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{{Short description|Sigmoid shape special function}}
{{Use dmy dates|date=March 2023}}
{{Distinguish|Loss function}}
In mathematics, the '''error function''' (also called the '''Gauss error function'''), often denoted by '''{{math|erf}}''', is a function <math>\mathrm{erf}: \mathbb{C} \to \mathbb{C}</math> defined as:<ref>{{cite book|last =Andrews|first = Larry C.|url = https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110 |title = Special functions of mathematics for engineers|page = 110|publisher = SPIE Press |date= 1998|isbn = 9780819426161}}</ref>
<math display="block">\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt.</math>
{{Infobox mathematical function
| name = Error function
| image = Error Function.svg
| imagesize = 400px
| imagealt = Plot of the error function over real numbers
| caption = Plot of the error function over real numbers
| general_definition = <math>\operatorname{erf}(z) = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,\mathrm dt</math>
| fields_of_application = Probability, thermodynamics, digital communications
| domain = <math>\mathbb{C}</math>
| range = <math>\left( -1,1 \right)</math>
| parity = Odd
| root = 0
| derivative = <math>\frac{d}{dz}\operatorname{erf}(z) = \frac{2}{\sqrt\pi} e^{-z^2} </math>
| antiderivative = <math>\int \operatorname{erf}(z)\,dz = z \operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt\pi} + C</math>
| taylor_series = <math>\operatorname{erf}(z) = \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-1)^n}{2n+1} \frac{z^{2n+1}}{n!}</math>
}}

The integral here is a complex [[Contour integration|contour integral]] which is path-independent because <math>\exp(-t^2)</math> is [[Holomorphic function|holomorphic]] on the whole complex plane <math>\mathbb{C}</math>. In many applications, the function argument is a [[real number]], in which case the function value is also real.

In some old texts,<ref>{{cite book |last1=Whittaker |first1=Edmund Taylor |title=A Course of Modern Analysis |title-link=A Course of Modern Analysis |last2=Watson |first2=George Neville |date=2021 |publisher=[[Cambridge University Press]] |isbn=978-1-316-51893-9 |editor-last=Moll |editor-first=Victor Hugo |editor-link=Victor Hugo Moll |edition=5th revised |page=358 |authorlink1=Edmund T. Whittaker |authorlink2=George N. Watson}}</ref>
the error function is defined without the factor of <math>\frac{2}{\sqrt{\pi}}</math>.
This [[nonelementary integral]] is a [[sigmoid function|sigmoid]] function that occurs often in [[probability]], [[statistics]], and [[partial differential equation]]s. 

In statistics, for non-negative real values of {{mvar|x}}, the error function has the following interpretation: for a real [[random variable]] {{mvar|Y}} that is [[normal distribution|normally distributed]] with [[mean]] 0 and [[standard deviation]] <math>\frac{1}{\sqrt{2}}</math>, {{math|erf(''x'')}} is the probability that {{mvar|Y}} falls in the range {{closed-closed|−''x'', ''x''}}.

Two closely related functions are the '''complementary error function''' <math>\mathrm{erfc}: \mathbb{C} \to \mathbb{C}</math> is defined as

<math display="block">\operatorname{erfc}(z) = 1 - \operatorname{erf}(z),</math>

and the '''imaginary error function''' <math>\mathrm{erfi}: \mathbb{C} \to \mathbb{C}</math> is defined as

<math display="block">\operatorname{erfi}(z) = -i\operatorname{erf}(iz),</math>

where {{mvar|i}} is the [[imaginary unit]].

==Name==
The name "error function" and its abbreviation {{math|erf}} were proposed by [[James Whitbread Lee Glaisher|J.&nbsp;W.&nbsp;L.&nbsp;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 |access-date=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 |access-date=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 display="block">
 f(x) = \left(\frac{c}{\pi}\right)^{1/2} e^{-c x^2}
</math>
(the [[normal distribution]]), Glaisher calculates the probability of an error lying between {{mvar|p}} and {{mvar|q}} as
<math display="block">
 \left(\frac{c}{\pi}\right)^\frac{1}{2} \int_p^qe^{-cx^2}\,dx =
 \frac{1}{2} \big(\operatorname{erf}(q\sqrt{c}) - \operatorname{erf}(p\sqrt{c})\big).
</math>

==Applications==
When the results of a series of measurements are described by a [[normal distribution]] with [[standard deviation]] {{mvar|σ}} and [[expected value]] 0, then {{math|erf ({{sfrac|''a''|''σ'' {{sqrt|2}}}})}} is the probability that the error of a single measurement lies between {{math|−''a''}} and {{math|+''a''}}, for positive {{mvar|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 a random variable {{math|''X'' ~ Norm[''μ'',''σ'']}} (a normal distribution with mean {{mvar|μ}} and standard deviation {{mvar|σ}}) and a constant {{math|''L'' > ''μ''}}, it can be shown via [[integration by substitution]]:
<math display="block">\begin{align}
\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)
\end{align}</math>

where {{mvar|A}} and {{mvar|B}} are certain numeric constants. If {{mvar|L}} is sufficiently far from the mean, specifically {{math|''μ'' − ''L'' ≥ ''σ''{{sqrt|ln(''k'')}}}}, then:

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

so the probability goes to 0 as {{math|''k'' → ∞}}.

The probability for {{mvar|X}} being in the interval {{closed-closed|''L<sub>a</sub>'', ''L<sub>b</sub>''}} can be derived as
<math display="block">\begin{align}
\Pr[L_a\leq X \leq L_b] &= \int_{L_a}^{L_b} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \, dx \\
&= \frac{1}{2}\left(\operatorname{erf}\left(\frac{L_b-\mu}{\sqrt{2}\sigma}\right) - \operatorname{erf}\left(\frac{L_a-\mu}{\sqrt{2}\sigma}\right)\right).\end{align}</math>

==Properties==
{{multiple image
 | header    =  Plots in the complex plane
 | direction = vertical
 | width     = 250
 | image1    = ComplexExp2.png
 | caption1  = Integrand {{math|exp(−''z''<sup>2</sup>)}}
 | image2    = ComplexErfz.png
 | caption2  = {{math|erf(''z'')}}
}}

The property {{math|1=erf (−''z'') = −erf(''z'')}} means that the error function is an [[even and odd functions|odd function]]. This directly results from the fact that the integrand {{math|''e''<sup>−''t''<sup>2</sup></sup>}} is an [[even function]] (the antiderivative of an even function which is zero at the origin is an odd function and vice versa).

Since the error function is an [[entire function]] which takes real numbers to real numbers, for any [[complex number]] {{mvar|z}}:
<math display="block">\operatorname{erf}(\overline{z}) = \overline{\operatorname{erf}(z)} </math>
where <math>\overline{z}
</math> denotes the [[complex conjugate]] of <math>z</math>.

The integrand {{math|1=''f'' = exp(−''z''<sup>2</sup>)}} and {{math|1=''f'' = erf(''z'')}} are shown in the complex {{mvar|z}}-plane in the figures at right with [[domain coloring]].

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

===Taylor series===
The error function is an [[entire function]]; it has no singularities (except that at infinity) and its [[Taylor expansion]] always converges. For {{math|''x'' >> 1}}, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated in [[Closed-form expression|closed form]] in terms of [[Elementary function (differential algebra)|elementary functions]] (see [[Liouville's theorem (differential algebra)|Liouville's theorem]]), but by expanding the [[integrand]] {{math|''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 display="block">\begin{align}
\operatorname{erf}(z)
&= \frac{2}{\sqrt\pi}\sum_{n=0}^\infty\frac{(-1)^n z^{2n+1}}{n! (2n+1)} \\[6pt]
&= \frac{2}{\sqrt\pi} \left(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\cdots\right)
\end{align}</math>
which holds for every [[complex number]]&nbsp;{{mvar|z}}. The denominator terms are sequence [[oeis:A007680|A007680]] in the [[OEIS]].

It is a special case of [[Confluent_hypergeometric_function|Kummer's function]]:

<math>
\operatorname{erf}(z) = \frac{2z}{\surd \pi}{}_1F_1(1/2;3/2;-z^2).
</math>

For iterative calculation of the above series, the following alternative formulation may be useful:
<math display="block">\begin{align}
\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) \\[6pt]
&= \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{z}{2n+1} \prod_{k=1}^n \frac{-z^2}{k}
\end{align}</math>
because {{math|{{sfrac|−(2''k'' − 1)''z''<sup>2</sup>|''k''(2''k'' + 1)}}}} expresses the multiplier to turn the {{mvar|k}}th term into the {{math|(''k''&nbsp;+&nbsp;1)}}th term (considering {{mvar|z}} as the first term).

The imaginary error function has a very similar Maclaurin series, which is:
<math display="block">\begin{align}
\operatorname{erfi}(z)
 &= \frac{2}{\sqrt\pi}\sum_{n=0}^\infty\frac{z^{2n+1}}{n! (2n+1)} \\[6pt]
 &=\frac{2}{\sqrt\pi} \left(z+\frac{z^3}{3}+\frac{z^5}{10}+\frac{z^7}{42}+\frac{z^9}{216}+\cdots\right)
\end{align}</math>
which holds for every [[complex number]]&nbsp;{{mvar|z}}.

===Derivative and integral===
The derivative of the error function follows immediately from its definition:
<math display="block">\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 display="block">\frac{d}{dz}\operatorname{erfi}(z) =\frac{2}{\sqrt\pi} e^{z^2}.</math>Higher order derivatives are given by
<math display="block">\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 {{mvar|H}} are the physicists' [[Hermite polynomials]].<ref>{{mathworld|title=Erf|urlname=Erf}}</ref>

An [[antiderivative]] of the error function, obtainable by [[integration by parts]], is
<math display="block">\int \operatorname{erf}(z) dz = z\operatorname{erf}(z) + \frac{e^{-z^2}}{\sqrt\pi}+C.</math>
An antiderivative of the imaginary error function, also obtainable by integration by parts, is
<math display="block">\int \operatorname{erfi}(z) dz = z\operatorname{erfi}(z) - \frac{e^{z^2}}{\sqrt\pi}+C.</math>

===Bürmann series===

An expansion,<ref>{{cite journal|first1=H. M. |last1=Schöpf |first2=P. H. |last2=Supancic |title=On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion |journal=The Mathematica Journal |year=2014 |volume=16 |doi=10.3888/tmj.16-11 |url=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/|doi-access=free }}</ref> which converges more rapidly for all real values of {{mvar|x}} than a Taylor expansion, is obtained by using [[Hans Heinrich Bürmann]]'s theorem:<ref>{{mathworld|urlname=BuermannsTheorem | title = Bürmann's Theorem }}</ref>
<math display="block">\begin{align}
\operatorname{erf}(x)
&= \frac{2}{\sqrt\pi} \sgn(x) \cdot \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) \cdot \sqrt{1-e^{-x^2}} \left(\frac{\sqrt\pi}{2} + \sum_{k=1}^\infty c_k e^{-kx^2} \right).
\end{align}</math>
where {{math|sgn}} is the [[sign function]]. By keeping only the first two coefficients and choosing {{math|1=''c''<sub>1</sub> = {{sfrac|31|200}}}} and {{math|1=''c''<sub>2</sub> = −{{sfrac|341|8000}}}}, the resulting approximation shows its largest [[Approximation error|relative error]] at {{math|1=''x'' = ±1.40587}}, where it is less than 0.0034361:
<math display="block">\operatorname{erf}(x) \approx \frac{2}{\sqrt\pi}\sgn(x) \cdot \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 a complex number {{mvar|z}}, there is not a ''unique'' complex number {{mvar|w}} satisfying {{math|1=erf(''w'') = ''z''}}, so a true inverse function would be multivalued. However, for {{math|−1 < ''x'' < 1}}, there is a unique ''real'' number denoted {{math|erf<sup>−1</sup>(''x'')}} satisfying
<math display="block">\operatorname{erf}\left(\operatorname{erf}^{-1}(x)\right) = x.</math>

