Error function

Error function
Plot of the error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt.

The complementary error function, denoted erfc, is defined as

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

The imaginary error function, denoted erfi, is defined as

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

The complex error function, denoted w(x) and also known as the Faddeeva function, is defined as

w(x) = e^{-x^2}{\operatorname{erfc}}(-ix) = e^{-x^2}[1+i\,\,\operatorname{erfi}(x)]



Plots in the complex plane

Integrand exp(−z2)

The error function is odd:

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

Also, for any complex number z:

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

where \overline{z} is the complex conjugate of z.

The integrand ƒ = exp(−z2) and ƒ = erf(z) are shown in the complex z-plane in figures 2 and 3. Level of Im(ƒ) = 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) = constant are shown with thin green lines. Intermediate levels of Re(ƒ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values.

At the real axis, erf(z) approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand ez2 into its Taylor series and integrating term by term, one obtains the error function's Taylor series as:

\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)

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

\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}

because \frac{-(2k-1) z^2}{k (2k+1)} expresses the multiplier to turn the kth term into the (k + 1)th term (considering z as the first term).

The error function at +∞ is exactly 1 (see Gaussian integral).

The derivative of the error function follows immediately from its definition:

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

An antiderivative of the error function is

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

Inverse function

The inverse error function has series

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

where c0 = 1 and

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},\ldots\right\}.

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

\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 ).\

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.) Note that the error function's value at ±∞ is equal to ±1.

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large x is

\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}.\,

This series diverges for every finite x, and its meaning as asymptotic expansion is that, for any N\in\N one has

\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)  \,

where the remainder, in Landau notation, is

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

Indeed, the exact value of the remainder is

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

which follows easily by induction, writing e^{-t^2}=-(2t)^{-1}(e^{-t^2})' and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc(x) (while for not too large values of x note that the above Taylor expansion at 0 provides a very fast convergence).

Continued fractions expansion

A continued fractions expansion of the complementaty error function is [1]:

\mathrm{erfc}(z) = \frac{z}{\sqrt{\pi}}e^{-z^2} 
\qquad a_1 = 1,\quad a_m = \frac{m-1}{2},\quad m \geq 2

Approximation with elementary functions

Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25-28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are:

\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+a_3x^3+a_4x^4)^4}    (maximum error: 5·10-4)

where a1=0.278393, a2=0.230389, a3=0.000972, a4=0.078108

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

where p=0.47047, a1=0.3480242, a2=-0.0958798, a3=0.7478556

\operatorname{erf}(x)\approx 1-\frac{1}{(1+a_1x+a_2x^2+...+a_6x^6)^{16}}    (maximum error: 3·10-7)

where a1=0.0705230784, a2=0.0422820123, a3=0.0092705272, a4=0.0001520143, a5=0.0002765672, a6=0.0000430638

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

where p=0.3275911, a1=0.254829592, a2=-0.284496736, a3=1.421413741, a4=-1.453152027, a5=1.061405429

All of these approximations are valid for x≥0. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x)=-erf(-x).

Another approximation is given by

\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.

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x. Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.[2]

This approximation can also be inverted to calculate the inverse error function:

\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)}


When the results of a series of measurements are described by a normal distribution with standard deviation \scriptstyle\sigma and expected value 0, then  \scriptstyle\operatorname{erf}\,\left(\,\frac{a}{\sigma \sqrt{2}}\,\right) is the probability that the error of a single measurement lies between −a and +a, for positive a. This is useful, for example, in determining the bit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

Related functions

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. Indeed,

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

or rearranged for erf and erfc:

\mathrm{erf}(x)  &= 2 \Phi \left ( x \sqrt{2} \right ) - 1 \\
\mathrm{erfc}(x) &= 2 \Phi \left ( - x \sqrt{2} \right ).

Consequently, the error function is also closely related to the Q-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as

Q(x) =\tfrac{1}{2} - \tfrac{1}{2} \operatorname{erf} \Bigl( \frac{x}{\sqrt{2}} \Bigr)=\tfrac{1}{2}\operatorname{erfc}(\frac{x}{\sqrt{2}}).

The inverse of \scriptstyle\Phi\, is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

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

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function):


It has a simple expression in terms of the Fresnel integral. In terms of the Regularized Gamma function P and the incomplete gamma function,

\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) \ is the sign function.

Generalized error functions

Graph of generalised error functions En(x):
grey curve: E1(x) = (1 − e −x)/\scriptstyle\sqrt{\pi}
red curve: E2(x) = erf(x)
green curve: E3(x)
blue curve: E4(x)
gold curve: E5(x).

Some authors discuss the more general functions:[citation needed]

E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}\,dt

Notable cases are:

  • E0(x) is a straight line through the origin: \scriptstyle E_0(x)=\frac{x}{e \sqrt{\pi}}
  • E2(x) is the error function, erf(x).

