error function

The name "error function" and its abbreviation {{math|erf}} were proposed by J. W. L. Glaisher in 1871 on account of its connection with "the theory of Probability, and notably the theory of Errors."{{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}} The error function complement was also discussed by Glaisher in a separate publication in the same year.{{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}}

For the "law of facility" of errors whose density is given by

$$f(x)\; =\; \backslash left(\backslash frac\{c\}\{\backslash pi\}\backslash right)^\{1/2\}\; e^\{-c\; x^2\}$$

(the normal distribution), Glaisher calculates the probability of an error lying between {{mvar|p}} and {{mvar|q}} as:

$$\backslash left(\backslash frac\{c\}\{\backslash pi\}\backslash right)^\backslash frac\{1\}\{2\}\; \backslash int\_p^qe^\{-cx^2\}\backslash ,\backslash mathrm\; dx\; =\; \backslash tfrac\{1\}\{2\}\backslash left(\backslash operatorname\{erf\}\; \backslash left(q\backslash sqrt\{c\}\backslash right)\; -\backslash operatorname\{erf\}\; \backslash left(p\backslash sqrt\{c\}\backslash right)\backslash right).$$

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

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 conditions 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:

$$\backslash begin\{align\}$$

\Pr[X\leq L] &= \frac{1}{2} + \frac{1}{2} \operatorname{erf}\frac{L-\mu}{\sqrt{2}\sigma} \\

&\approx A \exp \left(-B \left(\frac{L-\mu}{\sigma}\right)^2\right)

\end{align}

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:

$$\backslash Pr[X\backslash leq\; L]\; \backslash leq\; A\; \backslash exp\; (-B\; \backslash ln\{k\})\; =\; \backslash frac\{A\}\{k^B\}$$

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

The probability for {{mvar|X}} being in the interval {{closed-closed|L_a, L_b}} can be derived as

$$\backslash begin\{align\}$$

\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) \,\mathrm dx \\

&= \frac{1}{2}\left(\operatorname{erf}\frac{L_b-\mu}{\sqrt{2}\sigma} - \operatorname{erf}\frac{L_a-\mu}{\sqrt{2}\sigma}\right).\end{align}

{{multiple image

| header = Plots in the complex plane

| direction = vertical

| width = 250

| image1 = ComplexExp2.png

| caption1 = Integrand {{math|exp(−z²)}}

| image2 = ComplexErfz.png

| caption2 = {{math|erf z}}

}}

The property {{math|1=erf (−z) = −erf z}} means that the error function is an odd function. This directly results from the fact that the integrand {{math|e^−t2}} 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}}:

$$\backslash operatorname\{erf\}\; \backslash overline\{z\}\; =\; \backslash overline\{\backslash operatorname\{erf\}\; z\}$$

where $\backslash overline\{z\}$

denotes the complex conjugate of $z$.

The integrand {{math|1=f = exp(−z²)}} 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 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 in terms of elementary functions (see Liouville's theorem), but by expanding the integrand {{math|e^−z2}} into its Maclaurin series and integrating term by term, one obtains the error function's Maclaurin series as:

$$\backslash begin\{align\}$$

\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}

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

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

$$\backslash begin\{align\}$$

\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}

because {{math|{{sfrac|−(2k − 1)z²|k(2k + 1)}}}} expresses the multiplier to turn the {{mvar|k}}th term into the {{math|(k + 1)}}th term (considering {{mvar|z}} as the first term).

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

$$\backslash begin\{align\}$$

\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}

which holds for every complex number {{mvar|z}}.

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

$$\backslash frac\{\backslash mathrm\; d\}\{\backslash mathrm\; dz\}\backslash operatorname\{erf\}\; z\; =\backslash frac\{2\}\{\backslash sqrt\backslash pi\}\; e^\{-z^2\}.$$

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

$$\backslash frac\{d\}\{dz\}\backslash operatorname\{erfi\}\; z\; =\backslash frac\{2\}\{\backslash sqrt\backslash pi\}\; e^\{z^2\}.$$

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

$$z\backslash operatorname\{erf\}z\; +\; \backslash frac\{e^\{-z^2\}\}\{\backslash sqrt\backslash pi\}.$$

