30 Spheroidal Wave Functions Properties 30.11 Radial Spheroidal Wave Functions 30.13 Wave Equation in Prolate Spheroidal Coordinates

§30.12 Generalized and Coulomb Spheroidal Functions

ⓘ

Keywords:: Coulomb, Coulomb spheroidal functions, generalized, spheroidal wave functions
Referenced by:: §31.12
Permalink:: http://dlmf.nist.gov/30.12
See also:: Annotations for Ch.30

Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation

30.12.1		$\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+{\left(\lambda+\alpha z+\gamma^{2}(1-z^{2})-\frac{\mu^{2}}{1-z^{2}}% \right)w}=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter and $\mu$ : real parameter Referenced by: §30.12 Permalink: http://dlmf.nist.gov/30.12.E1 Encodings: TeX, pMML, png See also: Annotations for §30.12 and Ch.30

which reduces to (30.2.1) if $\alpha=0$ . Equation (30.12.1) appears in astrophysics and molecular physics. For the theory and computation of solutions of (30.12.1) see Falloon (2001), Judd (1975), Leaver (1986), and Komarov et al. (1976).

Another generalization is provided by the differential equation

30.12.2		$\frac{\mathrm{d}}{\mathrm{d}z}\left((1-z^{2})\frac{\mathrm{d}w}{\mathrm{d}z}% \right)+\left(\lambda+\gamma^{2}(1-z^{2})-\frac{\alpha(\alpha+1)}{z^{2}}-\frac% {\mu^{2}}{1-z^{2}}\right)w=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $w$ : solution to DE, $\lambda$ : real parameter, $\gamma^{2}$ : real parameter and $\mu$ : real parameter Permalink: http://dlmf.nist.gov/30.12.E2 Encodings: TeX, pMML, png See also: Annotations for §30.12 and Ch.30

which also reduces to (30.2.1) when $\alpha=0$ . See Leitner and Meixner (1960), Slepian (1964) with $\mu=0$ , and Meixner et al. (1980).

30.11 Radial Spheroidal Wave Functions 30.13 Wave Equation in Prolate Spheroidal Coordinates

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§30.12 Generalized and Coulomb Spheroidal Functions

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