After division by n!, all the En for odd n look similar (but not identical) to each other. Similarly, the En for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the Gamma function and incomplete Gamma function:

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

Therefore, we can define the error function in terms of the incomplete Gamma function:

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

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

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

They have the power series

\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)}\,,

from which follow the symmetry properties

\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)!}\,.


  • C: C99 provides the functions double erf(double x) and double erfc(double x) in the header math.h. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively.
  • C++: C++11 provides erf() and erfc() in the header cmath. Both functions are overloaded to accept arguments of type float, double, and long double.
  • Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement.
  • Python: Included since version 2.7 as math.erf(). For previous version, an implementation of erf for complex arguments is in SciPy as scipy.special.erf()[3] and also in the arbitrary-precision arithmetic mpmath library as mpmath.erf()
  • Mathematica: erf is implemented as Erf and Erfc in Mathematica
  • R: "The so-called 'error function'"[5] is not provided directly, but is detailed as an example of the normal cumulative distribution function (?pnorm), which is based on W. J. Cody's rational Chebyshev approximation algorithm.[6]
  • Matlab provides both erf and erfc, also via W. J. Cody's algorithm.
  • Ruby: Provides Math.erf() and Math.erfc().

Table of values

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 0.0786496
2 0.00233887
3 1.10452e-05
4 7.70863e-09
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

See also

Related functions

In probability

  • Normal distribution
  • Normal cumulative distribution function, a scaled and shifted form of error function
  • Probit, the inverse or quantile function of the normal CDF
  • Q-function, the tail probability of the normal distribution


  1. ^ Cuyt, A.A.M.; Petersen, V.; Verdonk, B.; Waadeland, H.; Jones, W.B. (2008). Handbook of Continued Fractions for Special Functions. Springer-Verlag. ISBN 978-1-4020-6948-2. 
  2. ^ Winitzki, Sergei (6 February 2008). "A handy approximation for the error function and its inverse" (PDF). Retrieved 2011-10-03. 
  3. ^
  4. ^
  5. ^ R Development Core Team (2011-02-25), R: The Normal Distribution, 
  6. ^ Cody, W. J. (1969). "Rational Chebyshev Approximations for the Error Function". Math. Comp. 23 (107): 631–637. 

External links

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Error-Function — Plot der Fehlerfunktion Als Fehlerfunktion bezeichnet man in der Approximationstheorie die Differenz zwischen einer Funktion und ihrer besten Approximation. In der Theorie der speziellen Funktionen versteht man unter Fehlerfunktion oder Gaußsche… …   Deutsch Wikipedia

  • Error function — Plot der Fehlerfunktion Als Fehlerfunktion bezeichnet man in der Approximationstheorie die Differenz zwischen einer Funktion und ihrer besten Approximation. In der Theorie der speziellen Funktionen versteht man unter Fehlerfunktion oder Gaußsche… …   Deutsch Wikipedia

  • error-function diffusion — difuzija pagal paklaidų funkciją statusas T sritis radioelektronika atitikmenys: angl. error function diffusion vok. Fehlerfunktionsdiffusion, f rus. диффузия в соответствии с функцией ошибок, f pranc. diffusion en fonction d erreurs, f …   Radioelektronikos terminų žodynas

  • error-function distribution — pasiskirstymas pagal paklaidų funkciją statusas T sritis radioelektronika atitikmenys: angl. error function distribution vok. Fehlerfunktionsverteilung, f rus. распределение в соответствии с функцией погрешностей, n pranc. distribution en… …   Radioelektronikos terminų žodynas

  • Gauss error function — Gauso funkcija statusas T sritis fizika atitikmenys: angl. Gauss error function; Gauss function vok. Gaußsche Funktion, f rus. гауссова функция, f pranc. fonction de Gauss, f …   Fizikos terminų žodynas

  • Error analysis — is the study of kind and quantity of error that occurs, particularly in the fields of applied mathematics (particularly numerical analysis), applied linguistics and statistics. Error analysis in numerical modelling In numerical simulation or… …   Wikipedia

  • Error hiding — is an anti pattern, in computer programming. The programmer hides error messages by overriding them with exception handling. As a result of this the root error message is hidden from the user (hence error hiding ) and so they will not be told… …   Wikipedia

  • Function overloading — or method overloading is a feature found in various programming languages such as Ada, C#, VB.NET, C++, D and Java that allows the creation of several methods with the same name which differ from each other in terms of the type of the input and… …   Wikipedia

  • Error bar — Error bars are used on graphs to indicate the error in a reported measurement. They give a general idea of how accurate a measurement is, or conversely, how far from the reported value the true (error free) value might be. Often, but not always,… …   Wikipedia

  • Error correction mode — (ECM) is an optional transmission mode built into Class 1 fax machines or fax modems. ECM automatically detects and corrects errors in the fax transmission process that are sometimes caused by telephone line noise. The page data is divided into… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”