complementary error function

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31: 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). …

32: 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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33: 13.6 Relations to Other Functions

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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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34: 7.8 Inequalities

§7.8 Inequalities

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35: Bibliography C

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Calcolo delle funzioni speciali $\mathrm{\Gamma }(x)$, $\log \mathrm{\Gamma }(x)$, $\beta (x,y)$, $\mathrm{erf}(x)$, $\mathrm{erfc}(x)$ alle alte precisioni. Atti Accad. Sci. Lett. Arti Palermo Ser. (5) 2(1981/82) (1), pp. 7–25 (Italian).

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Notes:

Translated title: Computation of the special functions $\mathrm{\Gamma }(x),\log \mathrm{\Gamma }(x),\beta (x,y),\mathrm{erf}(x),\mathrm{erfc}(x)$ to a high degree of precision

External Links:

MathReview, ZentralBlatt

Referenced by:

§5.24(ii).

Encodings:

BibTeX

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36: Errata

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Originally named as a complementary error function, $w\left(z\right)$ has been renamed as the Faddeeva (or Faddeyeva) function.

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To be consistent with the notation used in (8.12.16), Equation (8.12.5) was changed to read

8.12.5 $$\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}}{2\mathrm{i}\sin (\pi a)}Q(-a,z{\mathrm{e}}^{\pm \pi \mathrm{i}})=\pm \frac{1}{2}\mathrm{erfc}\left(\pm \mathrm{i}\eta \sqrt{a/2}\right)-\mathrm{i}T(a,\eta )$$

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

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 $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

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37: Bibliography E

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The zeros of the complementary error function. Numer. Algorithms 49 (1-4), pp. 153–157.

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External Links:

ISSN 1017-1398, Document, FullText, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§7.13(ii).

Encodings:

BibTeX

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38: Bibliography F

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Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.13(i), §7.13(ii), 1st item.

Encodings:

BibTeX

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39: 2.4 Contour Integrals

… ►For a coalescing saddle point and a pole see Wong (1989, Chapter 7) and van der Waerden (1951); in this case the uniform approximants are complementary error functions. For a coalescing saddle point and endpoint see Olver (1997b, Chapter 9) and Wong (1989, Chapter 7); if the endpoint is an algebraic singularity then the uniform approximants are parabolic cylinder functions with fixed parameter, and if the endpoint is not a singularity then the uniform approximants are complementary error functions. …

40: 29.16 Asymptotic Expansions

… ►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. The approximating functions are exponential, trigonometric, and parabolic cylinder functions.