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

$$z\backslash operatorname\{erfi\}z\; -\; \backslash frac\{e^\{z^2\}\}\{\backslash sqrt\backslash pi\}.$$

Higher order derivatives are given by

$$\backslash operatorname\{erf\}^\{(k)\}z\; =\; \backslash frac\{2\; (-1)^\{k-1\}\}\{\backslash sqrt\backslash pi\}\; \backslash mathit\{H\}\_\{k-1\}(z)\; e^\{-z^2\}\; =\; \backslash frac\{2\}\{\backslash sqrt\backslash pi\}\; \backslash frac\{\backslash mathrm\; d^\{k-1\}\}\{\backslash mathrm\; dz^\{k-1\}\}\; \backslash left(e^\{-z^2\}\backslash right),\backslash qquad\; k=1,\; 2,\; \backslash dots$$

where {{mvar|H}} are the physicists' Hermite polynomials.{{mathworld|title=Erf|urlname=Erf}}

An expansion,{{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 }} which converges more rapidly for all real values of {{mvar|x}} than a Taylor expansion, is obtained by using Hans Heinrich Bürmann's theorem:{{mathworld|urlname=BuermannsTheorem | title = Bürmann's Theorem }}

$$\backslash begin\{align\}$$

\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}

where {{math|sgn}} is the sign function. By keeping only the first two coefficients and choosing {{math|1=c₁ = {{sfrac|31|200}}}} and {{math|1=c₂ = −{{sfrac|341|8000}}}}, the resulting approximation shows its largest relative error at {{math|1=x = ±1.40587}}, where it is less than 0.0034361:

$$\backslash operatorname\{erf\}\; x\; \backslash approx\; \backslash frac\{2\}\{\backslash sqrt\backslash pi\}\backslash sgn\; x\; \backslash cdot\; \backslash sqrt\{1-e^\{-x^2\}\}\; \backslash left(\backslash frac\{\backslash sqrt\{\backslash pi\}\}\{2\}\; +\; \backslash frac\{31\}\{200\}e^\{-x^2\}-\backslash frac\{341\}\{8000\}\; e^\{-2x^2\}\backslash right).$$

File:Mplwp erf inv.svg

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⁻¹ x}} satisfying

$$\backslash operatorname\{erf\}\backslash left(\backslash operatorname\{erf\}^\{-1\}\; x\backslash right)\; =\; x.$$

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{{cite arXiv | last1 = Dominici | first1 = Diego | title = Asymptotic analysis of the derivatives of the inverse error function | eprint = math/0607230 | year = 2006}}

$$\backslash operatorname\{erf\}^\{-1\}\; z=\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{c\_k\}\{2k+1\}\backslash left\; (\backslash frac\{\backslash sqrt\backslash pi\}\{2\}z\backslash right\; )^\{2k+1\},$$

where {{math|1=c₀ = 1}} and

$$\backslash begin\{align\}$$

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}

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

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

(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 {{math|±∞}} is equal to {{math|±1}}.

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

The inverse complementary error function is defined as

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

For real {{mvar|x}}, there is a unique real number {{math|erfi⁻¹ x}} satisfying {{math|1=erfi(erfi⁻¹ x) = x}}. The inverse imaginary error function is defined as {{math|erfi⁻¹ x}}.{{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}}

For any real x, Newton's method can be used to compute {{math|erfi⁻¹ x}}, and for {{math|−1 ≤ x ≤ 1}}, the following Maclaurin series converges:

$$\backslash operatorname\{erfi\}^\{-1\}\; z\; =\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{(-1)^k\; c\_k\}\{2k+1\}\; \backslash left(\; \backslash frac\{\backslash sqrt\backslash pi\}\{2\}\; z\; \backslash right)^\{2k+1\},$$

where {{math|c_k}} is defined as above.