The '''inverse error function''' is usually defined with domain {{open-open|−1,1}}, and it is restricted to this domain in many [[computer algebra]] systems.  However, it can be extended to the disk {{math|{{abs|''z''}} < 1}} of the complex plane, using the Maclaurin series<ref>{{cite arXiv | last1 = Dominici | first1 = Diego | title = Asymptotic analysis of the derivatives of the inverse error function | eprint = math/0607230 | year = 2006 }}</ref>
<math display="block">\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt\pi}{2}z\right )^{2k+1},</math>
where {{math|1=''c''<sub>0</sub> = 1}} and
<math display="block">\begin{align}
c_k & =\sum_{m=0}^{k-1}\frac{c_m c_{k-1-m}}{(m+1)(2m+1)} \\[1ex]
&= \left\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\frac{34807}{16200},\ldots\right\}.
\end{align}</math>

So we have the series expansion (common factors have been canceled from numerators and denominators):
<math display="block">\operatorname{erf}^{-1}(z) = \frac{\sqrt{\pi}}{2} \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 and denominator values in {{oeis|A092676}} and {{oeis|A092677}} respectively; without cancellation the numerator terms are values in {{oeis|A002067}}.) The error function's value at&nbsp;{{math|±∞}} is equal to&nbsp;{{math|±1}}.

For {{math|{{abs|''z''}} < 1}}, we have {{math|1=erf(erf<sup>−1</sup>(''z'')) = ''z''}}.

The '''inverse complementary error function''' is defined as
<math display="block">\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1}(z).</math>
For real {{mvar|x}}, there is a unique ''real'' number {{math|erfi<sup>−1</sup>(''x'')}} satisfying {{math|1=erfi(erfi<sup>−1</sup>(''x'')) = ''x''}}.  The '''inverse imaginary error function''' is defined as {{math|erfi<sup>−1</sup>(''x'')}}.<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|erfi<sup>−1</sup>(''x'')}}, and for {{math|−1 ≤ ''x'' ≤ 1}}, the following Maclaurin series converges:
<math display="block">\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 {{math|''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 {{mvar|x}} is
<math display="block">\begin{align}
\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)}{\left(2x^2\right)^n}\right) \\[6pt] 
&= \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n},
\end{align}</math>
where {{math|(2''n'' − 1)!!}} is the [[double factorial]] of {{math|(2''n'' − 1)}}, which is the product of all odd numbers up to {{math|(2''n'' − 1)}}. This series diverges for every finite {{mvar|x}}, and its meaning as asymptotic expansion is that for any integer {{math|''N'' ≥ 1}} one has
<math display="block">\operatorname{erfc}(x) = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^{N-1} (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n} + R_N(x)</math>
where the remainder is
<math display="block">R_N(x) := \frac{(-1)^N \, (2 N - 1)!!}{\sqrt{\pi} \cdot 2^{N - 1}} \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,</math>
which follows easily by induction, writing
<math display="block">e^{-t^2} = -\frac{1}{2 t} \, \frac{\mathrm{d}}{\mathrm{d}t} e^{-t^2}</math>
and integrating by parts.

The asymptotic behavior of the remainder term, in [[Landau notation]], is
<math display="block">R_N(x) = O\left(x^{- (1 + 2N)} e^{-x^2}\right)</math>
as {{math|''x'' → ∞}}. This can be found by 
<math display="block">R_N(x) \propto \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt = e^{-x^2} \int_0^\infty (t+x)^{-2N}e^{-t^2-2tx}\,\mathrm dt\leq e^{-x^2} \int_0^\infty x^{-2N} e^{-2tx}\,\mathrm dt \propto x^{-(1+2N)}e^{-x^2}.</math>
For large enough values of {{mvar|x}}, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of {{math|erfc ''x''}} (while for not too large values of {{mvar|x}}, the above Taylor expansion at 0 provides a very fast convergence).

===Continued fraction expansion===
A [[continued fraction]] expansion of the complementary error function was found by [[Pierre-Simon Laplace|Laplace]]:<ref>[[Pierre-Simon Laplace]], [[Traité de mécanique céleste]], tome 4 (1805), livre X, page 255.</ref><ref>{{cite book| last1 = Cuyt | first1 = Annie A. M.|author1-link= Annie Cuyt | 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 display="block">\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>

===Factorial series===
The inverse [[factorial series]]:
<math display="block">\begin{align}
\operatorname{erfc}(z)
&= \frac{e^{-z^2}}{\sqrt{\pi}\,z} \sum_{n=0}^\infty \frac{\left(-1\right)^n Q_n}{{\left(z^2+1\right)}^{\bar{n}}} \\[1ex]
&= \frac{e^{-z^2}}{\sqrt{\pi}\,z} \left[1 -\frac{1}{2}\frac{1}{(z^2+1)} + \frac{1}{4}\frac{1}{\left(z^2+1\right) \left(z^2+2\right)} - \cdots \right]
\end{align}</math>
converges for {{math|Re(''z''<sup>2</sup>) > 0}}. Here
<math display="block">\begin{align}
Q_n
&\overset{\text{def}}{{}={}}
\frac{1}{\Gamma{\left(\frac{1}{2}\right)}} \int_0^\infty \tau(\tau-1)\cdots(\tau-n+1)\tau^{-\frac{1}{2}} e^{-\tau} \,d\tau \\[1ex]
&= \sum_{k=0}^n \left(\frac{1}{2}\right)^{\bar{k}} s(n,k),
\end{align}</math>
{{math|''z''<sup>{{overline|''n''}}</sup>}} denotes the [[rising factorial]], and {{math|''s''(''n'',''k'')}} denotes a signed [[Stirling number of the first kind]].<ref>{{cite journal|last=Schlömilch|first=Oskar Xavier | author-link=Oscar Schlömilch|year=1859|title=Ueber facultätenreihen|url=https://archive.org/details/zeitschriftfrma09runggoog | journal=[[:de:Zeitschrift für Mathematik und Physik|Zeitschrift für Mathematik und Physik]] | language=de | volume=4 | pages=390–415}}</ref><ref>{{cite book | last=Nielson | first=Niels | url=https://archive.org/details/handbuchgamma00nielrich | title=Handbuch der Theorie der Gammafunktion | date=1906 | publisher=B. G. Teubner | location=Leipzig|language=de|access-date=2017-12-04|at=p. 283 Eq. 3}}</ref>
The Taylor series can be written in terms of the [[double factorial]]:
<math display="block">\operatorname{erf}(z) =  \frac{2}{\sqrt\pi} \sum_{n=0}^\infty \frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1}</math>

