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Error function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Loss_function" title="Loss function">Loss function</a>.</div>

In mathematics, the error function (also called the Gauss error function), often denoted by erf, is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} }</annotation>

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</math><img src="./90673157b19ab6205b1dde5e120f594b318abd83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.717ex; height:2.176ex;" alt="{\displaystyle \mathrm {erf} :\mathbb {C} \to \mathbb {C} }" loading="lazy"> defined as:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t.}</annotation>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Error function</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><div class="infobox-caption">Plot of the error function over real numbers</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t}</annotation>

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</math><img src="./a16727082e99fb7dc4add2c215a3fcb428923e73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.535ex; height:6.343ex;" alt="{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\int _{0}^{z}e^{-t^{2}}\,\mathrm {d} t}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fields of application</th><td class="infobox-data">Probability, thermodynamics, digital communications</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain, codomain and image</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Domain_of_a_function" title="Domain of a function">Domain</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Image_(mathematics)" title="Image (mathematics)">Image</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(-1,1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(-1,1\right)}</annotation>

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</math><img src="./81ce4d89119be71661d461e6916a8a733ed9cd88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle \left(-1,1\right)}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Basic features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Even_and_odd_functions" title="Even and odd functions">Parity</a></th><td class="infobox-data">Odd</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Specific features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Zero_of_a_function" title="Zero of a function">Root</a></th><td class="infobox-data">0</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Derivative" title="Derivative">Derivative</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}}</annotation>

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</math><img src="./f3d408f3670e1c95916a098acaf9cbf5faff4531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.557ex; height:6.343ex;" alt="{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Antiderivative" title="Antiderivative">Antiderivative</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \operatorname {erf} (z)\,dz=z\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C}">

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<annotation encoding="application/x-tex">{\displaystyle \int \operatorname {erf} (z)\,dz=z\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C}</annotation>

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</math><img src="./b97c9352fa865e2c0725b82b3db8a97209a031e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.432ex; height:7.009ex;" alt="{\displaystyle \int \operatorname {erf} (z)\,dz=z\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C}" loading="lazy"></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Series definition</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="Taylor_series" title="Taylor series">Taylor series</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}{\frac {z^{2n+1}}{n!}}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}{\frac {z^{2n+1}}{n!}}}</annotation>

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</math><img src="./015bc156654e2cdaee214922af1e006c55372cb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.349ex; height:6.843ex;" alt="{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}{\frac {z^{2n+1}}{n!}}}" loading="lazy"></td></tr></tbody></table>

The integral here is a complex <a href="Contour_integration" title="Contour integration">contour integral</a> which is path-independent because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(-t^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle \exp(-t^{2})}</annotation>

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In some old texts,<a href="#cite_note-2">[2]</a>

the error function is defined without the factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{\sqrt {\pi }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\sqrt {\pi }}}}</annotation>

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This <a href="Nonelementary_integral" title="Nonelementary integral">nonelementary integral</a> is a <a href="Sigmoid_function" title="Sigmoid function">sigmoid</a> function that occurs often in <a href="Probability" title="Probability">probability</a>, <a href="Statistics" title="Statistics">statistics</a>, and <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.

In statistics, for non-negative real values of x, the error function has the following interpretation: for a real <a href="Random_variable" title="Random variable">random variable</a> Y that is <a href="Normal_distribution" title="Normal distribution">normally distributed</a> with <a href="Mean" title="Mean">mean</a> 0 and <a href="Standard_deviation" title="Standard deviation">standard deviation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2}}}}</annotation>

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Two closely related functions are the complementary error function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {erfc} :\mathbb {C} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {erfc} :\mathbb {C} \to \mathbb {C} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (z)=1-\operatorname {erf} (z),}</annotation>

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and the imaginary error function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {erfi} :\mathbb {C} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {erfi} :\mathbb {C} \to \mathbb {C} }</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfi} (z)=-i\operatorname {erf} (iz),}</annotation>

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where i is the <a href="Imaginary_unit" title="Imaginary unit">imaginary unit</a>.

The name "error function" and its abbreviation erf were proposed by <a href="James_Whitbread_Lee_Glaisher" title="James Whitbread Lee Glaisher">J. W. L. Glaisher</a> in 1871 on account of its connection with "the theory of Probability, and notably the theory of <a href="Errors_and_residuals" title="Errors and residuals">Errors</a>."<a href="#cite_note-Glaisher1871a-3">[3]</a> The error function complement was also discussed by Glaisher in a separate publication in the same year.<a href="#cite_note-Glaisher1871b-4">[4]</a>

For the "law of facility" of errors whose <a href="Probability_density" class="mw-redirect" title="Probability density">density</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle f(x)=\left({\frac {c}{\pi }}\right)^{1/2}e^{-cx^{2}}}</annotation>

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(the <a href="Normal_distribution" title="Normal distribution">normal distribution</a>), Glaisher calculates the probability of an error lying between p and q as:

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {c}{\pi }}\right)^{\frac {1}{2}}\int _{p}^{q}e^{-cx^{2}}\,\mathrm {d} x={\tfrac {1}{2}}\left(\operatorname {erf} \left(q{\sqrt {c}}\right)-\operatorname {erf} \left(p{\sqrt {c}}\right)\right).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

When the results of a series of measurements are described by a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> with <a href="Standard_deviation" title="Standard deviation">standard deviation</a> σ and <a href="Expected_value" title="Expected value">expected value</a> 0, then erf (<style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠a/σ √2⁠) 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 <a href="Bit_error_rate" title="Bit error rate">bit error rate</a> of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the <a href="Heat_equation" title="Heat equation">heat equation</a> when <a href="Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> are given by the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>.

The error function and its approximations can be used to estimate results that hold <a href="With_high_probability" title="With high probability">with high probability</a> or with low probability. Given a random variable X ~ Norm[μ,σ] (a normal distribution with mean μ and standard deviation σ) and a constant L > μ, it can be shown via integration by substitution:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

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where A and B are certain numeric constants. If L is sufficiently far from the mean, specifically μ − L ≥ σ√ln(k), then:

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<annotation encoding="application/x-tex">{\displaystyle \Pr[X\leq L]\leq A\exp(-B\ln(k))={\frac {A}{k^{B}}}}</annotation>

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so the probability goes to 0 as k → ∞.

The probability for X being in the interval [La, Lb] can be derived as

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</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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 {d} x\\&={\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{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:254px;max-width:254px"><div class="trow"><div class="theader">Plots in the complex plane</div></div><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"></div><div class="thumbcaption">Integrand exp(−z2)</div></div></div><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"></div><div class="thumbcaption">erf(z)</div></div></div></div></div>

The property erf (−z) = −erf(z) means that the error function is an <a href="Even_and_odd_functions" title="Even and odd functions">odd function</a>. This directly results from the fact that the integrand e−t2 is an <a href="Even_function" class="mw-redirect" title="Even function">even function</a> (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 <a href="Entire_function" title="Entire function">entire function</a> which takes real numbers to real numbers, for any <a href="Complex_number" title="Complex number">complex number</a> z:

<mover>

</mover>

</mrow>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} ({\overline {z}})={\overline {\operatorname {erf} (z)}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {z}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {z}}}</annotation>

</semantics>

</math><img src="./64281d029a1d4bef9545644f01821c713f876f76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.208ex; height:2.343ex;" alt="{\displaystyle {\overline {z}}}" loading="lazy"> denotes the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">.

The integrand f = exp(−z2) and f = erf(z) are shown in the complex z-plane in the figures at right with <a href="Domain_coloring" title="Domain coloring">domain coloring</a>.

The error function at +∞ is exactly 1 (see <a href="Gaussian_integral" title="Gaussian integral">Gaussian integral</a>). At the real axis, erf z approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

<div class="mw-heading mw-heading3"><h3 id="Taylor_series">Taylor series</h3></div>

The error function is an <a href="Entire_function" title="Entire function">entire function</a>; it has no singularities (except that at infinity) and its <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> always converges. For x >> 1, however, cancellation of leading terms makes the Taylor expansion unpractical.

The defining integral cannot be evaluated in <a href="Closed-form_expression" title="Closed-form expression">closed form</a> in terms of <a href="Elementary_function_(differential_algebra)" class="mw-redirect" title="Elementary function (differential algebra)">elementary functions</a> (see <a href="Liouville's_theorem_(differential_algebra)" title="Liouville's theorem (differential algebra)">Liouville's theorem</a>), but by expanding the <a href="Integrand" class="mw-redirect" title="Integrand">integrand</a> e−z2 into its <a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> and integrating term by term, one obtains the error function's Maclaurin series as:

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

which holds for every <a href="Complex_number" title="Complex number">complex number</a> z. The denominator terms are sequence <a href="https://oeis.org/A007680" class="extiw external" title="oeis:A007680">A007680</a> in the <a href="OEIS" class="mw-redirect" title="OEIS">OEIS</a>.

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

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

because ⁠−(2k − 1)z2/k(2k + 1)⁠ expresses the multiplier to turn the kth term into the (k + 1)th term (considering z as the first term).

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

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

which holds for every <a href="Complex_number" title="Complex number">complex number</a> z.

<div class="mw-heading mw-heading3"><h3 id="Derivative_and_integral">Derivative and integral</h3></div>

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

<mfrac>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}e^{-z^{2}}.}</annotation>

</semantics>

</math>

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

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dz}}\operatorname {erfi} (z)={\frac {2}{\sqrt {\pi }}}e^{z^{2}}.}</annotation>

</semantics>

</math>Higher order derivatives are given by

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} ^{(k)}(z)={\frac {2(-1)^{k-1}}{\sqrt {\pi }}}{\mathit {H}}_{k-1}(z)e^{-z^{2}}={\frac {2}{\sqrt {\pi }}}{\frac {\mathrm {d} ^{k-1}}{\mathrm {d} z^{k-1}}}\left(e^{-z^{2}}\right),\qquad k=1,2,\dots }</annotation>

</semantics>

</math>

where H are the physicists' <a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a>.<a href="#cite_note-5">[5]</a>

An <a href="Antiderivative" title="Antiderivative">antiderivative</a> of the error function, obtainable by <a href="Integration_by_parts" title="Integration by parts">integration by parts</a>, is

<mfrac>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \operatorname {erf} (z)dz=z\operatorname {erf} (z)+{\frac {e^{-z^{2}}}{\sqrt {\pi }}}+C.}</annotation>

</semantics>

</math>

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

<mfrac>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \operatorname {erfi} (z)dz=z\operatorname {erfi} (z)-{\frac {e^{z^{2}}}{\sqrt {\pi }}}+C.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Bürmann_series">Bürmann series</h3></div>

An expansion,<a href="#cite_note-6">[6]</a> which converges more rapidly for all real values of x than a Taylor expansion, is obtained by using <a href="Hans_Heinrich_B%C3%BCrmann" title="Hans Heinrich Bürmann">Hans Heinrich Bürmann</a>'s theorem:<a href="#cite_note-7">[7]</a>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msqrt>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msqrt>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erf} (x)&={\frac {2}{\sqrt {\pi }}}\operatorname {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}{276480}}\left(1-e^{-x^{2}}\right)^{4}-\cdots \right)\\[10pt]&={\frac {2}{\sqrt {\pi }}}\operatorname {sgn}(x)\cdot {\sqrt {1-e^{-x^{2}}}}\left({\frac {\sqrt {\pi }}{2}}+\sum _{k=1}^{\infty }c_{k}e^{-kx^{2}}\right).\end{aligned}}}</annotation>

</semantics>

</math>

where sgn is the <a href="Sign_function" title="Sign function">sign function</a>. By keeping only the first two coefficients and choosing c1 = ⁠31/200⁠ and c2 = −⁠341/8000⁠, the resulting approximation shows its largest relative error at x = ±1.40587, where it is less than 0.0034361:

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<msqrt>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx {\frac {2}{\sqrt {\pi }}}\operatorname {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).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Inverse_functions">Inverse functions</h3></div>

Given a complex number z, there is not a unique complex number w satisfying erf(w) = z, so a true inverse function would be multivalued. However, for −1 < x < 1, there is a unique real number denoted erf−1(x) satisfying

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} \left(\operatorname {erf} ^{-1}(x)\right)=x.}</annotation>

</semantics>

</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 |z| < 1 of the complex plane, using the Maclaurin series<a href="#cite_note-8">[8]</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<msub>

</mrow>

</msub>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} ^{-1}(z)=\sum _{k=0}^{\infty }{\frac {c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},}</annotation>

</semantics>

</math>

where c0 = 1 and

<mtr>

<mtd>

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<mtd>

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</munderover>

<mfrac>

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<msub>

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</msub>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

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</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math>

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

<msup>

</mrow>

</msup>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

<mrow>

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<msup>

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</msup>

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</mfrac>

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<msup>

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</msup>

<mfrac>

<mrow>

<msup>

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</msup>

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</mfrac>

</mrow>

<msup>

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</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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).}</annotation>

</semantics>

</math>

(After cancellation the numerator and denominator values in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A092676" class="extiw external" title="oeis:A092676">A092676</a> and <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A092677" class="extiw external" title="oeis:A092677">A092677</a> respectively; without cancellation the numerator terms are values in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A002067" class="extiw external" title="oeis:A002067">A002067</a>.) The error function's value at ±∞ is equal to ±1.

