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

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

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

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

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

## 数值近似

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

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

## 与其他函数的关系

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

{\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，即概率单位英语Probit函数，

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

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

${\displaystyle \mathrm {erf} (x)={\frac {2x}{\sqrt {\pi }}}\,_{1}F_{1}\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}$

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

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

### 广义误差函数

${\displaystyle E_{n}(x)={\frac {n!}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{n}}\,\mathrm {d} t={\frac {n!}{\sqrt {\pi }}}\sum _{p=0}^{\infty }(-1)^{p}{\frac {x^{np+1}}{(np+1)p!}}\,.}$

x > 0时，广义误差函数可以用Γ函数和 不完全Γ函数表示，

${\displaystyle E_{n}(x)={\frac {\Gamma (n)\left(\Gamma \left({\frac {1}{n}}\right)-\Gamma \left({\frac {1}{n}},x^{n}\right)\right)}{\sqrt {\pi }}},\quad \quad x>0.\ }$

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

### 互补误差函数的迭代积分

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

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

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

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

## 函数表

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. 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 978-0-521-43064-7), 1992, page 214, Cambridge University Press.