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 spheroidal functions, spheroidal wave functions
Referenced by:: ¶ ‣ §31.12
Permalink:: http://dlmf.nist.gov/30.12

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

30.12.1 $\frac{d}{dz}\left((1-z^{2})\frac{dw}{dz}\right)+{\left(\lambda+\alpha z+\gamma% ^{2}(1-z^{2})-\frac{\mu^{2}}{1-z^{2}}\right)w}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable, : 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

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 $\frac{d}{dz}\left((1-z^{2})\frac{dw}{dz}\right)+\left(\lambda+\gamma^{2}(1-z^{% 2})-\frac{\alpha(\alpha+1)}{z^{2}}-\frac{\mu^{2}}{1-z^{2}}\right)w=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , : complex variable, : 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

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).

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