wave functions

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11: 11.12 Physical Applications

§11.12 Physical Applications

►Applications of Struve functions occur in water-wave and surface-wave problems (Hirata (1975) and Ahmadi and Widnall (1985)), unsteady aerodynamics (Shaw (1985) and Wehausen and Laitone (1960)), distribution of fluid pressure over a vibrating disk (McLachlan (1934)), resistive MHD instability theory (Paris and Sy (1983)), and optical diffraction (Levine and Schwinger (1948)). …

12: 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}\neq p_{2}

m_{1}\neq m_{2}

or both, given by … ►

Asymptotic Behavior

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13: 30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

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§30.9(ii) Oblate Spheroidal Wave Functions

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§30.9(iii) Other Approximations and Expansions

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14: Hans Volkmer

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30 Spheroidal Wave Functions

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15: 30.16 Methods of Computation

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§30.16(ii) Spheroidal Wave Functions of the First Kind

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30.16.9

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\lim_{d\to\infty}\sum_{j=1}^{d}(-% 1)^{j-p}e_{j,d}\mathsf{P}^{m}_{n+2(j-p)}\left(x\right).

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Symbols:: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $x$ : real variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $d$ : dimension and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.16.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §30.16(ii), §30.16 and Ch.30

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§30.16(iii) Radial Spheroidal Wave Functions

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16: 30.7 Graphics

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§30.7(ii) Functions of the First Kind

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See accompanying text — Figure 30.7.21: $|\mathit{Qs}^{0}_{0}\left(x+\mathrm{i}y,-4\right)|$ , $-1.8\leq x\leq 1.8$ , $-2\leq y\leq 2$ . Magnify 3D Help

17: Bibliography U

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K. M. Urwin (1964) 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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K. M. Urwin (1965) 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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18: 31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

19: 30.18 Software

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§30.18(iii) Spheroidal Wave Functions

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20: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at

z=0

and

1

, and an irregular singularity of rank 1 at

z=\infty

. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …