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{{Use dmy dates|date=July 2013}}
[[File:Error Function.svg|thumb|right|400px|Plot of the error function]]
In [[mathematics]], the '''error function''' (also called the '''Gauss error function''') is a [[special function]] (non-[[Elementary function|elementary]]) of [[Sigmoid function|sigmoid]] shape that occurs in [[probability]], [[statistics]], and [[partial differential equation]]s describing diffusion. It is defined as:<ref>{{cite book|last =Andrews|first = Larry C.|url = https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110#v=onepage&q&f=false |title = Special functions of mathematics for engineers|page = 110|publisher = SPIE Press |date= 1998|isbn = 9780819426161}}</ref><ref name="Greene">Greene, William H.; ''Econometric Analysis'' (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11</ref>

:<math>\begin{align}
\operatorname{erf}(x) & = \frac{1}{\sqrt\pi}\int_{-x}^x e^{-t^2} \,dt \\[5pt]
&= \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,dt.
\end{align} </math>

In statistics, for nonnegative values of ''x'', the error function has the following interpretation: for a [[random variable]] ''Y'' that is [[normal distribution|normally distributed]] with [[mean]] 0 and [[variance]] 1/2, erf(''x'') describes the probability of ''Y'' falling in the range [−''x'',&nbsp;''x''].

There are several closely related functions, such as the complementary error function, the imaginary error function, and others.

==Name==
The name "error function" and its abbreviation {{math|erf}} were proposed by [[James Whitbread Lee Glaisher|J. W. L. Glaisher]] in 1871 on account of its connection with "the theory of Probability, and notably the theory of [[errors and residuals|Errors]]."<ref name="Glaisher1871a">{{cite journal|last1=Glaisher|first1=James Whitbread Lee|title= On a class of definite integrals|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=July 1871|volume=42|pages=294–302|accessdate=6 December 2017|url=https://books.google.com/books?id=8Po7AQAAMAAJ&pg=RA1-PA294|number=277|series=4|doi=10.1080/14786447108640568}}</ref>  The  error function complement was also discussed by Glaisher in a separate publication in the same year.<ref name="Glaisher1871b">{{cite journal|last1=Glaisher|first1=James Whitbread Lee|title=On a class of definite integrals. Part II|journal=London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science|date=September 1871|volume=42|pages=421–436|accessdate=6 December 2017|url=https://books.google.com/books?id=yJ1YAAAAcAAJ&pg=PA421|series=4|number=279|doi=10.1080/14786447108640600}}</ref>
For the "law of facility" of errors whose [[probability density|density]] is given by

:<math>f(x)=\left(\frac{c}{\pi}\right)^{\tfrac{1}{2}}e^{-cx^2}</math>

(the [[normal distribution]]), Glaisher calculates the chance of an error lying between <math>p</math> and <math>q</math> as:

:<math>\left(\frac{c}{\pi}\right)^{\tfrac{1}{2}}\int_p^qe^{-cx^2}dx=\tfrac{1}{2}\left(\operatorname{erf} (q\sqrt{c}) -\operatorname{erf} (p\sqrt{c})\right).</math>

==Applications==
When the results of a series of measurements are described by a [[normal distribution]] with [[standard deviation]] <math>\sigma</math> and [[expected value]] 0, then <math> \textstyle\operatorname{erf}\left(\frac{a}{\sigma \sqrt{2}}\right)</math>  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 condition]]s are given by the [[Heaviside step function]].

The error function and its approximations can be used to estimate results that hold [[with high probability]] or with low probability. Given random variable <math>X \sim \operatorname{Norm}[\mu,\sigma]</math> and constant <math>L<\mu</math>:

:<math>\Pr[X\leq L] = \frac{1}{2} + \frac{1}{2}\operatorname{erf}\left(\frac{L-\mu}{\sqrt{2}\sigma}\right) \approx A \exp \left(-B \left(\frac{L-\mu}{\sigma}\right)^2\right)</math>

where ''A'' and ''B'' are certain numeric constants. If ''L'' is sufficiently far from the mean, i.e. <math>\mu-L \geq \sigma\sqrt{\ln{k}}</math>, then:

:<math>\Pr[X\leq L] \leq A \exp (-B \ln{k}) = \frac{A}{k^B}</math>

so the probability goes to 0 as <math>k\to\infty</math>.