== Bounds and 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 display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + a_3x^3 + a_4x^4\right)^4}, \qquad x \geq 0</math>
(maximum error: {{val|5e-4}})
{{pb}}
where {{math|''a''<sub>1</sub> {{=}} 0.278393}}, {{math|''a''<sub>2</sub> {{=}} 0.230389}}, {{math|''a''<sub>3</sub> {{=}} 0.000972}}, {{math|''a''<sub>4</sub> {{=}} 0.078108}}

<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + a_3t^3\right)e^{-x^2},\quad t=\frac{1}{1 + px}, \qquad x \geq 0</math>
(maximum error: {{val|2.5e-5}})
{{pb}}
where {{math|''p'' {{=}} 0.47047}}, {{math|''a''<sub>1</sub> {{=}} 0.3480242}}, {{math|''a''<sub>2</sub> {{=}} −0.0958798}}, {{math|''a''<sub>3</sub> {{=}} 0.7478556}}

<math display="block">\operatorname{erf}(x) \approx 1 - \frac{1}{\left(1 + a_1x + a_2x^2 + \cdots + a_6x^6\right)^{16}}, \qquad x \geq 0</math>
(maximum error: {{val|3e-7}})
{{pb}}
where {{math|''a''<sub>1</sub> {{=}} 0.0705230784}}, {{math|''a''<sub>2</sub> {{=}} 0.0422820123}}, {{math|''a''<sub>3</sub> {{=}} 0.0092705272}}, {{math|''a''<sub>4</sub> {{=}} 0.0001520143}}, {{math|''a''<sub>5</sub> {{=}} 0.0002765672}}, {{math|''a''<sub>6</sub> {{=}} 0.0000430638}}

<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5\right)e^{-x^2},\quad t = \frac{1}{1 + px}</math>
(maximum error: {{val|1.5e-7}})
{{pb}}
where {{math|''p'' {{=}} 0.3275911}}, {{math|''a''<sub>1</sub> {{=}} 0.254829592}}, {{math|''a''<sub>2</sub> {{=}} −0.284496736}}, {{math|''a''<sub>3</sub> {{=}} 1.421413741}}, {{math|''a''<sub>4</sub> {{=}} −1.453152027}}, {{math|''a''<sub>5</sub> {{=}} 1.061405429}}
{{pb}}

One can improve the accuracy of the A&S approximation by extending it with three extra parameters,
<math display="block">\operatorname{erf}(x) \approx 1 - \left(a_1t + a_2t^2 + \cdots + a_5t^5+a_6t^6+a_7t^7\right)e^{-x^2},\quad t = \frac{1}{1 + p_1x+p_2x^2}</math>
where p1 = 0.406742016006509,
p2 = 0.0072279182302319,
a1 = 0.316879890481381,
a2 = -0.138329314150635,
a3 = 1.08680830347054,
a4 = -1.11694155120396,
a5 = 1.20644903073232,
a6 = -0.393127715207728,
a7 = 0.0382613542530727.
The maximum error of this approximation is about {{val|2e-9}}. The parameters are obtained by fitting the extended approximation to the accurate values of the error function using the following Python code.
{{collapse top|title=Python code to fit extended A&S approximation|collapsed=yes}}
<syntaxhighlight lang="python">
import numpy as np
from math import erf, exp, sqrt
from scipy.optimize import least_squares

#
# Extended A&S approximation:
# erf(x) ≈ 1 − t * exp(−x^2) * (a1 + a2*t + a3*t^2 + ... + a7*t^6)
# where now
#      t = 1 / (1 + p1*x + p2*x^2)
# We fit parameters p1, p2, a1..a7 over x in [0, 10].
#


def approx_erf(params, x):
    p1 = params[0]
    p2 = params[1]
    a = params[2:]

    t = 1.0 / (1.0 + p1 * x + p2 * x * x)

    poly = np.zeros_like(x)
    tt = np.ones_like(x)  # t^0

    # polynomial: a1*t^0 + a2*t^1 + ... + a7*t^6
    for ak in a:
        poly += ak * tt
        tt *= t

    return 1.0 - t * np.exp(-x * x) * poly


def residuals(params, xs, ys):
    return approx_erf(params, xs) - ys


#
# Prepare data for fitting
#

N = 300
xmin = 0
xmax = 10
xs = np.linspace(xmin, xmax, N)
ys = np.array([erf(x) for x in xs], dtype=float)

#
# Initial guess for parameters
# Start from original A&S values and extend them conservatively
#

p1_0 = 0.3275911  # original A&S p
p2_0 = 0.0  # new denominator parameter

# original A&S 5 coefficients, add two => 7 in total
a0 = [
    0.254829592,
    -0.284496736,
    1.421413741,
    -1.453152027,
    1.061405429,
    0.0,  # new term
    0.0,  # another new term
]

params0 = np.array([p1_0, p2_0] + a0, dtype=float)

