complementary error function

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21: 8.4 Special Values

… ►For $\mathrm{erf}\left(z\right)$, $\mathrm{erfc}\left(z\right)$, and $F\left(z\right)$, see §§7.2(i), 7.2(ii). … ►

8.4.6 $$\mathrm{\Gamma }(\frac{1}{2},z^2)=2{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=\sqrt{\pi }\mathrm{erfc}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.17

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

… ►

8.4.14 $$Q(n+\frac{1}{2},z^2)=\mathrm{erfc}\left(z\right)+\frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}\sum _{k=1}^n\frac{z^{2k-1}}{{\left(\frac{1}{2}\right)}_k},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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22: Software Index

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	Open Source													With Book						Commercial
…
7.25(ii) $\mathrm{erf}x$, $\mathrm{erfc}x$, ${\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $x\in ℝ$	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	✓		✓	✓	✓	✓	✓	✓	NMS
7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$	✓								✓			a	✓	✓							✓	✓	✓
…

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23: 2.11 Remainder Terms; Stokes Phenomenon

… ►Here $\mathrm{erfc}$ is the complementary error function (§7.2(i)), and … ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. On the other hand, when $\pi +\delta \le \theta \le 3\pi -\delta$, $c(\theta )$ is in the left half-plane and $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ differs from 2 by an exponentially-small quantity. In the transition through $\theta =\pi$, $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case $\rho =100$. ►

►►

Figure 2.11.1: Graph of $|\mathrm{erfc}\left(\sqrt{50}c(\theta )\right)|$. Magnify

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24: 7.12 Asymptotic Expansions

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§7.12(i) Complementary Error Function

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$\mathrm{erfc}z\sim {\displaystyle \frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}}{\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{(-1)}^m{\displaystyle \frac{{\left(\frac{1}{2}\right)}_m}{z^{2m+1}}},$

►

$\mathrm{erfc}\left(-z\right)\sim 2-{\displaystyle \frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}}{\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{(-1)}^m{\displaystyle \frac{{\left(\frac{1}{2}\right)}_m}{z^{2m+1}}},$

… ►For these and other error bounds see Olver (1997b, pp. 109–112), with $\alpha =\frac{1}{2}$ and $z$ replaced by $z^2$; compare (7.11.2). … ►(Note that some of these re-expansions themselves involve the complementary error function.) …

25: 7.5 Interrelations

… ►

7.5.10 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=\frac{1}{2}(1\pm \mathrm{i}){\mathrm{e}}^{{\zeta }^2}\mathrm{erfc}\zeta .$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable

A&S Ref:

7.3.22 (in different form)

Referenced by:

§7.12(ii), §7.13(iv), §7.5, §7.5

Permalink:

http://dlmf.nist.gov/7.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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26: 7.19 Voigt Functions

… ►

7.19.3 $$𝖴(x,t)+\mathrm{i}𝖵(x,t)=\sqrt{\frac{\pi }{4t}}{\mathrm{e}}^{z^2}\mathrm{erfc}z,$$ $z=(1-\mathrm{i}x)/(2\sqrt{t})$.

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Symbols:

$𝖴(x,t)$: Voigt function, $𝖵(x,t)$: Voigt function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable and $z$: complex variable

Referenced by:

§7.19(i), §7.22(iv)

Permalink:

http://dlmf.nist.gov/7.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.19(i), §7.19 and Ch.7

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27: 32.10 Special Function Solutions

… ►

32.10.22

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: real, $\epsilon =\pm 1$ and $C$: constant

Permalink:

http://dlmf.nist.gov/32.10.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §32.10(iv), §32.10 and Ch.32

►where $C$ is an arbitrary constant and $\mathrm{erfc}$ is the complementary error function (§7.2(i)). …

28: 8.18 Asymptotic Expansions of $I_x(a,b)$

… ►

8.18.9 $$I_x(a,b)\sim \frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi (a+b)}}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{c}_k(\eta )}{{(a+b)}^k},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

… ►For $\mathrm{erfc}$ see §7.2(i). …

29: 3.5 Quadrature

… ►

3.5.42 $$\mathrm{erfc}\lambda =\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{\zeta -2\lambda \sqrt{\zeta }}\frac{d\zeta }{\zeta },$$ $c>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral and $\lambda$: parameter

A&S Ref:

29.3.83 (in different form)

Referenced by:

§3.5(ix)

Permalink:

http://dlmf.nist.gov/3.5.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

►where $\mathrm{erfc}z$ is the complementary error function, and from (7.12.1) it follows that ►

3.5.43 $$\mathrm{erfc}\lambda \sim \frac{{\mathrm{e}}^{-{\lambda }^2}}{\sqrt{\pi }\lambda },$$ $\lambda \to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/3.5.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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3.5.45 $$\mathrm{erfc}\lambda =\frac{{\mathrm{e}}^{-{\lambda }^2}}{2\pi }{\int }_{-\pi }^{\pi }{\mathrm{e}}^{-{\lambda }^2{\tan }^2\left(\frac{1}{2}\theta \right)}d\theta .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\tan z$: tangent function and $\lambda$: parameter

Referenced by:

Table 3.5.20

Permalink:

http://dlmf.nist.gov/3.5.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

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Table 3.5.20: Composite trapezoidal rule for the integral (3.5.45) with $\lambda =10$.

► ►►

$h$		$\mathrm{erfc}\lambda$		$n$
…

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30: 13.18 Relations to Other Functions

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

Erratum (V1.0.3) for Equation (13.18.7)

Permalink:

http://dlmf.nist.gov/13.18.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.3):

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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