application to Rossby waves

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1: 30.1 Special Notation

… ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

►Flammer (1957) and Abramowitz and Stegun (1964) use ${\lambda }_{mn}(\gamma )$ for ${\lambda }_n^m\left({\gamma }^2\right)+{\gamma }^2$, $R_{mn}^{(j)}(\gamma ,z)$ for $S_n^{m(j)}(z,\gamma )$, and …

2: 30.11 Radial Spheroidal Wave Functions

§30.11 Radial Spheroidal Wave Functions

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§30.11(i) Definitions

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Connection Formulas

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§30.11(ii) Graphics

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§30.11(iv) Wronskian

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3: 31.17 Physical Applications

§31.17 Physical Applications

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§31.17(ii) Other Applications

… ►More applications—including those of generalized spheroidal wave functions and confluent Heun functions in mathematical physics, astrophysics, and the two-center problem in molecular quantum mechanics—can be found in Leaver (1986) and Slavyanov and Lay (2000, Chapter 4). For application of biconfluent Heun functions in a model of an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere see Boyd and Natarov (1998).

4: 6.17 Physical Applications

§6.17 Physical Applications

►Geller and Ng (1969) cites work with applications from diffusion theory, transport problems, the study of the radiative equilibrium of stellar atmospheres, and the evaluation of exchange integrals occurring in quantum mechanics. For applications in astrophysics, see also van de Hulst (1980). Lebedev (1965) gives an application to electromagnetic theory (radiation of a linear half-wave oscillator), in which sine and cosine integrals are used.

5: 26.19 Mathematical Applications

§26.19 Mathematical Applications

►Combinatorics has applications to analysis, algebra, and geometry. …Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). … ►These have applications in operations research, probability theory, and statistics. …

6: 26.20 Physical Applications

§26.20 Physical Applications

►An English translation of Pólya (1937) on applications of combinatorics to chemistry has been published as Pólya and Read (1987). …The latter reference also describes chemical applications of other combinatorial techniques. … ►For an application of statistical mechanics to combinatorics, see Bressoud (1999). ►Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).

7: 32.16 Physical Applications

§32.16 Physical Applications

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Statistical Physics

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Integrable Continuous Dynamical Systems

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Other Applications

… ►For applications in string theory see Seiberg and Shih (2005).

8: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … ►Ng and Geller (1969) cites work with applications from atomic physics and astrophysics. … ►These applications include astrophysics, plasma diagnostics, neutron diffraction, laser spectroscopy, and surface scattering. …

9: 9.15 Mathematical Applications

§9.15 Mathematical Applications

… ►For descriptions of, and references to, the underlying theory see §§2.4(v) and 2.8(iii).

10: 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)). …