#
# Fit using nonlinear least squares (Levenberg–Marquardt)
#

result = least_squares(
    residuals, params0, args=(xs, ys), xtol=1e-14, ftol=1e-14, gtol=1e-14, max_nfev=5000
)

params = result.x
p1_fit = params[0]
p2_fit = params[1]
a_fit = params[2:]

#
# Print fitted parameters
#

print("\nFitted parameters:")
print(f"p1 = {p1_fit:.15g},")
print(f"p2 = {p2_fit:.15g},")
for i, ai in enumerate(a_fit, 1):
    print(f"a{i} = {ai:.15g},")

#
# Evaluate approximation error
#

approx_vals = approx_erf(params, xs)
abs_err = np.abs(approx_vals - ys)

print(f"\nMaximum absolute error on [{xmin},{xmax}]:", np.max(abs_err))
print("RMS error:", np.sqrt(np.mean(abs_err**2)))
print("Done.")
</syntaxhighlight>
{{collapse bottom}}


All of these approximations are valid for {{math|''x'' ≥ 0}}.  To use these approximations for negative {{mvar|x}}, use the fact that {{math|erf(''x'')}} is an odd function, so {{math|erf(''x'') {{=}} −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 |bibcode= 2003ITWC....2..840C| citeseerx= 10.1.1.190.6761}}</ref>
<math display="block">\begin{align}
  \operatorname{erfc}(x) &\leq \frac{1}{2}e^{-2 x^2} + \frac{1}{2}e^{- x^2} \leq e^{-x^2}, &\quad x &> 0 \\[1.5ex]
  \operatorname{erfc}(x) &\approx \frac{1}{6}e^{-x^2} + \frac{1}{2}e^{-\frac{4}{3} x^2}, &\quad x &> 0 .
\end{align}</math>


The above have been generalized to sums of {{mvar|N}} exponentials<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902 |title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications |year=2020 |last1=Tanash |first1=I.M. |last2=Riihonen |first2=T. |volume=68 |issue=10 |pages=6514–6524 |arxiv=2007.06939 |bibcode=2020ITCom..68.6514T |s2cid=220514754}}</ref> with increasing accuracy in terms of {{mvar|N}} so that {{math|erfc(''x'')}} can be accurately approximated or bounded by {{math|2''Q̃''({{sqrt|2}}''x'')}}, where
<math display="block">\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math>
In particular, there is a systematic methodology to solve the numerical coefficients {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} that yield a [[minimax approximation algorithm|minimax]] approximation or bound for the closely related [[Q-function]]: {{math|''Q''(''x'') ≈ ''Q̃''(''x'')}}, {{math|''Q''(''x'') ≤ ''Q̃''(''x'')}}, or {{math|''Q''(''x'') ≥ ''Q̃''(''x'')}} for {{math|''x'' ≥ 0}}. The coefficients {{math|{(''a<sub>n</sub>'',''b<sub>n</sub>'')}{{su|b=''n'' {{=}} 1|p=''N''}}}} for many variations of the exponential approximations and bounds up to {{math|''N'' {{=}} 25}} have been released to open access as a comprehensive dataset.<ref>{{cite journal | doi=10.5281/zenodo.4112978 | title=Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set] | url=https://zenodo.org/record/4112978 | website=Zenodo | year=2020 | last1=Tanash | first1=I.M. | last2=Riihonen | first2=T.}}</ref>


A tight approximation of the complementary error function for {{math|''x'' ∈ [0,∞)}} is given by [[George Karagiannidis|Karagiannidis]] & Lioumpas (2007)<ref>{{cite journal|last1=Karagiannidis |first1=G. K. |last2=Lioumpas |first2=A. S. |url=http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf |title=An improved approximation for the Gaussian Q-function |date=2007 |journal=IEEE Communications Letters |volume=11 |issue=8 |pages=644–646|doi=10.1109/LCOMM.2007.070470 |s2cid=4043576 }}</ref> who showed for the appropriate choice of parameters {{math|{''A'',''B''}<nowiki/>}} that
<math display="block">\operatorname{erfc}(x) \approx \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x}.</math>
They determined {{math|{''A'',''B''} {{=}} {1.98,1.135}<nowiki/>}}, which gave a good approximation for all {{math|''x'' ≥ 0}}. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.<ref>{{cite journal |doi=10.1109/LCOMM.2021.3052257|title=Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function|journal=IEEE Communications Letters | year=2021 | last1=Tanash | first1=I.M.|last2=Riihonen|first2=T.|volume=25|issue=5|pages=1468–1471|arxiv=2101.07631|bibcode=2021IComL..25.1468T |s2cid=231639206}}</ref>


