LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle z\operatorname{erfi}(z) - \frac{e^{z^2}}{\sqrt{\pi}}.}

${\displaystyle z\operatorname {erfi} (z)-{\frac {e^{z^{2}}}{\sqrt {\pi }}}.}$

... is translated to the CAS output ...

Semantic latex: z \operatorname{erfi}(z) - \frac{\expe^{z^2}}{\sqrt{\cpi}}

Confidence: 0

Mathematica

Translation: z*erfi[z]-Divide[Exp[(z)^(2)],Sqrt[Pi]]

Information

Sub Equations

z*erfi[z]-Divide[Exp[(z)^(2)],Sqrt[Pi]]

Free variables

z

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Pi was translated to: Pi

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

SymPy

Translation: z*erfi(z)-(exp((z)**(2)))/(sqrt(pi))

Information

Sub Equations

z*erfi(z)-(exp((z)**(2)))/(sqrt(pi))

Free variables

z

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Pi was translated to: pi

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

Maple

Translation: z*erfi(z)-(exp((z)^(2)))/(sqrt(Pi))

Information

Sub Equations

z*erfi(z)-(exp((z)^(2)))/(sqrt(Pi))

Free variables

z

Symbol info

Recognizes e with power as the exponential function. It was translated as a function.

Pi was translated to: Pi

Was interpreted as a function call because of a leading \operatorname.

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$z$

$z)$

$e$

$z^{\bar {n}}$

$\operatorname {erfi} (x)$

Description

imaginary error function

integration by part

Complete translation information:

