/shop/%E2%99%9F%F0%9F%AA%90%F0%9F%A6%B8%20Buy%20DNP%20Online%20$0.99%20Per%20Capsule%3A%20%F0%9F%92%A3%20www.BuyDNPOnline.cc%20%F0%9F%92%A3%20Buy%20DNP%20UK%20%F0%9F%A6%B8%F0%9F%AA%90%E2%99%9F%20Buy%20DNP%20In%20Australia%20-%20Buy%20DNP-www.BuyDNPOnline.cc

SearchAdvancedHelp

(0.064 seconds)

11—20 of 983 matching pages

11: Bibliography K

… ►

Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(ii).

Encodings:

BibTeX

… ►

Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

ⓘ

External Links:

ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

… ►

Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.

ⓘ

External Links:

ISSN 0965-5425, Document, FullText

Referenced by:

§10.21(xiii), §10.21(xiii).

Encodings:

BibTeX

… ►

Two notes on notation. Amer. Math. Monthly 99 (5), pp. 403–422.

ⓘ

External Links:

ISSN 0002-9890, FullText, MathReview (W. Dörfler), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations, Notations.

Encodings:

BibTeX

… ►

On the zeros of the Fresnel integrals. Canad. J. Math. 9, pp. 118–131.

ⓘ

External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§7.13(iii).

Encodings:

BibTeX

…

12: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

13: Bibliography C

… ►

Generalized hypergeometric functions and the evaluation of scalar one-loop integrals in Feynman diagrams. J. Comput. Appl. Math. 115 (1-2), pp. 93–99.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Lorenzo L. Salcedo), ZentralBlatt

Referenced by:

§16.24(ii).

Encodings:

BibTeX

… ►

A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

ⓘ

External Links:

ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

… ►

Generalized incomplete gamma functions with applications. J. Comput. Appl. Math. 55 (1), pp. 99–124.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

… ►

Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

ⓘ

External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

4th item, 1st item.

Encodings:

BibTeX

… ►

Zeros of the Hankel function of real order out of the principal Riemann sheet. J. Comput. Appl. Math. 37 (1-3), pp. 89–99.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

…

14: Bibliography L

… ►

Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

ⓘ

Notes:

Double-precision Fortran

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

7th item.

Encodings:

BibTeX

… ►

Optimal cylindrical and spherical Bessel transforms satisfying bound state boundary conditions. Comput. Phys. Comm. 99 (2-3), pp. 297–306.

ⓘ

Notes:

Single-precision Fortran, maximum accuracy 8S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

… ►

Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.

ⓘ

Notes:

Course of Theoretical Physics , Vol. 9. Translated from the Russian by J. B. Sykes and M. J. Kearsley.

External Links:

ISBN 0-08-023073-3; 0-08-023072-5, MathReview

Referenced by:

§25.17.

Encodings:

BibTeX

… ►

Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

… ►

Deep water ship-waves. Phil. Mag. 9, pp. 733–757.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§36.13.

Encodings:

BibTeX

…

15: 16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

§16.10 Expansions in Series of ${}_p{}^{}F_q^{}$ Functions

►The following expansion, with appropriate conditions and together with similar results, is given in Fields and Wimp (1961): … ►The next expansion is given in Nørlund (1955, equation (1.21)): …When the series on the right-hand side converges in the half-plane . ►Expansions of the form ${\sum }_{n=1}^{\mathrm{\infty }}{(\pm 1)}^n{}_p{}^{}F_{p+1}^{}(𝐚;𝐛;-n^2{z}^2)$ are discussed in Miller (1997), and further series of generalized hypergeometric functions are given in Luke (1969b, Chapter 9), Luke (1975, §§5.10.2 and 5.11), and Prudnikov et al. (1990, §§5.3, 6.8–6.9).

16: Bibliography G

… ►

Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

ⓘ

Referenced by:

§18.39(iii).

Encodings:

BibTeX

… ►

Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

ⓘ

External Links:

ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

Encodings:

BibTeX

… ►

Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.

ⓘ

External Links:

Document, MathReview (E. Frank), ZentralBlatt

Referenced by:

§10.74(iv), §14.32, §3.10(iii), §3.6(iii).

Encodings:

BibTeX

… ►

Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.

ⓘ

External Links:

MathReview (M. S. Cheema), ZentralBlatt

Referenced by:

§26.21.

Encodings:

BibTeX

… ►

Error bounds in equilibrium statistical mechanics. J. Math. Phys. 9, pp. 655–663.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§18.39(iii), §18.40(ii).

Encodings:

BibTeX

…

17: 25.13 Periodic Zeta Function

… ►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

ⓘ

Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

… ► $F(x,s)$ is periodic in $x$ with period 1, and equals $\zeta \left(s\right)$ when $x$ is an integer. … ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

►

25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

18: 15.4 Special Cases

… ►See the final paragraph in §15.2(ii). …In (15.4.17), (15.4.18) and (15.4.19) when the third entry is a nonpositive integer one has to use the limit interpretation (15.2.6), rather than (15.2.5). Compare the final paragraph in §15.2(ii). … ►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) $a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\mathrm{\dots }$, and in (15.4.34) $a=0,-1,-2,\mathrm{\dots }$. Compare the final paragraph in §15.2(ii). …

19: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

…

20: 34.14 Tables

§34.14 Tables

►Tables of exact values of the squares of the $\mathit{3}j$ and $\mathit{6}j$ symbols in which all parameters are $\le 8$ are given in Rotenberg et al. (1959), together with a bibliography of earlier tables of $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols on pp. … ►Some selected $\mathit{9}j$ symbols are also given. … 16-17; for $\mathit{9}j$ symbols on p. … ► 310–332, and for the $\mathit{9}j$ symbols on pp. …