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1: 13.28 Physical Applications

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§13.28(i) Exact Solutions of the Wave Equation

►The reduced wave equation ${\nabla }^{2}w=k^{2}w$ in paraboloidal coordinates, $x=2\sqrt{\xi \eta }\cos \varphi$, $y=2\sqrt{\xi \eta }\sin \varphi$, $z=\xi -\eta$, can be solved via separation of variables $w=f_1(\xi )f_2(\eta ){\mathrm{e}}^{\mathrm{i}p\varphi }$, where …

2: 28.31 Equations of Whittaker–Hill and Ince

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§28.31(iii) Paraboloidal Wave Functions

►With (28.31.10) and (28.31.11), …are called paraboloidal wave functions. … ►More important are the double orthogonality relations for $p_1\ne p_2$ or $m_1\ne m_2$ or both, given by … ►

Asymptotic Behavior

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3: Bibliography U

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Integral equations for paraboloidal wave functions. I. Quart. J. Math. Oxford Ser. (2) 15, pp. 309–315.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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Integral equations for the paraboloidal wave functions. II. Quart. J. Math. Oxford Ser. (2) 16, pp. 257–262.

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External Links:

Document, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.31(iii), §28.31(iii).

Encodings:

BibTeX

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4: 28.32 Mathematical Applications

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§28.32(ii) Paraboloidal Coordinates

►The general paraboloidal coordinate system is linked with Cartesian coordinates via …

5: 12.17 Physical Applications

… ►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. … …

6: Bibliography

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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

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7: 37.18 Orthogonal Polynomials on Quadratic Domains

… ►For OPs with respect to classical weight functions, see below for the cone, see Xu (2021c) for the double cone and hyperboloid, and Xu (2022) for the paraboloid. …

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