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real {{mvar|x}} is

$$\backslash begin\{align\}$$

\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}

where {{math|(2n − 1)!!}} is the double factorial of {{math|(2n − 1)}}, which is the product of all odd numbers up to {{math|(2n − 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

$$\backslash operatorname\{erfc\}\; x\; =\; \backslash frac\{e^\{-x^2\}\}\{x\backslash sqrt\{\backslash pi\}\}\backslash sum\_\{n=0\}^\{N-1\}\; (-1)^n\; \backslash frac\{(2n\; -\; 1)!!\}\{\backslash left(2x^2\backslash right)^n\}\; +\; R\_N(x)$$

where the remainder is

$$R\_N(x)\; :=\; \backslash frac\{(-1)^N\; \backslash ,\; (2\; N\; -\; 1)!!\}\{\backslash sqrt\{\backslash pi\}\; \backslash cdot\; 2^\{N\; -\; 1\}\}\; \backslash int\_x^\backslash infty\; t^\{-2N\}e^\{-t^2\}\backslash ,\backslash mathrm\; dt,$$

which follows easily by induction, writing

$$e^\{-t^2\}\; =\; -\backslash frac\{1\}\{2\; t\}\; \backslash ,\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; e^\{-t^2\}$$

and integrating by parts.

The asymptotic behavior of the remainder term, in Landau notation, is

$$R\_N(x)\; =\; O\backslash left(x^\{-\; (1\; +\; 2N)\}\; e^\{-x^2\}\backslash right)$$

as {{math|x → ∞}}. This can be found by

$$R\_N(x)\; \backslash propto\; \backslash int\_x^\backslash infty\; t^\{-2N\}e^\{-t^2\}\backslash ,\backslash mathrm\; dt\; =\; e^\{-x^2\}\; \backslash int\_0^\backslash infty\; (t+x)^\{-2N\}e^\{-t^2-2tx\}\backslash ,\backslash mathrm\; dt\backslash leq\; e^\{-x^2\}\; \backslash int\_0^\backslash infty\; x^\{-2N\}\; e^\{-2tx\}\backslash ,\backslash mathrm\; dt\; \backslash propto\; x^\{-(1+2N)\}e^\{-x^2\}.$$

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).

A continued fraction expansion of the complementary error function was found by Laplace:Pierre-Simon Laplace, Traité de mécanique céleste, tome 4 (1805), livre X, page 255.{{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 }}

$$\backslash operatorname\{erfc\}\; z\; =\; \backslash frac\{z\}\{\backslash sqrt\backslash pi\}e^\{-z^2\}\; \backslash cfrac\{1\}\{z^2+\; \backslash cfrac\{a\_1\}\{1+\backslash cfrac\{a\_2\}\{z^2+\; \backslash cfrac\{a\_3\}\{1+\backslash dotsb\}\}\}\},\backslash qquad\; a\_m\; =\; \backslash frac\{m\}\{2\}.$$

The inverse factorial series:

$$\backslash begin\{align\}$$

\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}

converges for {{math|Re(z²) > 0}}. Here

$$\backslash begin\{align\}$$

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|z^{{{overline|n}}}}} denotes the rising factorial, and {{math|s(n,k)}} denotes a signed Stirling number of the first kind.{{cite journal|last=Schlömilch|first=Oskar Xavier | author-link=Oscar Schlömilch|year=1859|title=Ueber facultätenreihen|url=https://archive.org/details/zeitschriftfrma09runggoog | journal=Zeitschrift für Mathematik und Physik | language=de | volume=4 | pages=390–415}}{{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}}

There also exists a representation by an infinite sum containing the double factorial:

$$\backslash operatorname\{erf\}\; z\; =\; \backslash frac\{2\}\{\backslash sqrt\backslash pi\}\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(-2)^n(2n-1)!!\}\{(2n+1)!\}z^\{2n+1\}$$

• 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:

  $$\backslash operatorname\{erf\}\; x\; \backslash approx\; 1\; -\; \backslash frac\{1\}\{\backslash left(1\; +\; a\_1x\; +\; a\_2x^2\; +\; a\_3x^3\; +\; a\_4x^4\backslash right)^4\},\; \backslash qquad\; x\; \backslash geq\; 0$$

  (maximum error: {{val|5e-4}})

  {{pb}}

  where {{math|a₁ {{=}} 0.278393}}, {{math|a₂ {{=}} 0.230389}}, {{math|a₃ {{=}} 0.000972}}, {{math|a₄ {{=}} 0.078108}}

  $$\backslash operatorname\{erf\}\; x\; \backslash approx\; 1\; -\; \backslash left(a\_1t\; +\; a\_2t^2\; +\; a\_3t^3\backslash right)e^\{-x^2\},\backslash quad\; t=\backslash frac\{1\}\{1\; +\; px\},\; \backslash qquad\; x\; \backslash geq\; 0$$

