homogeneous equations

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11: Bibliography M

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On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.

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External Links:

Document, MathReview (W. R. Wasow), ZentralBlatt

Referenced by:

§12.2(vi), §2.7(iv), §2.7(iv).

Encodings:

BibTeX

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12: 9.16 Physical Applications

… ►An example from quantum mechanics is given in Landau and Lifshitz (1965), in which the exact solution of the Schrödinger equation for the motion of a particle in a homogeneous external field is expressed in terms of $\mathrm{Ai}\left(x\right)$. …

13: 19.7 Connection Formulas

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14: 14.30 Spherical and Spheroidal Harmonics

… ►In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). …As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): … ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations … ►

14.30.11_5 $${\mathrm{L}}_z{Y}_{l,m}=\mathrm{\hslash }m{Y}_{l,m},$$ $m=-l,-1+1,\mathrm{\dots },0,\mathrm{\dots },l-1,l$,

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Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and ${\mathrm{L}}_z$: $z$ component of the angular momentum operator

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E11_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

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

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15: 19.18 Derivatives and Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). …

16: 10.21 Zeros

… ►For describing the distribution of complex zeros by methods based on the Liouville–Green (WKB) approximation for linear homogeneous second-order differential equations, see Segura (2013). …

17: 28.33 Physical Applications

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Initial-value problems, in which only one equation (28.2.1) or (28.20.1) is involved. See §28.33(iii).

… ►We shall derive solutions to the uniform, homogeneous, loss-free, and stretched elliptical ring membrane with mass $\rho$ per unit area, and radial tension $\tau$ per unit arc length. The wave equation … ►The equation of motion is given by … ►In particular, the equation is stable for all sufficiently large values of $\omega$. …

18: 21.7 Riemann Surfaces

… ►Consider the set of points in ${ℂ}^2$ that satisfy the equation …Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. To accomplish this we write (21.7.1) in terms of homogeneous coordinates: …

19: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

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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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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:

MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

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

Encodings:

BibTeX

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Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen. Math. Ann. 63 (3), pp. 301–321.

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External Links:

ISSN 0025-5831, Document, ZentralBlatt

Referenced by:

§32.2(iv).

Encodings:

BibTeX

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20: 12.17 Physical Applications

§12.17 Physical Applications

►The main applications of PCFs in mathematical physics arise when solving the Helmholtz equation …The first two equations can be transformed into (12.2.2) or (12.2.3). ►In a similar manner coordinates of the paraboloid of revolution transform the Helmholtz equation into equations related to the differential equations considered in this chapter. … ►Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. …