For |z| < 1, we have erf(erf−1(z)) = z.

The inverse complementary error function is defined as

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} ^{-1}(1-z)=\operatorname {erf} ^{-1}(z).}</annotation>

</semantics>

</math>

For real x, there is a unique real number erfi−1(x) satisfying erfi(erfi−1(x)) = x. The inverse imaginary error function is defined as erfi−1(x).<a href="#cite_note-9">[9]</a>

For any real x, <a href="Newton's_method" title="Newton's method">Newton's method</a> can be used to compute erfi−1(x), and for −1 ≤ x ≤ 1, the following Maclaurin series converges:

<msup>

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</msup>

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</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

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</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfi} ^{-1}(z)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}c_{k}}{2k+1}}\left({\frac {\sqrt {\pi }}{2}}z\right)^{2k+1},}</annotation>

</semantics>

</math>

where ck is defined as above.

<div class="mw-heading mw-heading3"><h3 id="Asymptotic_expansion">Asymptotic expansion</h3></div>

A useful <a href="Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a> of the complementary error function (and therefore also of the error function) for large real x is

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</mfrac>

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<mrow>

<mrow>

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<msup>

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</mfrac>

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</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erfc} (x)&={\frac {e^{-x^{2}}}{x{\sqrt {\pi }}}}\left(1+\sum _{n=1}^{\infty }(-1)^{n}{\frac {1\cdot 3\cdot 5\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{aligned}}}</annotation>

</semantics>

</math>

where (2n − 1)!! is the <a href="Double_factorial" title="Double factorial">double factorial</a> of (2n − 1), which is the product of all odd numbers up to (2n − 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that for any integer N ≥ 1 one has

<mfrac>

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<msup>

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</msup>

<mfrac>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \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)}</annotation>

</semantics>

</math>

where the remainder is

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</mrow>

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</msubsup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle R_{N}(x):={\frac {(-1)^{N}\,(2N-1)!!}{{\sqrt {\pi }}\cdot 2^{N-1}}}\int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t,}</annotation>

</semantics>

</math>

which follows easily by induction, writing

<msup>

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</mfrac>

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<msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{-t^{2}}=-{\frac {1}{2t}}\,{\frac {\mathrm {d} }{\mathrm {d} t}}e^{-t^{2}}}</annotation>

</semantics>

</math>

and integrating by parts.

The asymptotic behavior of the remainder term, in <a href="Landau_notation" class="mw-redirect" title="Landau notation">Landau notation</a>, is

<msub>

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</msub>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle R_{N}(x)=O\left(x^{-(1+2N)}e^{-x^{2}}\right)}</annotation>

</semantics>

</math>

as x → ∞. This can be found by

<msub>

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<msup>

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<msup>

<msup>

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<msup>

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</msup>

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<msup>

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<msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle R_{N}(x)\propto \int _{x}^{\infty }t^{-2N}e^{-t^{2}}\,\mathrm {d} t=e^{-x^{2}}\int _{0}^{\infty }(t+x)^{-2N}e^{-t^{2}-2tx}\,\mathrm {d} t\leq e^{-x^{2}}\int _{0}^{\infty }x^{-2N}e^{-2tx}\,\mathrm {d} t\propto x^{-(1+2N)}e^{-x^{2}}.}</annotation>

</semantics>

</math>

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).

<div class="mw-heading mw-heading3"><h3 id="Continued_fraction_expansion">Continued fraction expansion</h3></div>

A <a href="Continued_fraction" title="Continued fraction">continued fraction</a> expansion of the complementary error function was found by <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</a>:<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a>

<mfrac>

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<msub>

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</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Factorial_series">Factorial series</h3></div>

The inverse factorial series:

<mtr>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

converges for Re(z2) > 0. Here

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math>

zn denotes the <a href="Rising_factorial" class="mw-redirect" title="Rising factorial">rising factorial</a>, and s(n,k) denotes a signed <a href="Stirling_number_of_the_first_kind" class="mw-redirect" title="Stirling number of the first kind">Stirling number of the first kind</a>.<a href="#cite_note-12">[12]</a><a href="#cite_note-13">[13]</a>

There also exists a representation by an infinite sum containing the <a href="Double_factorial" title="Double factorial">double factorial</a>:

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (z)={\frac {2}{\sqrt {\pi }}}\sum _{n=0}^{\infty }{\frac {(-2)^{n}(2n-1)!!}{(2n+1)!}}z^{2n+1}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Bounds_and_Numerical_approximations">Bounds and Numerical approximations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Approximation_with_elementary_functions">Approximation with elementary functions</h3></div>

<ul>

<li>

<a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Abramowitz and Stegun</a> 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:

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}\right)^{4}}},\qquad x\geq 0}</annotation>

</semantics>

</math>

(maximum error: 5×10−4)

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

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx 1-\left(a_{1}t+a_{2}t^{2}+a_{3}t^{3}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}},\qquad x\geq 0}</annotation>

</semantics>

</math>

(maximum error: 2.5×10−5)

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

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx 1-{\frac {1}{\left(1+a_{1}x+a_{2}x^{2}+\cdots +a_{6}x^{6}\right)^{16}}},\qquad x\geq 0}</annotation>

</semantics>

</math>

(maximum error: 3×10−7)

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

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx 1-\left(a_{1}t+a_{2}t^{2}+\cdots +a_{5}t^{5}\right)e^{-x^{2}},\quad t={\frac {1}{1+px}}}</annotation>

</semantics>

</math>

(maximum error: 1.5×10−7)

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

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

</li>

<li>

Exponential bounds and a pure exponential approximation for the complementary error function are given by<a href="#cite_note-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erfc} (x)&\leq {\frac {1}{2}}e^{-2x^{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{aligned}}}</annotation>

</semantics>

</math>

</li>

<li>

The above have been generalized to sums of N exponentials<a href="#cite_note-15">[15]</a> with increasing accuracy in terms of N so that erfc(x) can be accurately approximated or bounded by 2Q̃(√2x), where

