# §30.12 Generalized and Coulomb Spheroidal Functions

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

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

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

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).