Erfc

The complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$, the Gaussian's tail-probability integral.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaError function↗MathWorldErfc↗Erf↗WikidataQ579262↗DLMF7.2.E2↗7.2.E1↗FungrimErfc✓↗Fungrim entry7f355d✓↗ae3110↗bfc86e✓↗ec0205✓↗Wolfram LanguageErfc!↗

Erfc(z)the complementary error function $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$.

SageMatherfc(z)SymPyerfc(z)mpmatherfc(z)Juliaenumeratio_ball(erfc, z)Rusterfc(z)

Typed signatures

Type	Package	Applies
(complex | signed_infinity) -> complex	analyticreplaces compute-engine	always
(complex | signed_infinity) -> complex	compute-engine	always

Domain: Special functions

Details

• Defined as $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$. See Erf.
• $\operatorname{erfc}(0) = 1$, $\operatorname{erfc}(\infty) = 0$, $\operatorname{erfc}(-\infty) = 2$.
• $\operatorname{erfc}(-z) = 2 - \operatorname{erfc}(z)$, the mirror image of Erf's oddness.
• Computed directly from a continued fraction for large $|z|$ rather than as $1-\operatorname{erf}(z)$, since that subtraction would lose all precision once $\operatorname{erf}(z)$ rounds to $\pm1$.
• compute-engine requires an inexact (floating-point) argument to produce a numeric value under plain evaluation, the same convention as Erf.

Plot

ComplexPlot

real slice

Examples

Basic #

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In$\operatorname{erfc}(0)$Out$1$

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In$\operatorname{erfc}(∞)$Out$0$

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In$\operatorname{erfc}(-∞)$Out$2$

A floating-point argument evaluates directly to a high-precision decimal

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In$\operatorname{erfc}(0.5)$Out$0.479\,500\,122\,186\,953\,462\,317$

A machine-precision argument

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In$\operatorname{erfc}(1.5)$Out$0.033\,894\,853\,524\,689\,272\,933$

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In$\operatorname{Series}(\operatorname{erfc}(x), x, 0, 10)$Out$1-\frac{2x}{\sqrt{π}}+\frac{2x^3}{3\sqrt{π}}-\frac{x^5}{5\sqrt{π}}+\frac{x^7}{21\sqrt{π}}-\frac{x^9}{108\sqrt{π}}+O\left(x^{11}\right)$

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In$\operatorname{Normal}(\operatorname{Series}(\operatorname{erfc}(x), x, ∞, 5))$Out$\operatorname{Series}(\operatorname{erfc}(x), x, ∞, 5)$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\operatorname{erfc}(\bigl\lbrack0, 1\bigr\rbrack)$Out$\bigl\lbrack1, \operatorname{erfc}(1)\bigr\rbrack$

Complex arguments evaluate, as they do for Erf: $\operatorname{erfc}(z) = 1 - \operatorname{erf}(z)$

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In$\operatorname{erfc}(1.5-i)$Out$-0.078\,399\,207\,498\,933\,45-0.027\,963\,711\,238\,655\,847i$

Listable: threads over its limits at $\pm\infty$

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In$\operatorname{erfc}(\bigl\lbrack∞, -∞\bigr\rbrack)$Out$\bigl\lbrack0, 2\bigr\rbrack$

Interval arithmetic: Erfc is decreasing, so the endpoints swap

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In$\operatorname{erfc}(\mathrm{Interval}(0.2, 0.3))$Out$\mathrm{Interval}(0.671\,373\,240\,540\,872\,6, 0.777\,297\,410\,789\,521\,5)$

Uncertainty propagation

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In$\operatorname{erfc}(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.004\,677\,734\,981\,047\,266, 0.000\,206\,669\,853\,540\,920\,54)$

To 30 significant digits, every one correctly rounded

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In$N(\operatorname{erfc}(2), 30)$Out$0.004\,677\,734\,981\,047\,265\,837\,930\,743\,632\,75$

A 50-digit value of $\mathrm{erfc}(3/2)$

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In$N(\operatorname{erfc}(\frac{3}{2}), 50)$Out$0.033\,894\,853\,524\,689\,272\,933\,023\,738\,354\,052\,141\,318\,589\,520\,742\,363$

Matrix function of an off-diagonal complex matrix, expanded through its eigenvalues