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {Q}}(x)=\sum _{n=1}^{N}a_{n}e^{-b_{n}x^{2}}.}</annotation>

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</math>

In particular, there is a systematic methodology to solve the numerical coefficients {(an,bn)}N n = 1 that yield a <a href="Minimax_approximation_algorithm" title="Minimax approximation algorithm">minimax</a> approximation or bound for the closely related <a href="Q-function" title="Q-function">Q-function</a>: Q(x) ≈ Q̃(x), Q(x) ≤ Q̃(x), or Q(x) ≥ Q̃(x) for x ≥ 0. The coefficients {(an,bn)}N n = 1 for many variations of the exponential approximations and bounds up to N = 25 have been released to open access as a comprehensive dataset.<a href="#cite_note-16">[16]</a></li>

<li>

A tight approximation of the complementary error function for x ∈ [0,∞) is given by <a href="George_Karagiannidis" title="George Karagiannidis">Karagiannidis</a> & Lioumpas (2007)<a href="#cite_note-17">[17]</a> who showed for the appropriate choice of parameters {A,B} that

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)\approx {\frac {\left(1-e^{-Ax}\right)e^{-x^{2}}}{B{\sqrt {\pi }}x}}.}</annotation>

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They determined {A,B} = {1.98,1.135}, which gave a good approximation for all x ≥ 0. Alternative coefficients are also available for tailoring accuracy for a specific application or transforming the expression into a tight bound.<a href="#cite_note-18">[18]</a>

</li>

<li>

A single-term lower bound is<a href="#cite_note-19">[19]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)\geq {\sqrt {\frac {2e}{\pi }}}{\frac {\sqrt {\beta -1}}{\beta }}e^{-\beta x^{2}},\qquad x\geq 0,\quad \beta >1,}</annotation>

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

</li>

<li>

Another approximation is given by Sergei Winitzki using his "global Padé approximations":<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a>: 2–3

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)\approx \operatorname {sgn} x\cdot {\sqrt {1-\exp \left(-x^{2}{\frac {{\frac {4}{\pi }}+ax^{2}}{1+ax^{2}}}\right)}}}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle a={\frac {8(\pi -3)}{3\pi (4-\pi )}}\approx 0.140012.}</annotation>

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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.<a href="#cite_note-22">[22]</a>

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

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} ^{-1}(x)\approx \operatorname {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)}}.}</annotation>

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<li>

An approximation with a maximal error of 1.2×10−7 for any real argument is:<a href="#cite_note-23">[23]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)={\begin{cases}1-\tau &x\geq 0\\\tau -1&x<0\end{cases}}}</annotation>

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with

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

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and

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<annotation encoding="application/x-tex">{\displaystyle t={\frac {1}{1+{\frac {1}{2}}|x|}}.}</annotation>

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</li>

<li>An approximation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} }</annotation>

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</math><img src="./7e0f7e177f6099d318fa89a322106c82af17a8b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.688ex; height:2.176ex;" alt="{\displaystyle \operatorname {erfc} }" loading="lazy"> with a maximum relative error less than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{-53}}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{-53}}</annotation>

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</math><img src="./de7bec7a871756050a9e943662c2575cab7d0043.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.317ex; height:2.676ex;" alt="{\displaystyle 2^{-53}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\approx 1.1\times 10^{-16}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\approx 1.1\times 10^{-16}\right)}</annotation>

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</math><img src="./24f236f5f6d97a342fd4cac66145070468efee37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.875ex; height:3.343ex;" alt="{\displaystyle \left(\approx 1.1\times 10^{-16}\right)}" loading="lazy"> in absolute value is:<a href="#cite_note-24">[24]</a>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle x\geq 0}</annotation>

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</math><img src="./a2608e2b392b079f5b763f27bf52883dbee3b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 0}" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erfc} \left(x\right)&=\left({\frac {0.56418958354775629}{x+2.06955023132914151}}\right)\left({\frac {x^{2}+2.71078540045147805x+5.80755613130301624}{x^{2}+3.47954057099518960x+12.06166887286239555}}\right)\\&\left({\frac {x^{2}+3.47469513777439592x+12.07402036406381411}{x^{2}+3.72068443960225092x+8.44319781003968454}}\right)\left({\frac {x^{2}+4.00561509202259545x+9.30596659485887898}{x^{2}+3.90225704029924078x+6.36161630953880464}}\right)\\&\left({\frac {x^{2}+5.16722705817812584x+9.12661617673673262}{x^{2}+4.03296893109262491x+5.13578530585681539}}\right)\left({\frac {x^{2}+5.95908795446633271x+9.19435612886969243}{x^{2}+4.11240942957450885x+4.48640329523408675}}\right)e^{-x^{2}}\\\end{aligned}}}</annotation>

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and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0}">

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<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

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</math><img src="./1a4dbbf970b2d2863dcab589eafe006f08e727d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} \left(x\right)=2-\operatorname {erfc} \left(-x\right)}</annotation>

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<li>

A simple approximation for real-valued arguments could be done through <a href="Hyperbolic_functions" title="Hyperbolic functions">Hyperbolic functions</a>:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} \left(x\right)\approx z(x)=\tanh \left({\frac {2}{\sqrt {\pi }}}\left(x+{\frac {11}{123}}x^{3}\right)\right)}</annotation>

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which keeps the absolute difference <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|\operatorname {erf} \left(x\right)-z(x)\right|<0.000358,\,\forall x}">

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<annotation encoding="application/x-tex">{\displaystyle \left|\operatorname {erf} \left(x\right)-z(x)\right|<0.000358,\,\forall x}</annotation>

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</math><img src="./37e2c69cdb17118f3d5c64810c25fcce98693446.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.082ex; height:2.843ex;" alt="{\displaystyle \left|\operatorname {erf} \left(x\right)-z(x)\right|<0.000358,\,\forall x}" loading="lazy">.

</li>

<li>

Since the error function and the Gaussian Q-function are closely related through the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)=2Q({\sqrt {2}}x)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)=2Q({\sqrt {2}}x)}</annotation>

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</math><img src="./e970e2caa69cb8b15ce4637302b9938d7f9e78df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.164ex; height:3.176ex;" alt="{\displaystyle \operatorname {erfc} (x)=2Q({\sqrt {2}}x)}" loading="lazy"> or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle Q(x)={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)}</annotation>

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</math><img src="./6256a7fe36beb424d23c292985b23ca5f1accf29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.505ex; height:6.509ex;" alt="{\displaystyle Q(x)={\frac {1}{2}}\operatorname {erfc} \left({\frac {x}{\sqrt {2}}}\right)}" loading="lazy">, 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [0,\infty )}">

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<annotation encoding="application/x-tex">{\displaystyle x\in [0,\infty )}</annotation>

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</math><img src="./7fdcab24f8202e4a37d346b76fd6a80edcddc03f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.242ex; height:2.843ex;" alt="{\displaystyle x\in [0,\infty )}" loading="lazy"> was introduced by Abreu (2012)<a href="#cite_note-25">[25]</a> based on a simple algebraic expression with only two exponential terms:

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<annotation encoding="application/x-tex">{\displaystyle Q(x)\geq {\frac {1}{12}}e^{-x^{2}}+{\frac {1}{{\sqrt {2\pi }}(x+1)}}e^{-x^{2}/2},\qquad x\geq 0,}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle Q(x)\leq {\frac {1}{50}}e^{-x^{2}}+{\frac {1}{2(x+1)}}e^{-x^{2}/2},\qquad x\geq 0.}</annotation>

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These bounds stem from a unified form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}},}">

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<annotation encoding="application/x-tex">{\displaystyle Q_{\mathrm {B} }(x;a,b)={\frac {\exp(-x^{2})}{a}}+{\frac {\exp(-x^{2}/2)}{b(x+1)}},}</annotation>

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</math> where the parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are selected to ensure the bounding properties: for the lower bound, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {L} }=12}">

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {L} }=12}</annotation>

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</math><img src="./b5b9b11c3b77cc735e1345cbe20622f7b54c9a5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.913ex; height:2.509ex;" alt="{\displaystyle a_{\mathrm {L} }=12}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}}</annotation>

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</math><img src="./11c69d3140e2fcd13f86c0ba858de343f8b7897b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.786ex; height:3.009ex;" alt="{\displaystyle b_{\mathrm {L} }={\sqrt {2\pi }}}" loading="lazy">, and for the upper bound, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{\mathrm {U} }=50}">

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<annotation encoding="application/x-tex">{\displaystyle a_{\mathrm {U} }=50}</annotation>

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</math><img src="./40d96ee67b77ee38e288e21dfff4af8d72dc0e85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.118ex; height:2.509ex;" alt="{\displaystyle a_{\mathrm {U} }=50}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{\mathrm {U} }=2}">

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<annotation encoding="application/x-tex">{\displaystyle b_{\mathrm {U} }=2}</annotation>

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</math><img src="./2ce32971e532fef460efbfc0211550faf5623280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.723ex; height:2.509ex;" alt="{\displaystyle b_{\mathrm {U} }=2}" loading="lazy">.

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{n}(x)}">

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q^{n}(x)}</annotation>

</semantics>

</math><img src="./21d10ac7dc0a3efa8751bfa01d592977ec7c7cbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.196ex; height:2.843ex;" alt="{\displaystyle Q^{n}(x)}" loading="lazy"> for positive integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> via the binomial theorem, suggesting potential adaptability for powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erfc} (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x)}</annotation>

</semantics>

</math><img src="./0ce4532c4ac939dbe7ef6d2f1fee55f0978847cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.827ex; height:2.843ex;" alt="{\displaystyle \operatorname {erfc} (x)}" loading="lazy">, though this is less commonly required in error function applications.

</li>

</ul>

<div class="mw-heading mw-heading3"><h3 id="Table_of_values">Table of values</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Interval_estimation" title="Interval estimation">Interval estimation</a>, <a href="Coverage_probability" title="Coverage probability">Coverage probability</a>, and <a href="68%E2%80%9395%E2%80%9399.7_rule" title="68–95–99.7 rule">68–95–99.7 rule</a></div>

<th>1 − erf(x)

</th></tr>

<tr>

<td>1

</td></tr>

<tr>

<td>0.977435425

</td></tr>

<tr>

<td>0.954888894

</td></tr>

<tr>

<td>0.932378406

</td></tr>

<tr>

<td>0.909921874

</td></tr>

<tr>

<td>0.887537084

</td></tr>

<tr>

<td>0.777297411

</td></tr>

<tr>

<td>0.671373241

</td></tr>

<tr>

<td>0.571607645

</td></tr>

<tr>

<td>0.479500122

</td></tr>

<tr>

<td>0.396143909

</td></tr>

<tr>

<td>0.322198806

</td></tr>

<tr>

<td>0.257899035

</td></tr>

<tr>

<td>0.203091788

</td></tr>

<tr>

<td>0.157299207

</td></tr>

<tr>

<td>0.119794930

</td></tr>

<tr>

<td>0.089686022

</td></tr>

<tr>

<td>0.065992055

</td></tr>

<tr>

<td>0.047714880

</td></tr>

<tr>

<td>0.033894854

</td></tr>

<tr>

<td>0.023651617

</td></tr>

<tr>

<td>0.016209541

</td></tr>

<tr>

<td>0.010909498

</td></tr>

<tr>

<td>0.007209571

</td></tr>

<tr>

<td>0.004677735

</td></tr>

<tr>

<td>0.002979467

</td></tr>

<tr>

<td>0.001862846

</td></tr>

<tr>

<td>0.001143177

</td></tr>

<tr>

<td>0.000688514

</td></tr>

<tr>

<td>0.000406952

</td></tr>

<tr>

<td>0.000022090

</td></tr>

<tr>

<td>0.000000743

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Related_functions">Related functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Complementary_error_function">Complementary error function</h3></div>

The complementary error function, denoted erfc, is defined as

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<mi>erfcx</mi>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erfc} (x)&=1-\operatorname {erf} (x)\\[5pt]&={\frac {2}{\sqrt {\pi }}}\int _{x}^{\infty }e^{-t^{2}}\,\mathrm {d} t\\[5pt]&=e^{-x^{2}}\operatorname {erfcx} (x),\end{aligned}}}</annotation>

</semantics>

</math>

which also defines erfcx, the scaled complementary error function<a href="#cite_note-Cody93-26">[26]</a> (which can be used instead of erfc to avoid <a href="Arithmetic_underflow" title="Arithmetic underflow">arithmetic underflow</a><a href="#cite_note-Cody93-26">[26]</a><a href="#cite_note-Zaghloul07-27">[27]</a>). Another form of erfc x for x ≥ 0 is known as Craig's formula, after its discoverer:<a href="#cite_note-28">[28]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x\mid x\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}\right)\,\mathrm {d} \theta .}</annotation>