==Properties==
{{multiple image
   | header  =  Plots in the complex plane
   | direction = vertical
   | width     = 250
   | image1    = ComplexEx2.jpg
   | caption1  = Integrand exp(−''z''<sup>2</sup>)
   | image2    = ComplexErf.jpg
   | caption2  = erf(''z'')
  }}

The property <math>\operatorname{erf} (-z) = -\operatorname{erf} (z)</math> means that the error function is an [[even and odd functions|odd function]]. This directly results from the fact that the integrand <math>e^{-t^2}</math> is an [[even function]].

For any [[complex number]] ''z'':

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

where <math>\overline{z}</math> is the [[complex conjugate]] of ''z''.

The integrand ''f''&nbsp;=&nbsp;exp(−''z''<sup>2</sup>) and ''f''&nbsp;=&nbsp;erf(''z'') are shown in the complex ''z''-plane in figures 2 and 3.  Level of Im(''f'')&nbsp;=&nbsp;0 is shown with a thick green line. Negative integer values of Im(''f'') are shown with thick red lines. Positive integer values of Im(''f'') are shown with thick blue lines. Intermediate levels of Im(''f'')&nbsp;=&nbsp;constant are shown with thin green lines. Intermediate levels of Re(''f'')&nbsp;=&nbsp;constant are shown with thin red lines for negative values and with thin blue lines for positive values.

The error function at +∞ is exactly 1 (see [[Gaussian integral]]). At the real axis, erf(''z'') approaches unity at ''z''&nbsp;→&nbsp;+∞ and −1 at ''z''&nbsp;→&nbsp;−∞. At the imaginary axis, it tends to ±''i''∞.
<!-- ; the relation <math>\operatorname{erf}(-z)=-\operatorname{erf}(z)</math> holds.!-->

===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 expression|closed form]] in terms of [[Elementary function (differential algebra)|elementary functions]], but by expanding the [[integrand]] ''e''<sup>−''z''<sup>2</sup></sup> into its [[Maclaurin series]] and integrating term by term, one obtains the error function's Maclaurin series as:

:<math>\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)</math>

which holds for every [[complex number]]&nbsp;''z''. The denominator terms are sequence [[oeis:A007680|A007680]] in the [[OEIS]].

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

:<math>\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}</math>

because <math>\frac{-(2k-1) z^2}{k (2k+1)}</math> expresses the multiplier to turn the ''k''<sup>th</sup> term into the (''k''&nbsp;+&nbsp;1)<sup>th</sup> term (considering ''z'' as the first term).

The imaginary error function has a very similar Maclaurin series, which is:

:<math>\operatorname{erfi}(z)= \frac{2}{\sqrt{\pi}}\sum_{n=0}^\infty\frac{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>

which holds for every [[complex number]]&nbsp;''z''.

===Derivative and integral===
The derivative of the error function follows immediately from its definition:

:<math>\frac{d}{dz}\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} e^{-z^2}.</math>

From this, the derivative of the imaginary error function is also immediate:

:<math>\frac{d}{dz}\operatorname{erfi}(z)=\frac{2}{\sqrt{\pi}} e^{z^2}.</math>
An [[antiderivative]] of the error function, obtainable by [[integration by parts]], is

:<math>z\operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt{\pi}}.</math>

An antiderivative of the imaginary error function, also obtainable by integration by parts, is

:<math>z\operatorname{erfi}(z) - \frac{e^{z^2}}{\sqrt{\pi}}.</math>

Higher order derivatives are given by

:<math>\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}}  \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</math>

where <math> \mathit{H} </math> are the physicists' [[Hermite polynomials]].<ref>{{cite web|url = http://mathworld.wolfram.com/Erf.html |publisher =Wolfram|work = MathWorld|title = Erf|last = Weisstein|first = Eric W.}}</ref>

===Bürmann series===

An expansion,<ref>H. M. Schöpf and P. H. Supancic, "On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion," The Mathematica Journal, 2014. doi:10.3888/tmj.16–11.[http://www.mathematica-journal.com/2014/11/on-burmanns-theorem-and-its-application-to-problems-of-linear-and-nonlinear-heat-transfer-and-diffusion/#more-39602/ Schöpf, Supancic]</ref> which converges more rapidly for all real values of <math>x</math> than a Taylor expansion, is obtained by using [[Hans Heinrich Bürmann]]'s theorem:<ref>
{{cite web
 | url = http://mathworld.wolfram.com/BuermannsTheorem.html
 | title = Bürmann's Theorem
 | last = Weisstein
 | authorlink = Eric W. Weisstein
 | first = E. W.
 | date =
 | website = Wolfram MathWorld—A Wolfram Web Resource
 | publisher =
 | access-date =
 | quote = }}</ref>

