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31: 23 Weierstrass Elliptic and Modular
Functions

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32: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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33: Bibliography V

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An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

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Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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External Links:

ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.60(iii), §10.60(iii).

Encodings:

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

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34: Bibliography D

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

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35: Errata

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It is now mentioned that (25.2.5), defines the Stieltjes constants ${\gamma }_n$. Consequently, ${\gamma }_n$ in (25.2.4), (25.6.12) are now identified as the Stieltjes constants.

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These equations have been generalized to include the additional cases of $\partial J_{-\nu }\left(z\right)/\partial \nu$, $\partial I_{-\nu }\left(z\right)/\partial \nu$, respectively.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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36: 16.11 Asymptotic Expansions

… ►For subsequent use we define two formal infinite series, $E_{p,q}(z)$ and $H_{p,q}(z)$, as follows: ►

16.11.1 $$E_{p,q}(z)={(2\pi )}^{(p-q)/2}{\kappa }^{-\nu -(1/2)}{\mathrm{e}}^{\kappa z^{1/\kappa }}\sum _{k=0}^{\mathrm{\infty }}{c}_k{\left(\kappa z^{1/\kappa }\right)}^{\nu -k},$$ ,

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Defines:

$E_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $\kappa$ and $\nu$

Referenced by:

§16.11(i), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

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16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

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Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

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16.11.7 $$\left(\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_q{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_q}{b_1,\mathrm{\dots },b_q};z)\sim H_{q,q}(z{\mathrm{e}}^{\mp \pi \mathrm{i}})+E_{q,q}(z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $H_{p,q}(z)$: formal infinite series

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

►Here the upper or lower signs are chosen according as $z$ lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of $z{e}^{\mp \pi i}$ its phases are $\mathrm{ph}z\mp \pi$, respectively. …

37: 36 Integrals with Coalescing Saddles

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38: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

39: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

40: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

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Infinitely many Stokes smoothings in the gamma function. Proc. Roy. Soc. London Ser. A 434, pp. 465–472.

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External Links:

ISSN 0962-8444, FullText, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§5.11(ii).

Encodings:

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Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.73(iv).

Encodings:

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