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31: 2.1 Definitions and Elementary Properties

… ►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. … ►Condition (2.1.13) is equivalent to … ►means that for each $n$, the difference between $f(x)$ and the $n$th partial sum on the right-hand side is $O\left({(x-c)}^n\right)$ as $x\to c$ in $𝐗$. … ►Substitution, logarithms, and powers are also permissible; compare Olver (1997b, pp. 19–22). … ►The asymptotic property may also hold uniformly with respect to parameters. …

32: 7.2 Definitions

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§7.2(ii) Dawson’s Integral

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§7.2(iii) Fresnel Integrals

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Values at Infinity

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§7.2(iv) Auxiliary Functions

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§7.2(v) Goodwin–Staton Integral

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33: Errata

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Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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19.20.11 $$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$

as $y\to 0+$, $p$ ($\ne 0$) real, we have added the constant term $\frac{-3}{p}{R}_C(z,p)$ and the order term $O\left(y\ln y\right)$, and hence $\sim$ was replaced by $=$.

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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For consistency of notation across all chapters, the notation for logarithm has been changed to $\ln$ from $\log$ throughout Chapter 27.

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34: 18.27 $q$-Hahn Class

… ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. … ►

From Big $q$-Jacobi to Jacobi

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From Big $q$-Jacobi to Little $q$-Jacobi

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From Little $q$-Jacobi to Jacobi

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From Little $q$-Laguerre to Laguerre

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35: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Connection between quantum systems involving the fourth Painlevé transcendent and $k$-step rational extensions of the harmonic oscillator related to Hermite exceptional orthogonal polynomial. J. Math. Phys. 57 (5), pp. Paper 052101, 15 pp..

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External Links:

ISSN 0022-2488, Document, Link, MathReview Entry

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Recursion relations for the $3$-$j$ symbols. Nuclear Physics A 113 (1), pp. 215–220.

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External Links:

Document

Referenced by:

§34.3(iii).

Encodings:

BibTeX

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

Encodings:

BibTeX

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

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36: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

… ►With the notation of Koekoek et al. (2010, (9.13.1)) the left-hand side of (18.34.1) has to be replaced by $y_n(x;a-2)$. …where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►where primes denote derivatives with respect to $x$. …

37: 6.5 Further Interrelations

§6.5 Further Interrelations

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6.5.1 $$E_1\left(-x\pm \mathrm{i}0\right)=-\mathrm{Ei}\left(x\right)\mp \mathrm{i}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.4 $$\frac{1}{2}(\mathrm{Ei}\left(x\right)-E_1\left(x\right))=\mathrm{Chi}\left(x\right)=\mathrm{Ci}\left(\mathrm{i}x\right)-\frac{1}{2}\pi \mathrm{i}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.2.24 (in modified form)

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.5 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\mathrm{i}(E_1\left(-\mathrm{i}z\right)-E_1\left(\mathrm{i}z\right))+\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.21

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.7 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=E_1\left(\mp \mathrm{i}z\right){\mathrm{e}}^{\mp \mathrm{i}z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, §6.5, §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

38: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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

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5.17.4 $$\mathrm{Ln}G\left(z+1\right)=\frac{1}{2}z\ln \left(2\pi \right)-\frac{1}{2}z(z+1)+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-{\int }_0^z\mathrm{Ln}\mathrm{\Gamma }\left(t+1\right)dt.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

►In this equation (and in (5.17.5) below), the $\mathrm{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). ►When $z\to \mathrm{\infty }$ in , … ►

5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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39: 2.6 Distributional Methods

… ►This leads to integrals of the form … ►To assign a distribution to the function $f_n(t)$, we first let $f_{n,n}(t)$ denote the $n$th repeated integral (§1.4(v)) of $f_n$: … ►An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …

40: 9.11 Products

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§9.11(iii) Integral Representations

… ►For further integral representations see Reid (1995, 1997a, 1997b). ►

§9.11(iv) Indefinite Integrals

… ►For related integrals see Gordon (1969, Appendix B). … ►

§9.11(v) Definite Integrals

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