: <math>\begin{align}
\operatorname{erf}(x) &= \frac 2 {\sqrt{\pi}} \sgn(x) \sqrt{1-e^{-x^2}} \left( 1-\frac{1}{12} \left (1-e^{-x^2} \right ) -\frac{7}{480} \left (1-e^{-x^2} \right )^2 -\frac{5}{896} \left (1-e^{-x^2} \right )^3-\frac{787}{276 480} \left (1-e^{-x^2} \right )^4 - \cdots \right) \\[10pt]
&= \frac{2}{\sqrt{\pi}} \sgn(x) \sqrt{1-e^{-x^2}} \left(\frac{\sqrt{\pi}}{2} + \sum_{k=1}^\infty c_k e^{-kx^2} \right).
\end{align}</math>

By keeping only the first two coefficients and choosing <math>c_{1}=\frac{31}{200}</math> and <math>c_2 = -\frac{341}{8000},</math> the resulting approximation shows its largest relative error at <math>x=\pm 1.3796,</math> where it is less than <math> 3.6127\cdot 10^{-3}</math>:

: <math>\operatorname{erf}(x)\approx \frac{2}{\sqrt{\pi}}\sgn(x)\sqrt{1-e^{-x^2}} \left(\frac{\sqrt{\pi }}{2} + \frac{31}{200}e^{-x^2}-\frac{341}{8000} e^{-2x^2}\right). </math>

===Inverse functions===
[[File:Mplwp erf inv.svg|thumb|300px|Inverse error function]]

Given complex number ''z'', there is not a ''unique'' complex number ''w'' satisfying <math>\operatorname{erf}(w)=z</math>, so a true inverse function would be multivalued. However, for {{math|−1 < ''x'' < 1}}, there is a unique ''real'' number denoted <math>\operatorname{erf}^{-1}(x)</math> satisfying

:<math>\operatorname{erf}\left(\operatorname{erf}^{-1}(x)\right) = x.</math>

The '''inverse error function''' is usually defined with domain (−1,1), and it is restricted to this domain in many computer algebra systems.  However, it can be extended to the disk {{math|{{!}}''z''{{!}} < 1}} of the complex plane, using the Maclaurin series

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

where ''c''<sub>0</sub> = 1 and

:<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},\frac{34807}{16200},\ldots\right\}.</math>

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

:<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>

(After cancellation the numerator/denominator fractions are entries {{oeis|A092676}}/{{oeis|A092677}} in the [[OEIS]]; without cancellation the numerator terms are given in entry {{oeis|A002067}}.) The error function's value at&nbsp;±∞ is equal to&nbsp;±1.

For {{math|{{!}}''z''{{!}} < 1}}, we have <math>\operatorname{erf}\left(\operatorname{erf}^{-1}(z)\right)=z</math>.

The '''inverse complementary error function''' is defined as

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

For ''real'' ''x'', there is a unique ''real'' number <math>\operatorname{erfi}^{-1}(x)</math> satisfying <math>\operatorname{erfi}\left(\operatorname{erfi}^{-1}(x)\right)=x</math>.  The '''inverse imaginary error function''' is defined as <math>\operatorname{erfi}^{-1}(x)</math>.<ref>{{cite arXiv|last1=Bergsma|first1=Wicher|title=On a new correlation coefficient, its orthogonal decomposition and associated tests of independence|eprint=math/0604627|year=2006}}</ref>

For any real ''x'', [[Newton's method]] can be used to compute <math>\operatorname{erfi}^{-1}(x)</math>, and for <math>-1\leq x\leq 1</math>, the following Maclaurin series converges:
:<math>\operatorname{erfi}^{-1}(z)=\sum_{k=0}^\infty\frac{(-1)^k c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1},</math>

where ''c''<sub>''k''</sub> is defined as above.

