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31—40 of 87 matching pages

31: Staff

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Amparo Gil, Universidad de Cantabria, for Chap. 3

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Javier Segura, Universidad de Cantabria, for Chap. 3

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32: 30.13 Wave Equation in Prolate Spheroidal Coordinates

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30.13.7 $${\nabla }^{2}w+{\kappa }^{2}w=0,$$

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Defines:

$\kappa$: parameter (locally)

Symbols:

$w_j$: solution to DE

Referenced by:

§30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v)

Permalink:

http://dlmf.nist.gov/30.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

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30.13.8 $$w(\xi ,\eta ,\varphi )=w_1(\xi )w_2(\eta )w_3(\varphi ),$$

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Symbols:

$\xi$: prolate spheroidal coordinate, $\eta$: prolate spheroidal coordinate, $\varphi$: prolate spheroidal coordinate and $w_j$: solution to DE

Referenced by:

§30.13(iv), §30.13(v), §30.14(iv), §30.14(iv), §30.14(v)

Permalink:

http://dlmf.nist.gov/30.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

►where $w_1$, $w_2$, $w_3$ satisfy the differential equations … ►

30.13.11 $$\frac{d^2{w}_3}{{d\varphi }^2}+{\mu }^2{w}_3=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\varphi$: prolate spheroidal coordinate, $w_j$: solution to DE and $\mu$: real parameter

Referenced by:

§30.14(iv)

Permalink:

http://dlmf.nist.gov/30.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(iv), §30.13 and Ch.30

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30.13.16 $$w_1({\xi }_1)=w_1({\xi }_2)=0,$$

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Symbols:

$\xi$: prolate spheroidal coordinate and $w_j$: solution to DE

Permalink:

http://dlmf.nist.gov/30.13.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §30.13(v), §30.13 and Ch.30

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33: 5.13 Integrals

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de Branges–Wilson Beta Integral

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34: 23.21 Physical Applications

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§23.21(ii) Nonlinear Evolution Equations

… ►For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …

35: 29.14 Orthogonality

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29.14.6 $${\mathrm{𝑑𝐸}}_{2n+1}^m(s,k^2){\mathrm{𝑑𝐸}}_{2n+1}^m(K+\mathrm{i}t,k^2),$$

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Symbols:

${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$: imaginary unit, $m$: nonnegative integer, $n$: nonnegative integer and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.14 and Ch.29

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36: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Mémoire sur la diffraction de la lumière. Mém. de l’Académie des Sciences, pp. 247–382.

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Notes:

Œvres Complètes d’Augustin Fresnel, Paris, 1866

Referenced by:

Figure 7.3.4, Figure 7.3.4.

Encodings:

BibTeX

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37: Bibliography H

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La $q$-conjecture de Macdonald-Morris pour $G_2$ . C. R. Acad. Sci. Paris Sér. I Math. 303 (6), pp. 211–213 (French).

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External Links:

ISSN 0249-6291, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.14.

Encodings:

BibTeX

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Une $q$-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

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External Links:

ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§17.13.

Encodings:

BibTeX

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Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

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Notes:

Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.

External Links:

ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§25.10(i), §27.2(i).

Encodings:

BibTeX

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A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1), pp. 31–51.

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External Links:

ISSN 0003-9527, Document, MathReview (Richard Brown), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

BibTeX

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38: 7.21 Physical Applications

… ►Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …

39: 25.18 Methods of Computation

… ►For recent investigations see, for example, van de Lune et al. (1986) and Odlyzko (1987). …

40: Bibliography C

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Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

Encodings:

BibTeX

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Intégrales et Fonctions Elliptiques en Vue des Applications. Préface de Henri Villat. Publications Scientifiques et Techniques du Ministère de l’Air, No. 452, Centre de Documentation de l’Armement, Paris.

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External Links:

MathReview (B. Ć. Carlson)

Referenced by:

§19.11(i), §19.12, §19.14(i), §19.36(ii), §19.4(ii), §19.8(ii), §19.8(iii), Ch.19.

Encodings:

BibTeX

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Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée. Mem. Ac. Sc. St. Pétersbourg 6, pp. 141–157.

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Referenced by:

§25.16(i).

Encodings:

BibTeX

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Accélération de calcul de nombres de Bernoulli. J. Number Theory 28 (3), pp. 347–362 (French).

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External Links:

ISSN 0022-314X, Document, MathReview (Louis Shapiro), ZentralBlatt

Referenced by:

§24.19(i).

Encodings:

BibTeX

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Exact elliptic compactons in generalized Korteweg-de Vries equations. Complexity 11 (6), pp. 30–34.

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External Links:

ISSN 1076-2787, Document, MathReview

Referenced by:

§19.6(i).

Encodings:

BibTeX

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