A single-term lower bound is<ref>{{cite journal |last1=Chang |first1=Seok-Ho |last2=Cosman |first2=Pamela C. |author-link2 = Pamela Cosman |last3=Milstein |first3=Laurence B. |date=November 2011 |title=Chernoff-Type Bounds for the Gaussian Error Function |url=http://escholarship.org/uc/item/6hw4v7pg |journal=IEEE Transactions on Communications |volume=59 |issue=11 |pages=2939–2944 |doi=10.1109/TCOMM.2011.072011.100049 |bibcode=2011ITCom..59.2939C |s2cid=13636638}}</ref>
<math display="block" display="block">\operatorname{erfc}(x) \geq \sqrt{\frac{2 e}{\pi}} \frac{\sqrt{\beta - 1}}{\beta} e^{- \beta x^2}, \qquad x \ge 0,\quad \beta > 1,</math>
where the parameter {{mvar|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=Sergei |title=Computational Science and Its Applications – ICCSA 2003 |date=2003  |volume=2667 |chapter=Uniform approximations for transcendental functions |publisher=Springer, Berlin |pages=[https://archive.org/details/computationalsci0000iccs_a2w6/page/780 780–789] |isbn=978-3-540-40155-1 |doi=10.1007/3-540-44839-X_82 |chapter-url-access=registration |chapter-url=https://archive.org/details/computationalsci0000iccs_a2w6 |series=Lecture Notes in Computer Science }}</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 |quote=Indeed, Winitzki [32] provided the so-called global Padé approximation | arxiv=1310.5592 |s2cid=118148950 }}</ref>{{rp|2–3}}
<math display="block">\operatorname{erf}(x) \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2\frac{\frac{4}{\pi} + ax^2}{1 + ax^2}\right)}</math>
where
<math display="block">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 {{mvar|x}}. Using the alternate value {{math|''a'' ≈ 0.147}} reduces the maximum relative error to about 0.00013.<ref>{{Cite web <!-- Deny Citation Bot--> |url=https://www.academia.edu/9730974/A_handy_approximation_for_the_error_function_and_its_inverse |last=Winitzki |first=Sergei |date=6 February 2008 |title=A handy approximation for the error function and its inverse }}</ref>
{{pb}}

The extended "global Pade" approximation,
<math display="block">\operatorname{erf}(x) \approx \sgn x \cdot \sqrt{1 - \exp\left(-x^2
\frac{
4+0.880877880079853x^2+0.144026670907584x^4+0.0077581300270021x^6
}{
\pi+0.786235558186528x^2+0.128368576906837x^4+0.00773380006014367x^6}
\right)}\,,</math>
provides the maximum error of about {{val|2e-9}}, as demonstrated by the following Python script.
{{collapse top|title=Python script to fit extended "global Pade" approximation|collapsed=yes}}
<syntaxhighlight lang="python">
import numpy,math
from scipy.optimize import least_squares

# approximation to erf(x)

def approx_erf(p,x):
	frac=(4+p[0]*x**2+p[1]*x**4+p[2]*x**6)/(
		math.pi+p[3]*x**2+p[4]*x**4+p[5]*x**6)
	return numpy.sign(x)*numpy.sqrt(
		1-numpy.exp(-x*x*frac))

def residuals(params, xs, ys):
    return approx_erf(params, xs) - ys

# data for fitting

N = 200
xmin = 0
xmax = 9
xs = numpy.linspace(xmin, xmax, N)
ys = numpy.array([math.erf(x) for x in xs], dtype=float)
params0 = numpy.array([0.9,0.1,0.008,0.8,0.1,0.008], dtype=float)

# fitting

result = least_squares(
    residuals, params0, args=(xs, ys),
	xtol=1e-14, ftol=1e-14, gtol=1e-14, max_nfev=5000
)
params = result.x

# print out fitted parameters

print("\nFitted parameters:")
for i, pi in enumerate(params, 0):
    print(f"p{i} = {pi:.15g},")

# evaluate approximation error

approx_vals = approx_erf(params, xs)
abs_err = numpy.abs(approx_vals - ys)

print(f"\nMaximum absolute error on [{xmin},{xmax}]:", numpy.max(abs_err))
print("RMS error:", numpy.sqrt(numpy.mean(abs_err**2)))
print("Done.")
</syntaxhighlight>
{{collapse bottom}}


The Winitzki's approximation can be inverted to obtain an approximation for the inverse error function:
<math display="block">\operatorname{erf}^{-1}(x) \approx \sgn x \cdot \sqrt{\sqrt{\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)^2 - \frac{\ln\left(1 - x^2\right)}{a}} -\left(\frac{2}{\pi a} + \frac{\ln\left(1 - x^2\right)}{2}\right)}.</math>


An approximation with a maximal error of {{val|1.2e-7}} for any real argument is:<ref>{{cite book | last = Press | first = William H. | title = Numerical Recipes in Fortran 77: The Art of Scientific Computing | isbn = 0-521-43064-X | year = 1992 | page = 214 | publisher = Cambridge University Press }}</ref>
<math display="block">\operatorname{erf}(x) = \begin{cases}
1-\tau &  x\ge 0\\
\tau-1 &  x < 0
\end{cases}</math>
with
<math display="block">\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 display="block">t = \frac{1}{1 + \frac{1}{2}|x|}.</math>


An approximation of <math>\operatorname{erfc}</math> with a maximum relative error less than <math>2^{-53}</math> <math>\left(\approx 1.1 \times 10^{-16}\right)</math> in absolute value is:<ref>{{Cite journal | last = Dia | first = Yaya D. |date = 2023 | title = Approximate Incomplete Integrals, Application to Complementary Error Function | url = https://www.ssrn.com/abstract=4487559 | journal = SSRN Electronic Journal | language = en | doi = 10.2139/ssrn.4487559 | issn = 1556-5068}}</ref>
for {{nowrap|<math>x\ge 0</math>,}}
<math display="block">\begin{aligned}
\operatorname{erfc} \left(x\right)
& =
\left(\frac{0.56418958354775629}{x+2.06955023132914151}\right) \left(\frac{x^2+2.71078540045147805 x+5.80755613130301624}{x^2+3.47954057099518960 x+12.06166887286239555}\right) \\ & \left(\frac{x^2+3.47469513777439592 x+12.07402036406381411}{x^2+3.72068443960225092 x+8.44319781003968454}\right)
\left(\frac{x^2+4.00561509202259545 x+9.30596659485887898}{x^2+3.90225704029924078 x+6.36161630953880464}\right) \\
& \left(\frac{x^2+5.16722705817812584 x+9.12661617673673262}{x^2+4.03296893109262491 x+5.13578530585681539}\right)
\left(\frac{x^2+5.95908795446633271 x+9.19435612886969243}{x^2+4.11240942957450885 x+4.48640329523408675}\right) e^{-x^2} \\
\end{aligned}</math>
and for <math>x<0</math>
<math display="block">\operatorname{erfc} \left(x\right) = 2 - \operatorname{erfc} \left(-x\right)</math>