===Asymptotic expansion===
A useful [[asymptotic expansion]] of the complementary error function (and therefore also of the error function) for large real ''x'' is

:<math>\operatorname{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>

where (2''n''&nbsp;–&nbsp;1)!! is the [[double factorial]] of (2''n''&nbsp;–&nbsp;1), which is the product of all odd numbers up to (2''n''&nbsp;–&nbsp;1). This series diverges for every finite ''x'', and its meaning as asymptotic expansion is that, for any <math>N \in \N</math> one has

:<math>\operatorname{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>

where the remainder, in [[Landau notation]], is

:<math>R_N(x) = O\left(x^{1 - 2N} e^{-x^2}\right)</math>

as <math>x\to\infty.</math>

Indeed, the exact value of the remainder is

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

which follows easily by induction, writing

:<math>e^{-t^2} = -(2t)^{-1}\left(e^{-t^2}\right)'</math>

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'', the above Taylor expansion at 0 provides a very fast convergence).

===Continued fraction expansion===
A [[continued fraction]] expansion of the complementary error function is:<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>\operatorname{erfc}(z) = \frac{z}{\sqrt{\pi}}e^{-z^2} \cfrac{1}{z^2+ \cfrac{a_1}{1+\cfrac{a_2}{z^2+ \cfrac{a_3}{1+\dotsb}}}}\qquad a_m = \frac{m}{2}.</math>

===Integral of error function with Gaussian density function===

:<math>\int_{-\infty}^{\infty} \operatorname{erf} \left(ax+b \right) \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2 \sigma^2} }\,dx= \operatorname{erf} \left[ \frac{a\mu+b}{\sqrt{1+2 a^2 \sigma^2}} \right], \qquad a,b,\mu,\sigma \in \R</math>

===Factorial series===
The inverse [[factorial series]]

:<math>\begin{align}
\operatorname{erfc} z &= \frac{e^{-z^2}}{\sqrt{\pi}\,z} \sum_{n=0}^\infty \frac{(-1)^n Q_n}{{(z^2+1)}^{\bar{n}}}\\
   &= \frac{e^{-z^2}}{\sqrt{\pi}\,z}\left(1 -\frac{1}{2}\frac{1}{(z^2+1)} + \frac{1}{4}\frac{1}{(z^2+1)(z^2+2)} - \cdots \right)
\end{align}</math>

converges for <math>\operatorname{Re}(z^2) > 0.</math> Here

:<math>Q_n \stackrel{\text{def}}{=} \frac{1}{\Gamma(1/2)} \int_0^\infty \tau(\tau-1)\cdots(\tau-n+1)\tau^{-1/2} e^{-\tau} d\tau = \sum_{k=0}^n \left(\frac{1}{2}\right)^{\bar{k}} s(n,k),</math>

<math>z^{\bar{n}}</math> denotes the [[rising factorial]], and <math>s(n,k)</math> denotes a signed [[Stirling number of the first kind]].<ref>{{cite journal
| last       = Schlömilch
| first      = Oskar Xavier
| year       = 1859
| title      = Ueber facultätenreihen
| url        = https://archive.org/details/zeitschriftfrma09runggoog
| journal    = [[:de:Zeitschrift für Mathematik und Physik|Zeitschrift für Mathematik und Physik]]
| volume     = 4
| pages      = 390–415
| access-date = 2017-12-04
| language    = de
| author-link = Oscar Schlömilch
}}</ref><ref>Eq (3) on page 283 of {{cite book 
|last= Nielson
|first= Niels
|date= 1906
|title= Handbuch der theorie der gammafunktion 
|url= https://archive.org/details/handbuchgamma00nielrich 
|location= Leipzig
|publisher= B. G. Teubner
|language= de
| access-date = 2017-12-04
}}</ref>

== Numerical approximations ==

===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:
:: <math>\operatorname{erf}(x) \approx 1 - \frac{1}{(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4)^4}, \qquad x \geq 0</math> 
:(maximum error: 5&times;10<sup>−4</sup>)

:where ''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}, \qquad x \geq 0</math>&nbsp;&nbsp;&nbsp; (maximum error: 2.5&times;10<sup>−5</sup>)

: where ''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}}, \qquad x \geq 0</math>&nbsp;&nbsp;&nbsp; (maximum error: 3&times;10<sup>−7</sup>)

: where ''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&times;10<sup>−7</sup>)

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

: All of these approximations are valid for ''x''&nbsp;≥&nbsp;0.  To use these approximations for negative ''x'', use the fact that erf(x) is an odd function, so erf(''x'')&nbsp;=&nbsp;−erf(−''x'').

* Exponential bounds and a pure exponential approximation for the complementary error function are given by <ref>{{cite journal |url = http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf |last1= Chiani|first1= M.|last2= Dardari|first2= D.|last3= Simon|first3= M.K.|date = 2003 |title = New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels|journal = IEEE Transactions on Wireless Communications|volume = 2|number=4|pages = 840–845| doi=10.1109/TWC.2003.814350|citeseerx= 10.1.1.190.6761}}</ref>

:: <math>\begin{align}
  \operatorname{erfc}(x) &\leq \frac{1}{2}e^{-2 x^2} + \frac{1}{2}e^{- x^2} \leq e^{-x^2}, \qquad x > 0 \\
  \operatorname{erfc}(x) &\approx \frac{1}{6}e^{-x^2} + \frac{1}{2}e^{-\frac{4}{3} x^2}, \qquad x > 0 .
\end{align}</math>

* A tight approximation of the complementary error function for <math>x \in [0,\infty)</math> is given by Karagiannidis & Lioumpas (2007)<ref>Karagiannidis, G. K., & Lioumpas, A. S. [http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf ''An improved approximation for the Gaussian Q-function'']. 2007. IEEE Communications Letters, 11(8), pp. 644-646.</ref> who showed for the appropriate choice of parameters <math>\{A, B\}</math> that

:: <math>\operatorname{erfc}\left(x\right) \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math>

: They determined <math>\{A, B\} = \{1.98, 1.135\},</math> which gave a good approximation for all <math>x \ge 0.</math>

* A single-term lower bound is<ref>{{cite journal|last1=Chang|first1=Seok-Ho|last2=Cosman|first2=Pamela C.|last3=Milstein|first3=Laurence B.|title=Chernoff-Type Bounds for the Gaussian Error Function|journal=IEEE Transactions on Communications|date=November 2011|volume=59|issue=11|pages=2939–2944|doi=10.1109/TCOMM.2011.072011.100049|url=http://escholarship.org/uc/item/6hw4v7pg}}</ref>

:: <math>\operatorname{erfc}(x) \geq \sqrt{\frac{2 e}{\pi}} \frac{\sqrt{\beta - 1}}{\beta} e^{- \beta x^2}, \qquad x \ge 0, \beta > 1,</math>

:where the parameter ''β'' can be picked to minimize error on the desired interval of approximation.

* Another approximation is given by Sergei Winitzki using his "global Padé approximations":<ref>{{cite book |last=Winitzki |first=Serge |date=2003 |title=Lecture Notes in Comput. Sci. |volume=2667 |chapter=Uniform approximations for transcendental functions |chapterurl=https://link.springer.com/chapter/10.1007/3-540-44839-X_82 |location= |publisher=Spronger, Berlin |pages=780–789 |isbn=978-3-540-40155-1 |doi=10.1007/3-540-44839-X_82 |author-link= |url-access=registration |url=https://archive.org/details/computationalsci0000iccs_a2w6 }} (Sect. 3.1 "Error Function of Real Argument erf ''x''")</ref><ref>{{cite journal|last1=Zeng |first1=Caibin |last2=Chen |first2=Yang Cuan |title=Global Padé approximations of the generalized Mittag-Leffler function and its inverse |journal=Fractional Calculus and Applied Analysis |date=2015 |volume=18 |issue=6 |pages=1492–1506 |doi= 10.1515/fca-2015-0086 |url=https://www.degruyter.com/view/j/fca.2015.18.issue-6/fca-2015-0086/fca-2015-0086.xml |quote=Indeed, Winitzki [32] provided the so-called global Padé approximation|arxiv=1310.5592 }}</ref>{{rp|2–3}}

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

: where

:: <math>a = \frac{8(\pi - 3)}{3\pi(4 - \pi)} \approx 0.140012.</math>

: This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the ''relative'' error is less than 0.00035 for all real ''x''. Using the alternate value ''a''&nbsp;≈&nbsp;0.147 reduces the maximum relative error to about 0.00013.<ref>{{Cite web |last=Winitzki |first=Sergei |date=6 February 2008 |title=A handy approximation for the error function and its inverse |url=https://s3.amazonaws.com/academia.edu.documents/35916780/erf-approx.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1551564740&Signature=LV1QBs%2BzZZsP3SP5h1rVwKB5Wqo%3D&response-content-disposition=inline%3B%20filename%3DA_handy_approximation_for_the_error_func.pdf }}</ref>

: This approximation can be inverted to obtain an approximation for the inverse error function:

:: <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>

===Polynomial===
An approximation with a maximal error of <math>1.2\times 10^{-7}</math> for any real argument is:<ref>Numerical Recipes in Fortran 77: The Art of Scientific Computing ({{isbn|0-521-43064-X}}), 1992, page 214, Cambridge University Press.</ref>

:<math>\operatorname{erf}(x)=\begin{cases}
1-\tau &  x\ge 0\\
\tau-1 &  x < 0
\end{cases}</math>

with

:<math>\begin{align}
\tau &= t\cdot\exp\left(-x^2-1.26551223+1.00002368 t+0.37409196 t^2+0.09678418 t^3 -0.18628806 t^4\right.\\
 &\left. \qquad\qquad\qquad +0.27886807 t^5-1.13520398 t^6+1.48851587 t^7 -0.82215223 t^8+0.17087277 t^9\right)
\end{align}</math>

and

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

===Table of values===
{{further|Interval estimation|Coverage probability|68–95–99.7 rule}}
{| class="wikitable" style="text-align:left;margin-left:24pt"
|- "
! x !! erf(x) !! 1-erf(x)
|-
|0 || {{val|0}}  || {{val|1}}
|-
|0.02 || {{val|0.022564575}}  || {{val|0.977435425}}
|-
|0.04 || {{val|0.045111106}}  || {{val|0.954888894}}
|-
|0.06 || {{val|0.067621594}}  || {{val|0.932378406}}
|-
|0.08 || {{val|0.090078126}}  || {{val|0.909921874}}
|-
|0.1 || {{val|0.112462916}}  || {{val|0.887537084}}
|-
|0.2 || {{val|0.222702589}}  || {{val|0.777297411}}
|-
|0.3 || {{val|0.328626759}}  || {{val|0.671373241}}
|-
|0.4 || {{val|0.428392355}}  || {{val|0.571607645}}
|-
|0.5 || {{val|0.520499878}}  || {{val|0.479500122}}
|-
|0.6 || {{val|0.603856091}}  || {{val|0.396143909}}
|-
|0.7 || {{val|0.677801194}}  || {{val|0.322198806}}
|-
|0.8 || {{val|0.742100965}}  || {{val|0.257899035}}
|-
|0.9 || {{val|0.796908212}}  || {{val|0.203091788}}
|-
|1 || {{val|0.842700793}}  || {{val|0.157299207}}
|-
|1.1 || {{val|0.88020507}}  || {{val|0.11979493}}
|-
|1.2 || {{val|0.910313978}}  || {{val|0.089686022}}
|-
|1.3 || {{val|0.934007945}}  || {{val|0.065992055}}
|-
|1.4 || {{val|0.95228512}}  || {{val|0.04771488}}
|-
|1.5 || {{val|0.966105146}}  || {{val|0.033894854}}
|-
|1.6 || {{val|0.976348383}}  || {{val|0.023651617}}
|-
|1.7 || {{val|0.983790459}}  || {{val|0.016209541}}
|-
|1.8 || {{val|0.989090502}}  || {{val|0.010909498}}
|-
|1.9 || {{val|0.992790429}}  || {{val|0.007209571}}
|-
|2 || {{val|0.995322265}}  || {{val|0.004677735}}
|-
|2.1 || {{val|0.997020533}}  || {{val|0.002979467}}
|-
|2.2 || {{val|0.998137154}}  || {{val|0.001862846}}
|-
|2.3 || {{val|0.998856823}}  || {{val|0.001143177}}
|-
|2.4 || {{val|0.999311486}}  || {{val|0.000688514}}
|-
|2.5 || {{val|0.999593048}}  || {{val|0.000406952}}
|-
|3 || {{val|0.99997791}}  || {{val|0.00002209}}
|-
|3.5 || {{val|0.999999257}} ||  {{val|0.000000743}}
|-
|}

==Related functions==

===Complementary error function===
The '''complementary error function''', denoted <math>\mathrm{erfc}</math>, is defined as

