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21: Bibliography G

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Sur les équations différentielles du second ordre et du premier degré dont l’intégrale générale est a points critiques fixes. Acta Math. 33 (1), pp. 1–55.

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§32.10(ii), §32.10(iv), §32.7(ii).

Encodings:

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

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External Links:

ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

Encodings:

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

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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:

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

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Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

Encodings:

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22: Bibliography P

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Optical properties of layer antiferromagnets with ${\mathrm{K}}_2{\mathrm{NiF}}_4$ structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

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External Links:

Document

Referenced by:

§19.35(ii).

Encodings:

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Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.

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External Links:

ISSN 0090-5364, FullText, MathReview (Takeaki Kariya), ZentralBlatt

Referenced by:

§35.9.

Encodings:

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

Encodings:

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Algorithm 28. Algorithm for the computation of Bessel function integrals. J. Comput. Appl. Math. 11 (1), pp. 119–137.

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External Links:

ISSN 0377-0427, Document, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

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23: 36.5 Stokes Sets

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36.5.4 $$80x^5-40x^4-55x^3+5x^2+20x-1=0,$$

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Symbols:

$x$: real parameter

Permalink:

http://dlmf.nist.gov/36.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.7 $$X=\frac{9}{20}+20u^4-\frac{Y^2}{20u^2}+6u^2\mathrm{sign}\left(z\right),$$

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Symbols:

$\mathrm{sign}x$: sign of, $z$: real parameter, $X$: scaled coordinate and $Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E7

Encodings:

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See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.10 $$160u^6+40u^4=Y^2.$$

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Symbols:

$Y$: scaled coordinate

Permalink:

http://dlmf.nist.gov/36.5.E10

Encodings:

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See also:

Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

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36.5.14

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Permalink:

http://dlmf.nist.gov/36.5.E14

Encodings:

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See also:

Annotations for §36.5(iii), §36.5(iii), §36.5 and Ch.36

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$Y(u,X)=8u-24u^2+X{\displaystyle \frac{u-\frac{1}{6}}{{\left(u\left(u-\frac{1}{3}\right)\right)}^{1/2}}},$

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24: Bibliography V

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

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Über die Riemann’sche Primzahlfunction. Math. Ann. 55, pp. 441–464 (German).

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External Links:

ISSN 0025-5831; 1432-1807/e, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.12.

Encodings:

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On the localization and computation of zeros of Bessel functions. Z. Angew. Math. Mech. 77 (6), pp. 467–475.

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External Links:

ISSN 0044-2267, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

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25: 4.14 Definitions and Periodicity

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4.14.3 $$\cos z\pm \mathrm{i}\sin z={\mathrm{e}}^{\pm \mathrm{i}z},$$

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Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.14.E3

Encodings:

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See also:

Annotations for §4.14 and Ch.4

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26: Bibliography B

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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:

FullText, MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

§8.7.

Encodings:

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New “coherent” states associated with non-compact groups. Comm. Math. Phys. 21 (1), pp. 41–55.

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External Links:

ISSN 0010-3616, MathReview (R. Mirman), ZentralBlatt

Referenced by:

§13.28(iii).

Encodings:

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

Encodings:

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

Encodings:

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Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

Encodings:

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27: 8.21 Generalized Sine and Cosine Integrals

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8.21.4 $$\mathrm{si}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\sin tdt,$$ ,

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Symbols:

$dx$: differential of $x$ , $\mathrm{si}(a,z)$: generalized sine integral, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.8 (This function is called $S(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E4

Encodings:

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See also:

Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.5 $$\mathrm{ci}(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}\cos tdt,$$ ,

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{ci}(a,z)$: generalized cosine integral, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.7 (This function is called $C(z,a)$ in AMS 55.)

Referenced by:

§8.21(iii), §8.21(iii), §8.21(v), §8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E5

Encodings:

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See also:

Annotations for §8.21(iii), §8.21 and Ch.8

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8.21.18 $$f(a,z)=\mathrm{si}(a,z)\cos z-\mathrm{ci}(a,z)\sin z,$$

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Defines:

$f(a,z)$: auxiliary function (locally)

Symbols:

$\cos z$: cosine function, $\mathrm{ci}(a,z)$: generalized cosine integral, $\mathrm{si}(a,z)$: generalized sine integral, $\sin z$: sine function, $z$: complex variable and $a$: parameter

A&S Ref:

5.2.6 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E18

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.22 $$f(a,z)={\int }_0^{\mathrm{\infty }}\frac{\sin t}{{(t+z)}^{1-a}}dt,$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function

A&S Ref:

5.2.12 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E22

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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8.21.23 $$g(a,z)={\int }_0^{\mathrm{\infty }}\frac{\cos t}{{(t+z)}^{1-a}}dt.$$

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Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function

A&S Ref:

5.2.13 (This generalizes the form in AMS 55.)

Referenced by:

§8.21(vii)

Permalink:

http://dlmf.nist.gov/8.21.E23

Encodings:

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See also:

Annotations for §8.21(vii), §8.21 and Ch.8

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28: 22.15 Inverse Functions

… ►Comprehensive treatments are given by Carlson (2005), Lawden (1989, pp. 52–55), Bowman (1953, Chapter IX), and Erdélyi et al. (1953b, pp. 296–301). …

29: Bibliography M

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On the computation of zeroes of $J_n(z)-i{J}_{n+1}(z)=0$ . Quart. Appl. Math. 55 (4), pp. 623–633.

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External Links:

ISSN 0033-569X, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

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Spheroidal eigenfunctions of the tidal equation. Phys. Rev. Lett. 73 (11), pp. 1557–1560.

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External Links:

ISSN 0031-9007, Document, MathReview, ZentralBlatt

Referenced by:

§30.13(v).

Encodings:

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Zeros of the function $A{i}^{\prime }(z)-\sigma Ai(z)$ . Differential Equations 11, pp. 797–811.

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External Links:

ZentralBlatt

Referenced by:

§9.9(i).

Encodings:

BibTeX

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30: 10 Bessel Functions

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