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31: 28.33 Physical Applications

… ►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. … ►

28.33.1 $$\frac{{\partial }^{2}W}{{\partial x}^2}+\frac{{\partial }^{2}W}{{\partial y}^2}-\frac{\rho }{\tau }\frac{{\partial }^{2}W}{{\partial t}^2}=0,$$

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Symbols:

$\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $y$: real variable, $\rho$: mass, $\tau$: radial tension and $W(x,y,t)$: solution

Permalink:

http://dlmf.nist.gov/28.33.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.33(ii), §28.33 and Ch.28

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32: 28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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33: 18.39 Applications in the Physical Sciences

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a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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c) Spherical Radial Coulomb Wave Functions

… ►The radial Coulomb wave functions $R_{n,l}(r)$, solutions of … ►

d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►The radial operator (18.39.28) …

34: Bibliography B

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Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.

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External Links:

Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

1st item.

Encodings:

BibTeX

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Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (C. J. Bouwkamp)

Referenced by:

2nd item, 1st item.

Encodings:

BibTeX

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Methods of calculation of radial wave functions and new tables of Coulomb functions. Physical Rev. (2) 80, pp. 553–560.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§33.7.

Encodings:

BibTeX

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35: 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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36: Bibliography S

… ►

The accuracy of iterated JWBK approximations for Coulomb radial functions. Comput. Phys. Comm. 32 (2), pp. 115–119.

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External Links:

Document

Referenced by:

§33.23(vii).

Encodings:

BibTeX

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FGH, a code for the calculation of Coulomb radial wave functions from series expansions. Comput. Phys. Comm. 146 (2), pp. 250–253.

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Notes:

Single-precision Fortran code.

External Links:

Document, Code, ZentralBlatt

Referenced by:

10th item.

Encodings:

BibTeX

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37: Bibliography G

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Radial quadrature for multi exponential integrands. J. Comput. Chem. 24 (4), pp. 732–740.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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38: Bibliography K

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A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 7012 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

Encodings:

BibTeX

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39: Bibliography L

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New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

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External Links:

ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§28.24, §28.24.

Encodings:

BibTeX

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40: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The reduced mass is $m=m_1{m}_2/(m_1+m_2)$, and at energy of relative motion $E$ with relative orbital angular momentum $\mathrm{\ell }\mathrm{\hslash }$, the Schrödinger equation for the radial wave function $w(s)$ is given by …