A simple approximation for real-valued arguments could be done through [[Hyperbolic functions]]:
<math display="block">\operatorname{erf} \left(x\right) \approx z(x) = \tanh\left(\frac{2}{\sqrt{\pi}}\left(x+\frac{11}{123}x^3\right)\right)</math>
which keeps the absolute difference {{nowrap|<math>\left|\operatorname{erf} \left(x\right)-z(x)\right| < 0.000358,\, \forall x</math>.}}


Since the error function and the Gaussian Q-function are closely related through the identity <math>\operatorname{erfc}(x) = 2 Q(\sqrt{2} x)</math> or equivalently <math>Q(x) = \frac{1}{2} \operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)</math>, bounds developed for the Q-function can be adapted to approximate the complementary error function. A pair of tight lower and upper bounds on the Gaussian Q-function for positive arguments <math>x \in [0, \infty)</math> was introduced by Abreu (2012)<ref>{{cite journal |doi=10.1109/TCOMM.2012.080612.110075 |title=Very Simple Tight Bounds on the Q-Function |journal=IEEE Transactions on Communications |volume=60 |issue=9 |pages=2415–2420 |year=2012 |last=Abreu |first=Giuseppe |bibcode=2012ITCom..60.2415A }}</ref> based on a simple [[algebraic expression]] with only two exponential terms:
<math display="block">Q(x) \geq \frac{1}{12} e^{-x^2} + \frac{1}{\sqrt{2\pi} (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0,</math>
and
<math display="block">Q(x) \leq \frac{1}{50} e^{-x^2} + \frac{1}{2 (x + 1)} e^{-x^2 / 2}, \qquad x \geq 0.</math>

These bounds stem from a unified form <math display="block">Q_{\mathrm{B}}(x; a, b) = \frac{\exp(-x^2)}{a} + \frac{\exp(-x^2 / 2)}{b (x + 1)},</math> where the parameters <math>a</math> and <math>b</math> are selected to ensure the bounding properties: for the lower bound, <math>a_{\mathrm{L}} = 12</math> and <math>b_{\mathrm{L}} = \sqrt{2\pi}</math>, and for the upper bound, <math>a_{\mathrm{U}} = 50</math> and <math>b_{\mathrm{U}} = 2</math>. 
These expressions maintain simplicity and tightness, providing a practical trade-off between accuracy and ease of computation. They are particularly valuable in theoretical contexts, such as communication theory over fading channels, where both functions frequently appear. Additionally, the original Q-function bounds can be extended to <math>Q^n(x)</math> for positive integers <math>n</math> via the [[binomial theorem]], suggesting potential adaptability for powers of <math>\operatorname{erfc}(x)</math>, though this is less commonly required in error function applications.

===Table of values===
{{further|Interval estimation|Coverage probability|68–95–99.7 rule}}
{| class="wikitable" style="text-align:left;margin-left:24pt"
! {{math|''x''}}!! {{math|erf(''x'')}} !! {{math|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|0.880205070}} || {{val|0.119794930}}
|-
|1.2 || {{val|0.910313978}} || {{val|0.089686022}}
|-
|1.3 || {{val|0.934007945}} || {{val|0.065992055}}
|-
|1.4 || {{val|0.952285120}} || {{val|0.047714880}}
|-
|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|0.999977910}} || {{val|0.000022090}}
|-
|3.5 || {{val|0.999999257}} || {{val|0.000000743}}
|}

==Related functions==

===Complementary error function===
[[File:Plot of the error function Erf(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the error function erf(''z'') in the complex plane from {{nobr|−2 − 2''i''}} to {{nobr|2 + 2''i''}}]]
The '''complementary error function''', denoted {{math|erfc}}, is defined as
<math display="block">\begin{align}
 \operatorname{erfc}(x) &= 1 - \operatorname{erf}(x) \\
                        &= \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\,dt \\
                        &= e^{-x^2} \operatorname{erfcx}(x),
\end{align}</math>
which also defines {{math|erfcx}}, 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 |s2cid=5621105 }}</ref> (which can be used instead of {{math|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=1 March 2007 |doi=10.1111/j.1365-2966.2006.11377.x|bibcode=2007MNRAS.375.1043Z |doi-access=free }}</ref>). Another form of {{math|erfc ''x''}} for {{math|''x'' ≥ 0}} 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 display="block">
 \operatorname{erfc} (x \mid x\ge 0) = 
 \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp\left(-\frac{x^2}{\sin^2 \theta}\right) \,d\theta.
</math>
This expression is valid only for positive values of {{mvar|x}}, but can be used in conjunction with {{math|erfc(''x'') {{=}} 2 − erfc(−''x'')}} to obtain {{math|erfc(''x'')}} for negative values. This form is advantageous in that the range of integration is fixed and finite. An extension of this expression for the {{math|erfc}} of the sum of two non-negative variables is<ref>{{cite journal |doi=10.1109/TCOMM.2020.2986209 |title=A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis|journal=IEEE Transactions on Communications |volume=68 |issue=7 |pages=4117–4125 |year=2020 |last1=Behnad |first1=Aydin |bibcode=2020ITCom..68.4117B |s2cid=216500014}}</ref>
<math display="block">
 \operatorname{erfc}(x + y \mid x, y \ge 0) =
 \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp\left(-\frac{x^2}{\sin^2 \theta} - \frac{y^2}{\cos^2 \theta}\right) \,d\theta.
</math>

===Imaginary error function===
[[File:Plot of the imaginary error function Erfi(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|thumb|Plot of the imaginary error function erfi(''z'') in the complex plane from {{nobr|−2 − 2''i''}} to {{nobr|2 + 2''i''}}]]
The '''imaginary error function''', denoted {{math|erfi}}, is defined as
<math display="block">\begin{align}
 \operatorname{erfi}(x) &= -i\operatorname{erf}(ix) \\
                        &= \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,dt \\
                        &= \frac{2}{\sqrt\pi} e^{x^2} D(x),
\end{align}</math>
where {{math|''D''(''x'')}} is the [[Dawson function]] (which can be used instead of {{math|erfi}} to avoid [[arithmetic overflow]]<ref name=Cody93/>).

