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1: 30.14 Wave Equation in Oblate Spheroidal Coordinates

§30.14 Wave Equation in Oblate Spheroidal Coordinates

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§30.14(i) Oblate Spheroidal Coordinates

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§30.14(ii) Metric Coefficients

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§30.14(iii) Laplacian

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§30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids

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2: 30.4 Functions of the First Kind

… ►When ${\gamma }^2>0$ ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the prolate angular spheroidal wave function, and when ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ is the oblate angular spheroidal wave function. If $\gamma =0$, ${\mathrm{𝖯𝗌}}_n^m(x,0)$ reduces to the Ferrers function ${𝖯}_n^m\left(x\right)$: …

3: Bibliography V

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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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Tables of Angular Spheroidal Wave Functions, Vol. 1, Prolate, $m=0$; Vol. 2, Oblate, m=0. Naval Res. Lab. Reports, Washington, D.C..

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Referenced by:

5th item.

Encodings:

BibTeX

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4: 30.9 Asymptotic Approximations and Expansions

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§30.9(ii) Oblate Spheroidal Wave Functions

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5: 30.2 Differential Equations

… ►In applications involving prolate spheroidal coordinates ${\gamma }^2$ is positive, in applications involving oblate spheroidal coordinates ${\gamma }^2$ is negative; see §§30.13, 30.14. …

6: 14.30 Spherical and Spheroidal Harmonics

… ► $P_n^m\left(\mathrm{i}x\right)$ and $Q_n^m\left(\mathrm{i}x\right)$ ($x>0$) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$. …

7: 30.1 Special Notation

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8: Bibliography J

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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9: Bibliography D

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Uniform asymptotic expansions for oblate spheroidal functions I: Positive separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 121 (3-4), pp. 303–320.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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Uniform asymptotic expansions for oblate spheroidal functions II: Negative separation parameter $\lambda$ . Proc. Roy. Soc. Edinburgh Sect. A 125 (4), pp. 719–737.

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External Links:

ISSN 0308-2105, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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10: Bibliography H

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Tables of Radial Spheroidal Wave Functions, Vols. 1-3, Prolate, $m=0,1,2$; Vols. 4-6, Oblate, $m=0,1,2$ . Technical report Naval Research Laboratory, Washington, D.C..

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Notes:

NRL Reports 7088-7093

External Links:

ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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