:<math>\begin{align}
             \operatorname{erfc}(x) & = 1-\operatorname{erf}(x) \\[5pt]
                                    & = \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\, dt \\[5pt]
                                    & = e^{-x^2} \operatorname{erfcx}(x),
       \end{align} </math>

which also defines <math>\mathrm{erfcx}</math>, the '''scaled complementary error function'''<ref name=Cody93>{{Citation |first=W. J. |last=Cody |title=Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers |url=http://www.stat.wisc.edu/courses/st771-newton/papers/p22-cody.pdf |journal=[[ACM Trans. Math. Softw.]] |volume=19 |issue=1 |pages=22–32 |date=March 1993 |doi=10.1145/151271.151273|citeseerx=10.1.1.643.4394 }}</ref> (which can be used instead of erfc to avoid [[arithmetic underflow]]<ref name=Cody93/><ref name=Zaghloul07>{{Citation |first=M. R. |last=Zaghloul |title=On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=375 |issue=3 |pages=1043–1048 |date=March 1, 2007 |url=http://mnras.oxfordjournals.org/content/375/3/1043 |doi=10.1111/j.1365-2966.2006.11377.x}}</ref>). Another form of <math>\operatorname{erfc}(x)</math> for non-negative <math>x</math> is known as Craig's formula, after its discoverer:<ref>John W. Craig, [http://wsl.stanford.edu/~ee359/craig.pdf ''A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations''] {{Webarchive|url=https://web.archive.org/web/20120403231129/http://wsl.stanford.edu/~ee359/craig.pdf |date=3 April 2012 }}, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.</ref>
:<math>
\operatorname{erfc} (x \mid x\ge 0) = \frac 2 \pi \int_0^{\pi/2} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) \, d\theta.</math>

This expression is valid only for positive values of ''x'', but it can be used in conjunction with erfc(''x'') = 2 − erfc(−''x'') to obtain erfc(''x'') for negative values. This form is advantageous in that the range of integration is fixed and finite.

===Imaginary error function===
The '''imaginary error function''', denoted ''erfi'', is defined as

:<math>\begin{align}
             \operatorname{erfi}(x) & = -i\operatorname{erf}(ix) \\[5pt]
                                    & = \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,dt \\[5pt]
                                    & = \frac{2}{\sqrt{\pi}} e^{x^2} D(x),
       \end{align} </math>

where ''D''(''x'') is the [[Dawson function]] (which can be used instead of erfi to avoid [[arithmetic overflow]]<ref name=Cody93/>).

Despite the name "imaginary error function", <math>\operatorname{erfi}(x)</math> is real when ''x'' is real.

When the error function is evaluated for arbitrary [[complex number|complex]] arguments ''z'', the resulting '''complex error function''' is usually discussed in scaled form as the [[Faddeeva function]]:

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

===Cumulative distribution function===
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,

: <math>\Phi(x) =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^\tfrac{-t^2}{2}\,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>

or rearranged for erf and erfc:

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

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

:<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>

The [[inverse function|inverse]] of <math>\Phi</math> is known as the [[Quantile function|normal quantile function]], or [[probit]] function and may be expressed in terms of the inverse error function as

:<math>\operatorname{probit}(p) = \Phi^{-1}(p) = \sqrt{2}\operatorname{erf}^{-1}(2p-1) = -\sqrt{2}\operatorname{erfc}^{-1}(2p).</math>

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):

:<math>\operatorname{erf}(x) = \frac{2x}{\sqrt{\pi}} M\left(\frac12,\frac32,-x^2\right).</math>

It has a simple expression in terms of the [[Fresnel integral]].{{Elucidate|date=May 2012}}

In terms of the [[regularized gamma function]] P and the [[incomplete gamma function]],

:<math>\operatorname{erf}(x) = \sgn(x) P\left(\frac12, x^2\right) = \frac{\sgn(x)}{\sqrt{\pi}}\gamma\left(\frac12, x^2\right).</math>

<math>\sgn(x)</math> is the [[sign function]].

