derivatives of the error function

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21: 7.20 Mathematical Applications

… ►

§7.20(i) Asymptotics

►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►

§7.20(iii) Statistics

… ►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

22: 9.9 Zeros

… ►They are denoted by

a_{k}

a^{\prime}_{k}

b_{k}

b^{\prime}_{k}

, respectively, arranged in ascending order of absolute value for

k=1,2,\ldots.

… ►They lie in the sectors

\tfrac{1}{3}\pi<\operatorname{ph}z<\tfrac{1}{2}\pi

and

-\tfrac{1}{2}\pi<\operatorname{ph}z<-\tfrac{1}{3}\pi

, and are denoted by

\beta_{k}

\beta^{\prime}_{k}

, respectively, in the former sector, and by

\overline{\beta_{k}}

\overline{\beta^{\prime}_{k}}

, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for

k=1,2,\ldots.

See §9.3(ii) for visualizations. … ►

§9.9(iii) Derivatives With Respect to $k$

… ►For error bounds for the asymptotic expansions of

a_{k}

b_{k}

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). … ►Tables 9.9.1 and 9.9.2 give 10D values of the first ten real zeros of

\operatorname{Ai}

\operatorname{Ai}'

\operatorname{Bi}

\operatorname{Bi}'

, together with the associated values of the derivative or the function. …

23: 10.67 Asymptotic Expansions for Large Argument

… ►

§10.67(i) $\operatorname{ber}_{\nu}x,\operatorname{bei}_{\nu}x,\operatorname{ker}_{\nu}x,% \operatorname{kei}_{\nu}x$ , and Derivatives

… ►

10.67.3

\operatorname{ber}_{\nu}x\sim\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\*% \sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\cos\left(\frac{x}{\sqrt{2}}+\left(% \frac{\nu}{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi\right)-\frac{1}{\pi}(\sin% \left(2\nu\pi\right)\operatorname{ker}_{\nu}x+\cos\left(2\nu\pi\right)% \operatorname{kei}_{\nu}x),

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Symbols:: $\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 9.10.1, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v), §10.67(i), §10.67(i), §10.68(iii), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(i), §10.67 and Ch.10

►

10.67.4

\operatorname{bei}_{\nu}x\sim\frac{e^{x/\sqrt{2}}}{(2\pi x)^{\frac{1}{2}}}\*% \sum_{k=0}^{\infty}\frac{a_{k}(\nu)}{x^{k}}\sin\left(\frac{x}{\sqrt{2}}+\left(% \frac{\nu}{2}+\frac{3k}{4}-\frac{1}{8}\right)\pi\right)+\frac{1}{\pi}(\cos% \left(2\nu\pi\right)\operatorname{ker}_{\nu}x-\sin\left(2\nu\pi\right)% \operatorname{kei}_{\nu}x).

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Symbols:: $\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{kei}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Kelvin function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $x$ : real variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 9.10.2, 9.10.5--9.10.7 (modified)
Referenced by:: §10.61(v), §10.67(i), §10.67(i), §10.67(ii), §10.68(iii), §10.69
Permalink:: http://dlmf.nist.gov/10.67.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.67(i), §10.67 and Ch.10

… ► ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu=0$

…

24: 3.5 Quadrature

… ►In particular, when

k=\infty

the error term is an exponentially-small function of

1/h

, and in these circumstances the composite trapezoidal rule is exceptionally efficient. … ►An interpolatory quadrature rule … ►Equation (3.5.36), without the error term, becomes … ►

Example

… ►where

\operatorname{erfc}z

is the complementary error function, and from (7.12.1) it follows that …

25: 10.19 Asymptotic Expansions for Large Order

… ►

§10.19(i) Asymptotic Forms

… ►

§10.19(ii) Debye’s Expansions

… ►For error bounds for the first of (10.19.6) see Olver (1997b, p. 382). … ►For proofs and also for the corresponding expansions for second derivatives see Olver (1952). … ►See also §10.20(i).

26: 9.18 Tables

… ►

•

Miller (1946) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ . Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

►

•

Sherry (1959) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $k=1(1)50$ ; 20S.