  (maximum error: {{val|2.5e-5}})

  {{pb}}

  where {{math|p {{=}} 0.47047}}, {{math|a₁ {{=}} 0.3480242}}, {{math|a₂ {{=}} −0.0958798}}, {{math|a₃ {{=}} 0.7478556}}

  $$\backslash operatorname\{erf\}\; x\; \backslash approx\; 1\; -\; \backslash frac\{1\}\{\backslash left(1\; +\; a\_1x\; +\; a\_2x^2\; +\; \backslash cdots\; +\; a\_6x^6\backslash right)^\{16\}\},\; \backslash qquad\; x\; \backslash geq\; 0$$

  (maximum error: {{val|3e-7}})

  {{pb}}

  where {{math|a₁ {{=}} 0.0705230784}}, {{math|a₂ {{=}} 0.0422820123}}, {{math|a₃ {{=}} 0.0092705272}}, {{math|a₄ {{=}} 0.0001520143}}, {{math|a₅ {{=}} 0.0002765672}}, {{math|a₆ {{=}} 0.0000430638}}

  $$\backslash operatorname\{erf\}\; x\; \backslash approx\; 1\; -\; \backslash left(a\_1t\; +\; a\_2t^2\; +\; \backslash cdots\; +\; a\_5t^5\backslash right)e^\{-x^2\},\backslash quad\; t\; =\; \backslash frac\{1\}\{1\; +\; px\}$$

  (maximum error: {{val|1.5e-7}})

  {{pb}}

  where {{math|p {{=}} 0.3275911}}, {{math|a₁ {{=}} 0.254829592}}, {{math|a₂ {{=}} −0.284496736}}, {{math|a₃ {{=}} 1.421413741}}, {{math|a₄ {{=}} −1.453152027}}, {{math|a₅ {{=}} 1.061405429}}

  {{pb}}

  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{{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}}

  $$\backslash begin\{align\}$$

  \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}

• The above have been generalized to sums of {{mvar|N}} exponentials{{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}} with increasing accuracy in terms of {{mvar|N}} so that {{math|erfc x}} can be accurately approximated or bounded by {{math|2Q̃({{sqrt|2}}x)}}, where

  $$\backslash tilde\{Q\}(x)\; =\; \backslash sum\_\{n=1\}^N\; a\_n\; e^\{-b\_n\; x^2\}.$$

  In particular, there is a systematic methodology to solve the numerical coefficients {{math|{(a_n,b_n)}{{su|b=n {{=}} 1|p=N}}}} that yield a 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_n,b_n)}{{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.{{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.}}

• A tight approximation of the complementary error function for {{math|x ∈ [0,∞)}} is given by Karagiannidis & Lioumpas (2007){{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 }} who showed for the appropriate choice of parameters {{math|{A,B}}} that

  $$\backslash operatorname\{erfc\}\; x\; \backslash approx\; \backslash frac\{\backslash left(1\; -\; e^\{-Ax\}\backslash right)e^\{-x^2\}\}\{B\backslash sqrt\{\backslash pi\}\; x\}.$$

  They determined {{math|{A,B} {{=}} {1.98,1.135}}}, 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.{{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|s2cid=231639206}}

• A single-term lower bound is{{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 |s2cid=13636638}}

  $$\backslash operatorname\{erfc\}\; x\; \backslash geq\; \backslash sqrt\{\backslash frac\{2\; e\}\{\backslash pi\}\}\; \backslash frac\{\backslash sqrt\{\backslash beta\; -\; 1\}\}\{\backslash beta\}\; e^\{-\; \backslash beta\; x^2\},\; \backslash qquad\; x\; \backslash ge\; 0,\backslash quad\; \backslash beta\; >\; 1,$$

  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":{{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 }}{{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 }}{{rp|2–3}}

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

  where

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

  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.{{Cite web |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 }}