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In$\mathrm{FunctionExpand}(\mathrm{MatrixFunction}(\mathrm{Erfc}, \bigl\lbrack\bigl\lbrack0, (1/2)i π\bigr\rbrack, \bigl\lbrack π, 0\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack\frac{1}{2}\operatorname{erfc}((-1/2-(1/2)i)π)+\frac{1}{2}\operatorname{erfc}((1/2+(1/2)i)π), (-\frac{1}{4}-\frac{1}{4}i)\operatorname{erfc}((-1/2-(1/2)i)π)+(\frac{1}{4}+\frac{1}{4}i)\operatorname{erfc}((1/2+(1/2)i)π)\bigr\rbrack, \bigl\lbrack(\frac{1}{2}-\frac{1}{2}i)\operatorname{erfc}((1/2+(1/2)i)π)+(-\frac{1}{2}+\frac{1}{2}i)\operatorname{erfc}((-1/2-(1/2)i)π), \frac{1}{2}\operatorname{erfc}((-1/2-(1/2)i)π)+\frac{1}{2}\operatorname{erfc}((1/2+(1/2)i)π)\bigr\rbrack\bigr\rbrack$

The reflection identity $\mathrm{erfc}(-z) = 2 - \mathrm{erfc}(z)$ holds

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In$\mathrm{FullSimplify}(\operatorname{erfc}(-z)=2-\operatorname{erfc}(z))$Out$\mathrm{True}$

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In$\operatorname{Series}(\operatorname{erfc}(x), x, 0, 7)$Out$1-\frac{2x}{\sqrt{π}}+\frac{2x^3}{3\sqrt{π}}-\frac{x^5}{5\sqrt{π}}+\frac{x^7}{21\sqrt{π}}+O\left(x^9\right)$

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In$\operatorname{Normal}(\operatorname{Series}(\operatorname{erfc}(x), x, ∞, 1))$Out$\operatorname{Series}(\operatorname{erfc}(x), x, ∞, 1)$

Properties #

$\operatorname{erfc}(-z) = 2 - \operatorname{erfc}(z)$, here at $z = 1$

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In$\operatorname{erfc}(-1)+\operatorname{erfc}(1)=2$Out$\mathrm{True}$

Definition: $\operatorname{erf}(z) + \operatorname{erfc}(z) = 1$. See Erf

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In$\operatorname{erf}(2)+\operatorname{erfc}(2)=1$Out$\mathrm{True}$

$\operatorname{erfc}(-x) = 2 - \operatorname{erfc}(x)$ applied symbolically

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In$\operatorname{erfc}(-x)$Out$2-\operatorname{erfc}(x)$

$\frac{d}{dx}\operatorname{erfc}(x) = -\frac{2}{\sqrt\pi}e^{-x^2}$

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In$\frac{\mathrm{d}\operatorname{erfc}(x)}{\mathrm{d}x}$Out$-(2e^{-x^2})/\sqrt{π}$

Possible issues #

An exact integer argument is left unevaluated under plain evaluation -- pair with N() or use an inexact argument like 0.5 for a decimal

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In$\operatorname{erfc}(1)$Out$\operatorname{erfc}(1)$

Far in the tail Erfc keeps full relative precision: $\operatorname{erfc}(9.5) \approx 3.8\times10^{-41}$...

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In$\operatorname{erfc}(9.5)$Out$3.769\,214\,485\,654\,879\,941\,68\cdot10^{-41}$

...whereas $1 - \operatorname{erf}(9.5)$ cancels to exactly 0 in floating point

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In$1-\operatorname{erf}(9.5)$Out$0.0$

Erfc(ComplexInfinity) is Indeterminate, inheriting Erf's direction-dependent growth off the real axis

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Known value · DLMF 7.2.2

Neat examples #

The Gaussian tail shrinks fast: only about $7\times10^{-7}$ of the area lies beyond $3.5$ standard deviations

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In$\operatorname{erfc}(3.5)$Out$0.000\,000\,743\,098\,372\,341\,412\,745\,524$

About

Examples	27
Checked against	Wolfram Language 22 agree3 differ2 unevaluatedSageMath 20 agree1 differ3 unevaluatedSymPy 21 agree2 differ2 unevaluatedmpmath 15 agreeJulia 13 agree2 errorRust 12 agree
Crosswalk	WikipediaMathWorld ×2WikidataDLMF ×2FungrimFungrim entry ×4Wolfram LanguageSageMathSymPympmathJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: Erf