</semantics>

</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 erfc of the sum of two non-negative variables is as follows:<a href="#cite_note-29">[29]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erfc} (x+y\mid x,y\geq 0)={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\exp \left(-{\frac {x^{2}}{\sin ^{2}\theta }}-{\frac {y^{2}}{\cos ^{2}\theta }}\right)\,\mathrm {d} \theta .}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Imaginary_error_function">Imaginary error function</h3></div>

The imaginary error function, denoted erfi, is defined as

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<mtd>

</mtd>

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<mtr>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {erfi} (x)&=-i\operatorname {erf} (ix)\\[5pt]&={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{t^{2}}\,\mathrm {d} t\\[5pt]&={\frac {2}{\sqrt {\pi }}}e^{x^{2}}D(x),\end{aligned}}}</annotation>

</semantics>

</math>

where D(x) is the <a href="Dawson_function" title="Dawson function">Dawson function</a> (which can be used instead of erfi to avoid <a href="Arithmetic_overflow" class="mw-redirect" title="Arithmetic overflow">arithmetic overflow</a><a href="#cite_note-Cody93-26">[26]</a>).

Despite the name "imaginary error function", erfi(x) is real when x is real.

When the error function is evaluated for arbitrary <a href="Complex_number" title="Complex number">complex</a> arguments z, the resulting complex error function is usually discussed in scaled form as the <a href="Faddeeva_function" title="Faddeeva function">Faddeeva function</a>:

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<mi>erfcx</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w(z)=e^{-z^{2}}\operatorname {erfc} (-iz)=\operatorname {erfcx} (-iz).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Cumulative_distribution_function">Cumulative distribution function</h3></div>

The error function is essentially identical to the standard <a href="Normal_cumulative_distribution_function" class="mw-redirect" title="Normal cumulative distribution function">normal cumulative distribution function</a>, denoted Φ, also named norm(x) by some software languages, as they differ only by scaling and translation. Indeed,

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Phi (x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{\tfrac {-t^{2}}{2}}\,\mathrm {d} t\\[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{aligned}}}</annotation>

</semantics>

</math>

or rearranged for erf and erfc:

<mtr>

<mtd>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\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{aligned}}}</annotation>

</semantics>

</math>

Consequently, the error function is also closely related to the <a href="Q-function" title="Q-function">Q-function</a>, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as

<mtr>

<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

</semantics>

</math>

The <a href="Inverse_function" title="Inverse function">inverse</a> of Φ is known as the <a href="Quantile_function" title="Quantile function">normal quantile function</a>, or <a href="Probit" title="Probit">probit</a> function and may be expressed in terms of the inverse error function as

<mi>probit</mi>

<msup>

</mrow>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {probit} (p)=\Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)=-{\sqrt {2}}\operatorname {erfc} ^{-1}(2p).}</annotation>

</semantics>

</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 <a href="Mittag-Leffler_function" title="Mittag-Leffler function">Mittag-Leffler function</a>, and can also be expressed as a <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a> (Kummer's function):

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)={\frac {2x}{\sqrt {\pi }}}M\left({\tfrac {1}{2}},{\tfrac {3}{2}},-x^{2}\right).}</annotation>

</semantics>

</math>

It has a simple expression in terms of the <a href="Fresnel_integral" title="Fresnel integral">Fresnel integral</a>.

In terms of the <a href="Regularized_gamma_function" class="mw-redirect" title="Regularized gamma function">regularized gamma function</a> P and the <a href="Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a>,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)=\operatorname {sgn}(x)\cdot P\left({\tfrac {1}{2}},x^{2}\right)={\frac {\operatorname {sgn}(x)}{\sqrt {\pi }}}\gamma {\left({\tfrac {1}{2}},x^{2}\right)}.}</annotation>

</semantics>

</math>sgn(x) is the <a href="Sign_function" title="Sign function">sign function</a>.

<div class="mw-heading mw-heading3"><h3 id="Iterated_integrals_of_the_complementary_error_function">Iterated integrals of the complementary error function</h3></div>

The iterated integrals of the complementary error function are defined by<a href="#cite_note-30">[30]</a>

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<mtd>

<mi>ierfc</mi>

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</mrow>

<mrow>

<mrow>

<mi>ierfc</mi>

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</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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)-2z\operatorname {ierfc} (z)\right)\\\end{aligned}}}</annotation>

</semantics>

</math>

The general recurrence formula is

<msup>

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<annotation encoding="application/x-tex">{\displaystyle 2n\cdot i^{n}\!\operatorname {erfc} (z)=i^{n-2}\!\operatorname {erfc} (z)-2z\cdot i^{n-1}\!\operatorname {erfc} (z)}</annotation>

</semantics>

</math>

They have the power series

<msup>

</mrow>

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</mrow>

</mrow>

</munderover>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle i^{n}\!\operatorname {erfc} (z)=\sum _{j=0}^{\infty }{\frac {(-z)^{j}}{2^{n-j}j!\,\Gamma \left(1+{\frac {n-j}{2}}\right)}},}</annotation>

</semantics>

</math>

from which follow the symmetry properties

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 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)!}}}</annotation>

</semantics>

</math>

and

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle 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)!}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="As_real_function_of_a_real_argument">As real function of a real argument</h3></div>

<ul><li>In <a href="POSIX" title="POSIX">POSIX</a>-compliant operating systems, the header <code><a href="Math.h" class="mw-redirect" title="Math.h">math.h</a></code> shall declare and the mathematical library <code><a href="Libm" class="mw-redirect" title="Libm">libm</a></code> shall provide the functions <code>erf</code> and <code>erfc</code> (<a href="Double_precision" class="mw-redirect" title="Double precision">double precision</a>) as well as their <a href="Single_precision" class="mw-redirect" title="Single precision">single precision</a> and <a href="Extended_precision" title="Extended precision">extended precision</a> counterparts <code>erff</code>, <code>erfl</code> and <code>erfcf</code>, <code>erfcl</code>.<a href="#cite_note-31">[31]</a></li>

<li>The <a href="GNU_Scientific_Library" title="GNU Scientific Library">GNU Scientific Library</a> provides <code>erf</code>, <code>erfc</code>, <code>log(erf)</code>, and scaled error functions.<a href="#cite_note-32">[32]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="As_complex_function_of_a_complex_argument">As complex function of a complex argument</h3></div>

<ul><li><code><a rel="nofollow" class="external text" href="https://jugit.fz-juelich.de/mlz/libcerf">libcerf</a></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 <a href="Faddeeva_function" title="Faddeeva function">Faddeeva function</a> as implemented in the <a rel="nofollow" class="external text" href="http://ab-initio.mit.edu/Faddeeva">MIT Faddeeva Package</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFAndrews1998" class="citation book cs1">Andrews, Larry C. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2CAqsF-RebgC&pg=PA110">Special functions of mathematics for engineers</a>. SPIE Press. p. 110. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780819426161</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFWhittakerWatson2021" class="citation book cs1"><a href="Edmund_T._Whittaker" class="mw-redirect" title="Edmund T. Whittaker">Whittaker, Edmund Taylor</a>; <a href="George_N._Watson" class="mw-redirect" title="George N. Watson">Watson, George Neville</a> (2021). <a href="Victor_Hugo_Moll" class="mw-redirect" title="Victor Hugo Moll">Moll, Victor Hugo</a> (ed.). <a href="A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a> (5th revised ed.). <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 358. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-316-51893-9</bdi>.</cite>

</li>

<li id="cite_note-Glaisher1871a-3"><a href="#cite_ref-Glaisher1871a_3-0">^</a> <cite id="CITEREFGlaisher1871" class="citation journal cs1">Glaisher, James Whitbread Lee (July 1871). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8Po7AQAAMAAJ&pg=RA1-PA294">"On a class of definite integrals"</a>. London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4. 42 (277): 294–302. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786447108640568">10.1080/14786447108640568</a>. Retrieved 6 December 2017.</cite>

</li>

<li id="cite_note-Glaisher1871b-4"><a href="#cite_ref-Glaisher1871b_4-0">^</a> <cite id="CITEREFGlaisher1871" class="citation journal cs1">Glaisher, James Whitbread Lee (September 1871). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yJ1YAAAAcAAJ&pg=PA421">"On a class of definite integrals. Part II"</a>. London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 4. 42 (279): 421–436. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786447108640600">10.1080/14786447108640600</a>. Retrieved 6 December 2017.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Erf.html">"Erf"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFSchöpfSupancic2014" class="citation journal cs1">Schöpf, H. M.; Supancic, P. H. (2014). <a rel="nofollow" class="external text" href="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/">"On Bürmann's Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion"</a>. The Mathematica Journal. 16. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3888%2Ftmj.16-11">10.3888/tmj.16-11</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/BuermannsTheorem.html">"Bürmann's Theorem"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFDominici2006" class="citation arxiv cs1">Dominici, Diego (2006). "Asymptotic analysis of the derivatives of the inverse error function". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0607230">math/0607230</a>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFBergsma2006" class="citation arxiv cs1">Bergsma, Wicher (2006). "On a new correlation coefficient, its orthogonal decomposition and associated tests of independence". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0604627">math/0604627</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, <a href="Trait%C3%A9_de_m%C3%A9canique_c%C3%A9leste" title="Traité de mécanique céleste">Traité de mécanique céleste</a>, tome 4 (1805), livre X, page 255.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFCuytPetersenVerdonkWaadeland2008" class="citation book cs1"><a href="Annie_Cuyt" title="Annie Cuyt">Cuyt, Annie A. M.</a>; Petersen, Vigdis B.; Verdonk, Brigitte; Waadeland, Haakon; Jones, William B. (2008). Handbook of Continued Fractions for Special Functions. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-6948-2</bdi>.</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite id="CITEREFSchlömilch1859" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Oscar_Schl%C3%B6milch" class="mw-redirect" title="Oscar Schlömilch">Schlömilch, Oskar Xavier</a> (1859). <a rel="nofollow" class="external text" href="https://archive.org/details/zeitschriftfrma09runggoog">"Ueber facultätenreihen"</a>. <a href="https://de.wikipedia.org/wiki/Zeitschrift_f%C3%BCr_Mathematik_und_Physik" class="extiw external" title="de:Zeitschrift für Mathematik und Physik">Zeitschrift für Mathematik und Physik</a> (in German). 4: 390–415.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFNielson1906" class="citation book cs1 cs1-prop-foreign-lang-source">Nielson, Niels (1906). <a rel="nofollow" class="external text" href="https://archive.org/details/handbuchgamma00nielrich">Handbuch der Theorie der Gammafunktion</a> (in German). Leipzig: B. G. Teubner. p. 283 Eq. 3. Retrieved 4 December 2017.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFChianiDardariSimon2003" class="citation journal cs1">Chiani, M.; Dardari, D.; Simon, M.K. (2003). <a rel="nofollow" class="external text" href="http://campus.unibo.it/85943/1/mcddmsTranWIR2003.pdf">"New Exponential Bounds and Approximations for the Computation of Error Probability in Fading Channels"</a> (PDF). IEEE Transactions on Wireless Communications. 2 (4): 840–845. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.190.6761">10.1.1.190.6761</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTWC.2003.814350">10.1109/TWC.2003.814350</a>.</cite>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFTanashRiihonen2020" class="citation journal cs1">Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514–6524. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.06939">2007.06939</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOMM.2020.3006902">10.1109/TCOMM.2020.3006902</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220514754">220514754</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFTanashRiihonen2020" class="citation journal cs1">Tanash, I.M.; Riihonen, T. (2020). <a rel="nofollow" class="external text" href="https://zenodo.org/record/4112978">"Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]"</a>. Zenodo. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5281%2Fzenodo.4112978">10.5281/zenodo.4112978</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFKaragiannidisLioumpas2007" class="citation journal cs1">Karagiannidis, G. K.; Lioumpas, A. S. (2007). <a rel="nofollow" class="external text" href="http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf">"An improved approximation for the Gaussian Q-function"</a> (PDF). IEEE Communications Letters. 11 (8): 644–646. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLCOMM.2007.070470">10.1109/LCOMM.2007.070470</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4043576">4043576</a>.</cite>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFTanashRiihonen2021" class="citation journal cs1">Tanash, I.M.; Riihonen, T. (2021). "Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function". IEEE Communications Letters. 25 (5): 1468–1471. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2101.07631">2101.07631</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLCOMM.2021.3052257">10.1109/LCOMM.2021.3052257</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:231639206">231639206</a>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFChangCosmanMilstein2011" class="citation journal cs1">Chang, Seok-Ho; <a href="Pamela_Cosman" title="Pamela Cosman">Cosman, Pamela C.</a>; Milstein, Laurence B. (November 2011). <a rel="nofollow" class="external text" href="http://escholarship.org/uc/item/6hw4v7pg">"Chernoff-Type Bounds for the Gaussian Error Function"</a>. IEEE Transactions on Communications. 59 (11): 2939–2944. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOMM.2011.072011.100049">10.1109/TCOMM.2011.072011.100049</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13636638">13636638</a>.</cite>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite id="CITEREFWinitzki2003" class="citation book cs1">Winitzki, Sergei (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/computationalsci0000iccs_a2w6">"Uniform approximations for transcendental functions"</a>. Computational Science and Its Applications – ICCSA 2003. Lecture Notes in Computer Science. Vol. 2667. Springer, Berlin. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/computationalsci0000iccs_a2w6/page/780">780–789</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-44839-X_82">10.1007/3-540-44839-X_82</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-40155-1</bdi>.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFZengChen2015" class="citation journal cs1">Zeng, Caibin; Chen, Yang Cuan (2015). "Global Padé approximations of the generalized Mittag-Leffler function and its inverse". Fractional Calculus and Applied Analysis. 18 (6): 1492–1506. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1310.5592">1310.5592</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Ffca-2015-0086">10.1515/fca-2015-0086</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118148950">118148950</a>. <q>Indeed, Winitzki [32] provided the so-called global Padé approximation</q></cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFWinitzki2008" class="citation web cs1">Winitzki, Sergei (6 February 2008). <a rel="nofollow" class="external text" href="https://www.academia.edu/9730974/A_handy_approximation_for_the_error_function_and_its_inverse">"A handy approximation for the error function and its inverse"</a>.</cite>

