solution of triangles and spherical triangles

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21: 18.38 Mathematical Applications

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Differential Equations: Spectral Methods

… ►Schneider et al. (2016) discuss DVR/Finite Element solutions of the time-dependent Schrödinger equation. … ►

Zonal Spherical Harmonics

►Ultraspherical polynomials are zonal spherical harmonics. … ►has a solution …

22: 14.31 Other Applications

… ►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

23: 1.9 Calculus of a Complex Variable

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Triangle Inequality

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24: 23.5 Special Lattices

… ►The rhombus $0$, $2{\omega }_1-2{\omega }_3$, $2{\omega }_1$, $2{\omega }_3$ can be regarded as the union of two equilateral triangles: see Figure 23.5.2. …

25: 30.2 Differential Equations

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30.2.1 $$\frac{d}{dz}\left((1-z^2)\frac{dw}{dz}\right)+\left(\lambda +{\gamma }^2(1-z^2)-\frac{{\mu }^2}{1-z^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter and $\mu$: real parameter

Referenced by:

§30.11(i), §30.12, §30.12, §30.13(iv), §30.14(iv), §30.16(ii), §30.2(ii), §30.2(iii), §30.3(i), §30.4(i), §30.5, §30.6, §30.6

Permalink:

http://dlmf.nist.gov/30.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(i), §30.2 and Ch.30

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30.2.4 $$({\zeta }^2-{\gamma }^2)\frac{d^{2}w}{{d\zeta }^2}+2\zeta \frac{dw}{d\zeta }+\left({\zeta }^2-\lambda -{\gamma }^2-\frac{{\gamma }^2{\mu }^2}{{\zeta }^2-{\gamma }^2}\right)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $w$: solution to DE, $\lambda$: real parameter, ${\gamma }^2$: real parameter, $\mu$: real parameter and $\zeta$: parameter

Referenced by:

§30.2(iii)

Permalink:

http://dlmf.nist.gov/30.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.2(ii), §30.2 and Ch.30

… ►If $\gamma =0$, Equation (30.2.4) is satisfied by spherical Bessel functions; see (10.47.1).

26: 18.39 Applications in the Physical Sciences

… ►The solution, (18.39.29), of the spherical radial equation (18.39.28), now expressed in terms of the Bohr quantum number $n$, is …

27: 31.10 Integral Equations and Representations

… ►If $w(z)$ is a solution of Heun’s equation, then another solution $W(z)$ (possibly a multiple of $w(z)$) can be represented as … ►

Kernel Functions

… ►If $w(z)$ is a solution of Heun’s equation, then another solution $W(z)$ (possibly a multiple of $w(z)$) can be represented as … ►

Kernel Functions

… ►A further change of variables, to spherical coordinates, …

28: 10.22 Integrals

… ►(Thus if $a,b,c$ are the sides of a triangle, then $A^{\frac{1}{2}}$ is the area of the triangle.) …

29: Bibliography Z

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

Encodings:

BibTeX

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

Encodings:

BibTeX

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Tables and formulae for the spherical functions $P_{-1/2+i\tau }^m(z)$ . Translated by E. L. Albasiny, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy i formuly dlya sfericheskikh funktsii $P_{-1/2+i\tau }^m(z)$ , Vyčisl. Centr Akad Nauk SSSR, Moscow. 1962

External Links:

MathReview, ZentralBlatt

Referenced by:

§14.20(vii).

Encodings:

BibTeX

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30: Bibliography M

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Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

Ch.14.

Encodings:

BibTeX

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

BibTeX

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On the evaluation of the integral over the product of two spherical Bessel functions. J. Math. Phys. 32 (3), pp. 642–648.

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External Links:

ISSN 0022-2488, Document, MathReview (S. K. Chatterjea), ZentralBlatt

Referenced by:

§1.17(ii), §10.59.

Encodings:

BibTeX

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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:

BibTeX

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Analytic expressions for integrals of products of spherical Bessel functions. J. Phys. A 24 (7), pp. 1435–1453.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.59.

Encodings:

BibTeX

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