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Specify your own input

Formula input
Wikitext context	{{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'''), often denoted by '''erf''', is a complex function of a complex variable 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><ref name="Greene">Greene, William H.; ''Econometric Analysis'' (fifth edition), Prentice-Hall, 1993, p. 926, fn. 11</ref> :<math>\operatorname{erf} z = \frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt.</math> This integral is a [[special function\|special]] (non-[[elementary function\|elementary]]) and [[sigmoid function\|sigmoid]] function that occurs often in [[probability]], [[statistics]], and [[partial differential equation]]s. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of {{mvar\|x}}, the error function has the following interpretation: for a [[random variable]] {{mvar\|Y}} that is [[normal distribution\|normally distributed]] with [[mean]] 0 and [[variance]] 1/2, {{math\|erf ''x''}} is the probability that {{mvar\|Y}} falls in the range {{math\|[−''x'', ''x'']}}. Two closely related functions are the '''complementary error function''' ('''erfc''') defined as :<math>\operatorname{erfc} z = 1 - \operatorname{erf} z,</math> and the '''imaginary error function''' ('''erfi''') defined as :<math>\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. 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'' = exp(−''z''<sup>2</sup>) and ''f'' = erf(''z'') are shown in the complex ''z''-plane in figures 2 and 3. Level of Im(''f'') = 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'') = constant are shown with thin green lines. Intermediate levels of Re(''f'') = 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'' → +∞ and −1 at ''z'' → −∞. 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, but is famously known "[...] for its bad convergence if x > 1."<ref>{{Cite web\|url=https://oeis.org/A007680\|title=A007680 - OEIS\|website=oeis.org\|access-date=2020-04-02}}</ref> 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]] ''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'' + 1)<sup>st</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]] ''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 a 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 ±∞ is equal to ±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'' – 1)!! is the [[double factorial]] of (2''n'' – 1), which is the product of all odd numbers up to (2''n'' – 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\|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\|access-date=2017-12-04}}</ref><ref>Eq (3) on page 283 of {{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}}</ref> * Representation by an infinite sum containing the [[double factorial]]: ::<math>\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^\infty \frac{(-2)^n(2n-1)!!}{(2n+1)!}z^{2n+1}</math> == 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: 5×10<sup>−4</sup>) :where ''a''<sub>1</sub> = 0.278393, ''a''<sub>2</sub> = 0.230389, ''a''<sub>3</sub> = 0.000972, ''a''<sub>4</sub> = 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>    (maximum error: 2.5×10<sup>−5</sup>) : where ''p'' = 0.47047, ''a''<sub>1</sub> = 0.3480242, ''a''<sub>2</sub> = −0.0958798, ''a''<sub>3</sub> = 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>    (maximum error: 3×10<sup>−7</sup>) : where ''a''<sub>1</sub> = 0.0705230784, ''a''<sub>2</sub> = 0.0422820123, ''a''<sub>3</sub> = 0.0092705272, ''a''<sub>4</sub> = 0.0001520143, ''a''<sub>5</sub> = 0.0002765672, ''a''<sub>6</sub> = 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>    (maximum error: 1.5×10<sup>−7</sup>) : where ''p'' = 0.3275911, ''a''<sub>1</sub> = 0.254829592, ''a''<sub>2</sub> = −0.284496736, ''a''<sub>3</sub> = 1.421413741, ''a''<sub>4</sub> = −1.453152027, ''a''<sub>5</sub> = 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''). * 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> The above have been generalized to sums of <math>N</math> 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\|s2cid=220514754}}</ref> with increasing accuracy in terms of <math>N</math> so that <math>\operatorname{erfc}(x)</math> can be accurately approximated or bounded by <math>2\tilde{Q}(\sqrt{2}x)</math>, where ::<math>\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_n,b_n)\}_{n=1}^N</math> that yield a [[minimax approximation algorithm\|minimax]] approximation or bound for the closely related [[Q-function]]: <math>Q(x) \approx \tilde{Q}(x)</math>, <math>Q(x) \leq \tilde{Q}(x)</math>, or <math>Q(x) \geq \tilde{Q}(x)</math> for <math>x\geq0</math>. The coefficients <math>\{(a_n,b_n)\}_{n=1}^N</math> for many variations of the exponential approximations and bounds up to <math>N = 25</math> have been released to open access as a comprehensive dataset.<ref>{{cite web \|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 \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\|s2cid=13636638\|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 \|location= \|publisher=Spronger, 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 \|author-link= \|chapter-url-access=registration \|chapter-url=https://archive.org/details/computationalsci0000iccs_a2w6 \|series=Lecture Notes in Computer Science \|url=https://archive.org/details/computationalsci0000iccs_a2w6/page/780 }} (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 \|quote=Indeed, Winitzki [32] provided the so-called global Padé approximation\|arxiv=1310.5592 \|s2cid=118148950 }}</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'' ≈ 0.147 reduces the maximum relative error to about 0.00013.<ref>{{Cite document \|last=Winitzki \|first=Sergei \|date=6 February 2008 \|title=A handy approximation for the error function and its inverse }}</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 \|s2cid=5621105 }}</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\|doi-access=free }}</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. An extension of this expression for the <math>\mathrm{erfc}</math> of the sum of two non-negative variables is as follows:<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\|s2cid=216500014}}</ref> :<math>\operatorname{erfc} (x+y \mid x,y\ge 0) = \frac 2 \pi \int_0^{\pi/2} \exp \left( - \frac{x^2}{\sin^2 \theta} - \frac{y^2}{\cos^2 \theta} \right) \, d\theta.</math> ===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 some software languages{{Citation needed\|date=July 2020}}, 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 − e<sup> −''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'' > 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]]: :<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\|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}}, 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> ==Implementations== ===As real function of a real argument=== * In [[Posix]] compliant operating systems, the header [[math.h]] shall declare and the mathematical library [[libm]] shall provide the functions ''erf'' and ''erfc'' ([[double precision]]) as well as their [[single precision]] and [[extended precision]] counterparts ''erff'', ''erfl'' and ''erfc'', ''erfcl''.<ref>https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/math.h.html</ref> * The [[GNU Scientific Library]] provides ''erf'', ''erfc'', ''log(erf)'', and scaled error functions.<ref>https://www.gnu.org/software/gsl/doc/html/specfunc.html#error-functions</ref> ===As complex function of a complex argument=== * [http://jugit.fz-juelich.de/mlz/libcerf libcerf], numeric C library for complex error functions, provides the complex functions ''cerf'', ''cerfc'', ''cerfcx'' and the real functions ''erfi'', ''erfcx'' with approximately 13–14 digits precision, based on the [[Faddeeva function]] as implemented in the [http://ab-initio.mit.edu/Faddeeva MIT Faddeeva Package] ==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]]
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