Despite the name "imaginary error function", {{math|erfi(''x'')}} is real when {{mvar|x}} is real.

When the error function is evaluated for arbitrary [[complex number|complex]] arguments {{mvar|z}}, the resulting '''complex error function''' is usually discussed in scaled form as the [[Faddeeva function]]:
<math display="block">
 w(z) = e^{-z^2} \operatorname{erfc}(-iz) = \operatorname{erfcx}(-iz).
</math>

===Cumulative distribution function===
[[File:Normal cumulative distribution function complex plot in Mathematica 13.1 with ComplexPlot3D.svg|thumb|The normal cumulative distribution function plotted in the complex plane]]
The error function is essentially identical to the standard [[normal cumulative distribution function]], denoted {{math|Φ}}, also named {{math|norm(''x'')}} by some software languages{{Citation needed|date=July 2020}}, as they differ only by scaling and translation. Indeed,
<math display="block">\begin{align}
\Phi(x) 
&= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^\tfrac{-t^2}{2}\,dt\\[6pt] 
&= \frac{1}{2} \left(1+\operatorname{erf}\left(\frac{x}{\sqrt 2}\right)\right)\\[6pt]
&= \frac{1}{2} \operatorname{erfc}\left(-\frac{x}{\sqrt 2}\right)
\end{align}</math>
or rearranged for {{math|erf}} and {{math|erfc}}:
<math display="block">\begin{align}
  \operatorname{erf}(x)  &= 2 \Phi{\left ( x \sqrt{2} \right )} - 1 \\[6pt]
  \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 display="block">\begin{align}
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).
\end{align}</math>

The [[inverse function|inverse]] of {{math|Φ}} 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 display="block">\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 display="block">\operatorname{erf}(x) = \frac{2x}{\sqrt\pi} M\left(\tfrac{1}{2},\tfrac{3}{2},-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]] {{mvar|P}} and the [[incomplete gamma function]],
<math display="block">\operatorname{erf}(x)
= \sgn(x) \cdot P\left(\tfrac{1}{2}, x^2\right)
= \frac{\sgn(x)}{\sqrt\pi} \gamma{\left(\tfrac{1}{2}, x^2\right)}.</math>{{math|sgn(''x'')}} is the [[sign function]].

===Iterated integrals of the complementary error function===
The iterated integrals of the complementary error function are defined by<ref>{{cite book | last1 = Carslaw | first1 = 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 | page = 484}}</ref>
<math display="block">\begin{align}
i^n\!\operatorname{erfc}(z) &= \int_z^\infty  i^{n-1}\!\operatorname{erfc}(\zeta)\,d\zeta \\[6pt]
i^0\!\operatorname{erfc}(z) &= \operatorname{erfc}(z) \\
i^1\!\operatorname{erfc}(z) &= \operatorname{ierfc}(z) = \frac{1}{\sqrt\pi} e^{-z^2} - z \operatorname{erfc}(z) \\
i^2\!\operatorname{erfc}(z) &= \tfrac{1}{4} \left( \operatorname{erfc}(z) -2 z \operatorname{ierfc}(z) \right) \\
\end{align}</math>

The general recurrence formula is
<math display="block">2 n \cdot i^n\!\operatorname{erfc}(z) = i^{n-2}\!\operatorname{erfc}(z) -2 z \cdot i^{n-1}\!\operatorname{erfc}(z)</math>

They have the power series
<math display="block">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 display="block">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 display="block">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>

==Implementations==

===As real function of a real argument===
* In [[POSIX]]-compliant operating systems, the header <code>[[math.h]]</code> shall declare and the mathematical library <code>[[libm]]</code> shall provide the functions <code>erf</code> and <code>erfc</code> ([[double precision]]) as well as their [[single precision]] and [[extended precision]] counterparts <code>erff</code>, <code>erfl</code> and <code>erfcf</code>, <code>erfcl</code>.<ref>{{cite web | url = https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/math.h.html | access-date = 21 April 2023 | website = opengroup.org | title = math.h - mathematical declarations | year = 2018 | issue = 7}}</ref>
* The [[GNU Scientific Library]] provides <code>erf</code>, <code>erfc</code>, <code>log(erf)</code>, and scaled error functions.<ref>{{Cite web|url=https://www.gnu.org/software/gsl/doc/html/specfunc.html#error-functions|title = Special Functions – GSL 2.7 documentation}}</ref>

===As complex function of a complex argument===

* <code>[https://jugit.fz-juelich.de/mlz/libcerf libcerf]</code>, numeric C library for complex error functions, provides the complex functions <code>cerf</code>, <code>cerfc</code>, <code>cerfcx</code> and the real functions <code>erfi</code>, <code>erfcx</code> with approximately 13–14 digits precision, based on the [[Faddeeva function]] as implemented in the [http://ab-initio.mit.edu/Faddeeva MIT Faddeeva Package]

==References==
{{Reflist}}

==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 |access-date=9 August 2011 |archive-date=11 August 2011 |archive-url=https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=259 |url-status=dead }}
*{{dlmf|id=7|title=Error Functions, Dawson’s and Fresnel Integrals|first=Nico M. |last=Temme }}

==External links==
* [http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf A Table of Integrals of the Error Functions]

{{Nonelementary Integral}}
{{Authority control}}

[[Category:Special hypergeometric functions]]
[[Category:Gaussian function]]
[[Category:Functions related to probability distributions]]
[[Category:Analytic functions]]
[[Category:Sigmoid functions]]