  {{pb}}

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

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

• An approximation with a maximal error of {{val|1.2e-7}} for any real argument is:{{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 }}

  $$\backslash operatorname\{erf\}\; x\; =\; \backslash begin\{cases\}$$

  1-\tau & x\ge 0\\

  \tau-1 & x < 0

  \end{cases}

  with

  $$\backslash begin\{align\}$$

  \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}

  and

  $$t\; =\; \backslash frac\{1\}\{1\; +\; \backslash frac\{1\}\{2\}|x|\}.$$

• An approximation of $\backslash operatorname\{erfc\}$ with a maximum relative error less than $2^\{-53\}$ $\backslash left(\backslash approx\; 1.1\; \backslash times\; 10^\{-16\}\backslash right)$ in absolute value is:{{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}}

  for {{nowrap|$x\backslash ge\; 0$,}}

  $$\backslash begin\{aligned\}$$

  \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}

  and for

  $$\backslash operatorname\{erfc\}\; \backslash left(x\backslash right)\; =\; 2\; -\; \backslash operatorname\{erfc\}\; \backslash left(-x\backslash right)$$

• A simple approximation for real-valued arguments could be done through Hyperbolic functions:

  $$\backslash operatorname\{erf\}\; \backslash left(x\backslash right)\; \backslash approx\; z(x)\; =\; \backslash tanh\backslash left(\backslash frac\{2\}\{\backslash sqrt\{\backslash pi\}\}\backslash left(x+\backslash frac\{11\}\{123\}x^3\backslash right)\backslash right)$$

  which keeps the absolute difference {{nowrap|.}}

• Since the error function and the Gaussian Q-function are closely related through the identity $\backslash operatorname\{erfc\}(x)\; =\; 2\; Q(\backslash sqrt\{2\}\; x)$ or equivalently $Q(x)\; =\; \backslash frac\{1\}\{2\}\; \backslash operatorname\{erfc\}\backslash left(\backslash frac\{x\}\{\backslash sqrt\{2\}\}\backslash right)$, 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 $x\; \backslash in\; [0,\; \backslash infty)$ was introduced by Abreu (2012){{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}} based on a simple algebraic expression with only two exponential terms:

  $$Q(x)\; \backslash geq\; \backslash frac\{1\}\{12\}\; e^\{-x^2\}\; +\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\; (x\; +\; 1)\}\; e^\{-x^2\; /\; 2\},\; \backslash qquad\; x\; \backslash geq\; 0,$$

  and

  $$Q(x)\; \backslash leq\; \backslash frac\{1\}\{50\}\; e^\{-x^2\}\; +\; \backslash frac\{1\}\{2\; (x\; +\; 1)\}\; e^\{-x^2\; /\; 2\},\; \backslash qquad\; x\; \backslash geq\; 0.$$

  These bounds stem from a unified form $$Q\_\{\backslash mathrm\{B\}\}(x;\; a,\; b)\; =\; \backslash frac\{\backslash exp(-x^2)\}\{a\}\; +\; \backslash frac\{\backslash exp(-x^2\; /\; 2)\}\{b\; (x\; +\; 1)\},$$ where the parameters $a$ and $b$ are selected to ensure the bounding properties: for the lower bound, $a\_\{\backslash mathrm\{L\}\}\; =\; 12$ and $b\_\{\backslash mathrm\{L\}\}\; =\; \backslash sqrt\{2\backslash pi\}$, and for the upper bound, $a\_\{\backslash mathrm\{U\}\}\; =\; 50$ and $b\_\{\backslash mathrm\{U\}\}\; =\; 2$.

  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 $Q^n(x)$ for positive integers $n$ via the binomial theorem, suggesting potential adaptability for powers of $\backslash operatorname\{erfc\}(x)$, though this is less commonly required in error function applications.

{{further|Interval estimation|Coverage probability|68–95–99.7 rule}}

The complementary error function, denoted {{math|erfc}}, is defined as

File:Plot of the complementary error function Erfc(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg

$$\backslash begin\{align\}$$

\operatorname{erfc} x

& = 1-\operatorname{erf} x \\[5pt]

& = \frac{2}{\sqrt\pi} \int_x^\infty e^{-t^2}\,\mathrm dt \\[5pt]

& = e^{-x^2} \operatorname{erfcx} x,

\end{align}

which also defines {{math|erfcx}}, the scaled complementary error function{{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 }} (which can be used instead of {{math|erfc}} to avoid arithmetic underflow{{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 }}). Another form of {{math|erfc x}} for {{math|x ≥ 0}} is known as Craig's formula, after its discoverer: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.

$$\backslash operatorname\{erfc\}\; (x\; \backslash mid\; x\backslash ge\; 0)$$

= \frac{2}{\pi} \int_0^\frac{\pi}{2} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) \, \mathrm d\theta.