►

•

Zhang and Jin (1996, p. 339) tabulates $a_{k}$ , $\operatorname{Ai}'\left(a_{k}\right)$ , $a^{\prime}_{k}$ , $\operatorname{Ai}\left(a^{\prime}_{k}\right)$ , $b_{k}$ , $\operatorname{Bi}'\left(b_{k}\right)$ , $b^{\prime}_{k}$ , $\operatorname{Bi}\left(b^{\prime}_{k}\right)$ , $k=1(1)20$ ; 8D.

… ►

§9.18(vi) Scorer Functions

… ►

•

Smirnov (1960) tabulates $U_{1}(x,\alpha)$ , $U_{2}(x,\alpha)$ , defined by (9.13.20), (9.13.21), and also $\ifrac{\partial U_{1}(x,\alpha)}{\partial x}$ , $\ifrac{\partial U_{2}(x,\alpha)}{\partial x}$ , for $\alpha=1$ , $x=-6(.01)10$ to 5D or 5S, and also for $\alpha=\pm\tfrac{1}{4}$ , $\pm\tfrac{1}{3}$ , $\pm\tfrac{1}{2}$ , $\pm\tfrac{2}{3}$ , $\pm\tfrac{3}{4}$ , $\tfrac{5}{4}$ , $\tfrac{4}{3}$ , $\tfrac{3}{2}$ , $\tfrac{5}{3}$ , $\tfrac{7}{4}$ , 2, $x=0(.01)6$ ; 4D.

27: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

§28.28(i) Equations with Elementary Kernels

… ►

§28.28(ii) Integrals of Products with Bessel Functions

… ►

§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

… ►

§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

… ►

§28.28(v) Compendia

…

28: 10.75 Tables

… ►

§10.75(ii) Bessel Functions and their Derivatives

… ►

§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

… ►

§10.75(vi) Zeros of Modified Bessel Functions and their Derivatives

… ►

§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

… ►

§10.75(xii) Zeros of Kelvin Functions and their Derivatives

…

29: 5.17 Barnes’ $G$ -Function (Double Gamma Function)

§5.17 Barnes’ $G$ -Function (Double Gamma Function)

… ►When

z\to\infty

|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)

, ►

5.17.5

\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.

ⓘ

Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $B_{\NVar{n}}$ : Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\sim$ : Poincaré asymptotic expansion, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer, $z$ : complex variable and $A$ : Glaisher’s constant
Referenced by:: §5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)
Permalink:: http://dlmf.nist.gov/5.17.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.1.11):: For consistency we have replaced $\operatorname{Ln}z$ by $\ln z$ .
Errata (effective with 1.0.7):: Originally the term $z\operatorname{Ln}\Gamma\left(z+1\right)$ was incorrectly stated as $z\Gamma\left(z+1\right)$ . (This error was reported subsequently by Nick Jones on December 11, 2013.)
Reported 2013-08-01 by Gergő Nemes
See also:: Annotations for §5.17 and Ch.5

►For error bounds and an exponentially-improved extension, see Nemes (2014a). …and

\zeta'

is the derivative of the zeta function (Chapter 25). …

30: Bibliography G

… ►

W. Gautschi (1969) Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

ⓘ

Notes:: Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466
External Links:: ISSN 0001-0782, Document
Referenced by:: §7.25(iii).
Encodings:: BibTeX

… ►

W. Gautschi (1961) Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

ⓘ

External Links:: FullText, MathReview (J. C. P. Miller), ZentralBlatt
Referenced by:: §7.22(iii).
Encodings:: BibTeX

… ►

W. Gautschi (1970) Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

ⓘ

External Links:: Document, MathReview (P. Barrucand), ZentralBlatt
Referenced by:: §7.22(i), §7.25(iii).
Encodings:: BibTeX

… ►

M. Geller and E. W. Ng (1971) A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

ⓘ

Notes:: See for the first part same journal, Sect. B 73, 1-20 (1969)
External Links:: MathReview, ZentralBlatt
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

A. Gil and J. Segura (2014) On the complex zeros of Airy and Bessel functions and those of their derivatives. Anal. Appl. (Singap.) 12 (5), pp. 537–561.

ⓘ

External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §9.9(i), §9.9(i).
Encodings:: BibTeX

…