error functions

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31: 7.12 Asymptotic Expansions

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§7.12(i) Complementary Error Function

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

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

… ►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.) …

32: 13.6 Relations to Other Functions

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§13.6(ii) Incomplete Gamma Functions

… ►Special cases are the error functions ►

13.6.7 $$M(\frac{1}{2},\frac{3}{2},-z^2)=\frac{\sqrt{\pi }}{2z}\mathrm{erf}\left(z\right),$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erf}z$: error function and $z$: complex variable

A&S Ref:

13.6.19

Referenced by:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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13.6.8 $$U(\frac{1}{2},\frac{1}{2},z^2)=\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}\left(z\right).$$

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Symbols:

$U(a,b,z)$: Kummer 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:

§13.6(ii)

Permalink:

http://dlmf.nist.gov/13.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.6(ii), §13.6 and Ch.13

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33: 7.19 Voigt Functions

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7.19.2 $$𝖵(x,t)=\frac{1}{\sqrt{4\pi t}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{y{\mathrm{e}}^{-{(x-y)}^2/(4t)}}{1+y^2}dy.$$

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Defines:

$𝖵(x,t)$: Voigt function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.19(iii)

Permalink:

http://dlmf.nist.gov/7.19.E2

Encodings:

pMML, png, TeX

See also:

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

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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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34: 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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35: 32.10 Special Function Solutions

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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)). …

36: 8.18 Asymptotic Expansions of $I_x(a,b)$

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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). …

37: 3.5 Quadrature

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Example

… ►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

… ►

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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38: 12.11 Zeros

… ►When $a=\frac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathrm{erfc}(z/\sqrt{2})$; compare (12.7.5). …

39: Nico M. Temme

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7 Error Functions, Dawson’s and Fresnel Integrals

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40: 8.11 Asymptotic Approximations and Expansions

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8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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