This expression is valid only for positive values of {{mvar|x}}, but it 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 as follows:{{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}}

$$\backslash operatorname\{erfc\}\; (x+y\; \backslash mid\; x,y\backslash ge\; 0)\; =\; \backslash frac\{2\}\{\backslash pi\}\; \backslash int\_0^\backslash frac\{\backslash pi\}\{2\}\; \backslash exp\; \backslash left(\; -\; \backslash frac\{x^2\}\{\backslash sin^2\; \backslash theta\}\; -\; \backslash frac\{y^2\}\{\backslash cos^2\; \backslash theta\}\; \backslash right)\; \backslash ,\backslash mathrm\; d\backslash theta.$$

The imaginary error function, denoted {{math|erfi}}, is defined as

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

$$\backslash begin\{align\}$$

\operatorname{erfi} x

& = -i\operatorname{erf} ix \\[5pt]

& = \frac{2}{\sqrt\pi} \int_0^x e^{t^2}\,\mathrm dt \\[5pt]

& = \frac{2}{\sqrt\pi} e^{x^2} D(x),

\end{align}

where {{math|D(x)}} is the Dawson function (which can be used instead of {{math|erfi}} to avoid arithmetic overflow).

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 arguments {{mvar|z}}, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

$$w(z)\; =\; e^\{-z^2\}\backslash operatorname\{erfc\}(-iz)\; =\; \backslash operatorname\{erfcx\}(-iz).$$

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,

File:Normal cumulative distribution function complex plot in Mathematica 13.1 with ComplexPlot3D.svg

$$\backslash begin\{align\}$$

\Phi(x)

&= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^\tfrac{-t^2}{2}\,\mathrm dt\\[6pt]

&= \frac{1}{2} \left(1+\operatorname{erf}\frac{x}{\sqrt 2}\right)\\[6pt]

&= \frac{1}{2} \operatorname{erfc}\left(-\frac{x}{\sqrt 2}\right)

\end{align}

or rearranged for {{math|erf}} and {{math|erfc}}:

$$\backslash begin\{align\}$$

\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}

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

$$\backslash begin\{align\}$$

Q(x) &= \frac{1}{2} - \frac{1}{2} \operatorname{erf} \frac{x}{\sqrt 2}\\

&= \frac{1}{2}\operatorname{erfc}\frac{x}{\sqrt 2}.

\end{align}

The inverse of {{math|Φ}} is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

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

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

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

$$\backslash operatorname\{erf\}\; x\; =\; \backslash frac\{2x\}\{\backslash sqrt\backslash pi\}\; M\backslash left(\backslash tfrac\{1\}\{2\},\backslash tfrac\{3\}\{2\},-x^2\backslash right).$$

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,

$$\backslash operatorname\{erf\}\; x$$

= \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|sgn x}} is the sign function.

The iterated integrals of the complementary error function are defined by{{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}}

$$\backslash begin\{align\}$$

i^n\!\operatorname{erfc} z &= \int_z^\infty i^{n-1}\!\operatorname{erfc} \zeta\,\mathrm 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}

The general recurrence formula is

$$2\; n\; \backslash cdot\; i^n\backslash !\backslash operatorname\{erfc\}\; z\; =\; i^\{n-2\}\backslash !\backslash operatorname\{erfc\}\; z\; -2\; z\; \backslash cdot\; i^\{n-1\}\backslash !\backslash operatorname\{erfc\}\; z$$

They have the power series

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

from which follow the symmetry properties

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

and

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

• 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 erfcf, erfcl.{{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}}
• The GNU Scientific Library provides erf, erfc, log(erf), and scaled error functions.{{Cite web|url=https://www.gnu.org/software/gsl/doc/html/specfunc.html#error-functions|title = Special Functions – GSL 2.7 documentation}}

• [https://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]

{{Reflist}}

• {{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 }}

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

{{Nonelementary Integral}}

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Category:Special hypergeometric functions

Category:Gaussian function

Category:Functions related to probability distributions

Category:Analytic functions