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21: 13.14 Definitions and Basic Properties

§13.14 Definitions and Basic Properties

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Standard Solutions

►Standard solutions are: … ►

§13.14(v) Numerically Satisfactory Solutions

►Fundamental pairs of solutions of (13.14.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are …

22: 14.30 Spherical and Spheroidal Harmonics

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§14.30(ii) Basic Properties

… ►where $𝐚=(\frac{1}{2\lambda }-\frac{\lambda }{2},-\frac{\mathrm{i}}{2\lambda }-\frac{\mathrm{i}\lambda }{2},1)$ and $𝐱=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )$. … ►

14.30.10 $$\frac{1}{{\rho }^2}\frac{\partial }{\partial \rho }\left({\rho }^2\frac{\partial W}{\partial \rho }\right)+\frac{1}{{\rho }^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial W}{\partial \theta }\right)+\frac{1}{{\rho }^2{\sin }^2\theta }\frac{{\partial }^{2}W}{{\partial \varphi }^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function and $W(\rho ,\theta ,\varphi )$: solution

Permalink:

http://dlmf.nist.gov/14.30.E10

Encodings:

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See also:

Annotations for §14.30(iv), §14.30 and Ch.14

►has solutions $W(\rho ,\theta ,\varphi )={\rho }^l{Y}_{l,m}(\theta ,\varphi )$, which are everywhere one-valued and continuous. ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations …

23: Bibliography F

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Basic Hypergeometric Series and Applications. Mathematical Surveys and Monographs, Vol. 27, American Mathematical Society, Providence, RI.

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Notes:

With a foreword by George E. Andrews

External Links:

ISBN 0-8218-1524-5, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§17.1, §17.16, §17.6(ii), Ch.17, Notations F.

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

Encodings:

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Solution of the transcendental equation $w{e}^w=x$ . Comm. ACM 16 (2), pp. 123–124.

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Notes:

Includes Fortran code.

External Links:

ISSN 0001-0782, Document

Referenced by:

§4.48(iv).

Encodings:

BibTeX

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24: Bibliography S

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Very high accuracy Chebyshev expansions for the basic trigonometric functions. Math. Comp. 34 (149), pp. 237–244.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (Claude Carasso), ZentralBlatt

Referenced by:

§4.47(i).

Encodings:

BibTeX

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Spline Functions: Basic Theory. John Wiley & Sons Inc., New York.

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Notes:

Pure and Applied Mathematics, A Wiley-Interscience Publication. Reprinted by Robert E. Krieger Publishing Co. Inc., 1993.

External Links:

ISBN 0-471-76475-2, MathReview (D. D. Stancu), ZentralBlatt

Referenced by:

§24.17(ii), §3.11(vi).

Encodings:

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Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent. Phys. D 3 (1-2), pp. 165–184.

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External Links:

Document, ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

Encodings:

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An Introduction to Basic Fourier Series. Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-1221-7, MathReview (P. D. F. Ion), ZentralBlatt

Referenced by:

§17.3(i).

Encodings:

BibTeX

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25: 16.4 Argument Unity

… ►The basic transformation is given by … ►Relations between three solutions of three-term recurrence relations are given by Masson (1991). …