===Generalized error functions===
[[File:Error Function Generalised.svg|right|thumb|400px|Graph of generalised error functions ''E<sub>n</sub>''(''x''):<br />
grey curve: ''E''<sub>1</sub>(''x'') = (1&nbsp;−&nbsp;e<sup>&nbsp;−''x''</sup>)/<math>\scriptstyle\sqrt{\pi}</math><br />
red curve: ''E''<sub>2</sub>(''x'') = erf(''x'')<br />
green curve: ''E''<sub>3</sub>(''x'')<br />
blue curve: ''E''<sub>4</sub>(''x'')<br />
gold curve: ''E''<sub>5</sub>(''x'').]]
Some authors discuss the more general functions:{{citation needed|date=August 2011}}

:<math>E_n(x) = \frac{n!}{\sqrt{\pi}} \int_0^x e^{-t^n}\,dt =\frac{n!}{\sqrt{\pi}}\sum_{p=0}^\infty(-1)^p\frac{x^{np+1}}{(np+1)p!}.
</math>

Notable cases are:

*''E''<sub>0</sub>(''x'') is a straight line through the origin: <math>\textstyle E_0(x)=\dfrac{x}{e \sqrt{\pi}}</math>
*''E''<sub>2</sub>(''x'') is the error function, erf(''x'').

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

These generalised functions can equivalently be expressed for ''x''&nbsp;>&nbsp;0 using the [[gamma function]] and [[incomplete gamma function]]:

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

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

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

===Iterated integrals of the complementary error function===
The iterated integrals of the complementary error function are defined by<ref>{{citation|last=Carslaw|first=H. S.|author1-link=Horatio Scott Carslaw|last2=Jaeger|first2=J. C.|author2-link=John Conrad Jaeger|year=1959|title=Conduction of Heat in Solids|edition=2nd|publisher=Oxford University Press|isbn=978-0-19-853368-9}}, p 484</ref>

:<math>\begin{align}
\operatorname{i^{n}erfc}(z) &= \int_z^\infty  \operatorname{i^{n-1}erfc}(\zeta)\, d\zeta \\
\operatorname{i^{0}erfc}(z) &= \operatorname{erfc}(z) \\
\operatorname{i^{1}erfc}(z) &= \operatorname{ierfc}(z) = \frac{1}{\sqrt\pi} e^{-z^2} - z \operatorname{erfc}(z) \\
\operatorname{i^{2}erfc}(z) &= \frac{1}{4} \left[ \operatorname{erfc}(z) -2 z \operatorname{ierfc}(z) \right] \\
\end{align}</math>

The general recurrence formula is

:<math>2 n \operatorname{i^{n}erfc}(z) = \operatorname{i^{n-2}erfc}(z) -2 z \operatorname{i^{n-1}erfc}(z)</math>

They have the power series

:<math>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>

from which follow the symmetry properties

:<math>i^{2m} \operatorname{erfc} (-z) =-i^{2m} \operatorname{erfc}(z) +\sum_{q=0}^m \frac{z^{2q}}{2^{2(m-q)-1}(2q)! (m-q)!}</math>

and

:<math>i^{2m+1} \operatorname{erfc}(-z) =i^{2m+1} \operatorname{erfc}(z) +\sum_{q=0}^m \frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)! (m-q)!}.
</math>

==See also==

=== Related functions ===
* [[Gaussian integral]], over the whole real line
* [[Gaussian function]], derivative
* [[Dawson function]], renormalized imaginary error function
* [[Goodwin–Staton integral]]

===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

==References==
{{Reflist|30em}}

==Further reading==
* {{AS ref |7|297}}
*{{Citation |last1=Press |first1=William H. |last2=Teukolsky |first2=Saul A. |last3=Vetterling |first3=William T. |last4=Flannery |first4=Brian P. |year=2007 |title=Numerical Recipes: The Art of Scientific Computing |edition=3rd |publisher=Cambridge University Press |location=New York |isbn=978-0-521-88068-8 |chapter=Section 6.2. Incomplete Gamma Function and Error Function |chapter-url=http://apps.nrbook.com/empanel/index.html#pg=259 }}
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=Nico M. |last=Temme }}

==External links==
* [http://mathworld.wolfram.com/Erf.html MathWorld – Erf]
* [http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf  A Table of Integrals of the Error Functions]
{{Authority control}}
{{Nonelementary Integral}}
{{DEFAULTSORT:Error Function}}
[[Category:Special hypergeometric functions]]
[[Category:Gaussian function]]
[[Category:Functions related to probability distributions]]
[[Category:Analytic functions]]