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFPress1992" class="citation book cs1">Press, William H. (1992). Numerical Recipes in Fortran 77: The Art of Scientific Computing. Cambridge University Press. p. 214. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-43064-X</bdi>.</cite>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <cite id="CITEREFDia2023" class="citation journal cs1">Dia, Yaya D. (2023). <a rel="nofollow" class="external text" href="https://www.ssrn.com/abstract=4487559">"Approximate Incomplete Integrals, Application to Complementary Error Function"</a>. SSRN Electronic Journal. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2139%2Fssrn.4487559">10.2139/ssrn.4487559</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1556-5068">1556-5068</a>.</cite>

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <cite id="CITEREFAbreu2012" class="citation journal cs1">Abreu, Giuseppe (2012). "Very Simple Tight Bounds on the Q-Function". IEEE Transactions on Communications. 60 (9): 2415–2420. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOMM.2012.080612.110075">10.1109/TCOMM.2012.080612.110075</a>.</cite>

</li>

<li id="cite_note-Cody93-26">^ <a href="#cite_ref-Cody93_26-0">a</a> <a href="#cite_ref-Cody93_26-1">b</a> <a href="#cite_ref-Cody93_26-2">c</a> <cite id="CITEREFCody1993" class="citation cs2">Cody, W. J. (March 1993), <a rel="nofollow" class="external text" href="http://www.stat.wisc.edu/courses/st771-newton/papers/p22-cody.pdf">"Algorithm 715: SPECFUN—A portable FORTRAN package of special function routines and test drivers"</a> (PDF), <a href="ACM_Trans._Math._Softw." class="mw-redirect" title="ACM Trans. Math. Softw.">ACM Trans. Math. Softw.</a>, 19 (1): 22–32, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.643.4394">10.1.1.643.4394</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F151271.151273">10.1145/151271.151273</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5621105">5621105</a></cite>

</li>

<li id="cite_note-Zaghloul07-27"><a href="#cite_ref-Zaghloul07_27-0">^</a> <cite id="CITEREFZaghloul2007" class="citation cs2">Zaghloul, M. R. (1 March 2007), "On the calculation of the Voigt line profile: a single proper integral with a damped sine integrand", <a href="Monthly_Notices_of_the_Royal_Astronomical_Society" title="Monthly Notices of the Royal Astronomical Society">Monthly Notices of the Royal Astronomical Society</a>, 375 (3): 1043–1048, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007MNRAS.375.1043Z">2007MNRAS.375.1043Z</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-2966.2006.11377.x">10.1111/j.1365-2966.2006.11377.x</a></cite>

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> John W. Craig, <a rel="nofollow" class="external text" href="http://wsl.stanford.edu/~ee359/craig.pdf">A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120403231129/http://wsl.stanford.edu/~ee359/craig.pdf">Archived</a> 3 April 2012 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, Proceedings of the 1991 IEEE Military Communication Conference, vol. 2, pp. 571–575.

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFBehnad2020" class="citation journal cs1">Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117–4125. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCOMM.2020.2986209">10.1109/TCOMM.2020.2986209</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:216500014">216500014</a>.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFCarslawJaeger1959" class="citation book cs1"><a href="Horatio_Scott_Carslaw" title="Horatio Scott Carslaw">Carslaw, H. S.</a>; <a href="John_Conrad_Jaeger" title="John Conrad Jaeger">Jaeger, J. C.</a> (1959). Conduction of Heat in Solids (2nd ed.). Oxford University Press. p. 484. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-853368-9</bdi>.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/math.h.html">"math.h - mathematical declarations"</a>. opengroup.org. 2018. Retrieved 21 April 2023.</cite>

</li>

<li id="cite_note-32"><a href="#cite_ref-32">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.gnu.org/software/gsl/doc/html/specfunc.html#error-functions">"Special Functions – GSL 2.7 documentation"</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_297.htm">"Chapter 7"</a>. <a href="Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</a>. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-61272-0</bdi>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://www.loc.gov/item/65012253">65-12253</a>.</cite></li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811154417/http://apps.nrbook.com/empanel/index.html#pg=259">"Section 6.2. Incomplete Gamma Function and Error Function"</a>, Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=259">the original</a> on 11 August 2011, retrieved 9 August 2011</cite></li>

<li><cite id="CITEREFTemme2010" class="citation cs2">Temme, Nico M. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/7">"Error Functions, Dawson's and Fresnel Integrals"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2723248">2723248</a></cite>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn1p1_A1b.pdf">A Table of Integrals of the Error Functions</a></li></ul>

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</style><div id="Nonelementary_integrals27" style="font-size:114%;margin:0 4em"><a href="Nonelementary_integral" title="Nonelementary integral">Nonelementary integrals</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Elliptic_integral" title="Elliptic integral">Elliptic integral</a></li>

<li><a href="Exponential_integral" title="Exponential integral">Exponential integral</a></li>

<li><a href="Fresnel_integral" title="Fresnel integral">Fresnel integral</a></li>

<li><a href="Logarithmic_integral_function" title="Logarithmic integral function">Logarithmic integral function</a></li>

<li><a href="Trigonometric_integral" title="Trigonometric integral">Trigonometric integral</a></li></ul>

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