# spheroidal wave functions

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##### 1: 30.1 Special Notation
The main functions treated in this chapter are the eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$, $\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right)$, $\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right)$, and $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathsf{Ps}$, $\mathsf{Qs}$, $\mathit{Ps}$, $\mathit{Qs}$, respectively.
###### Other Notations
Flammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$, $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$, and …
##### 5: 30.12 Generalized and Coulomb Spheroidal Functions
###### §30.12 Generalized and Coulomb SpheroidalFunctions
Generalized spheroidal wave functions and Coulomb spheroidal functions are solutions of the differential equation …
##### 6: 30.5 Functions of the Second Kind
###### §30.5 Functions of the Second Kind
30.5.1 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$.
30.5.2 $\mathsf{Qs}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m+1}\mathsf{Qs}^{m}_{n}% \left(x,\gamma^{2}\right),$
30.5.3 $\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$
30.5.4 $\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
##### 7: 30.6 Functions of Complex Argument
###### §30.6 Functions of Complex Argument
The solutions …
30.6.4 $\mathit{Ps}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)=(\mp\mathrm{i})^{m}% \mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),$
30.6.5 $\mathit{Qs}^{m}_{n}\left(x\pm\mathrm{i}0,\gamma^{2}\right)={(\mp\mathrm{i})^{m% }\left(\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)\mp\tfrac{1}{2}\mathrm{i}% \pi\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)\right)}.$
##### 8: 30.4 Functions of the First Kind
###### §30.4(i) Definitions
If $\gamma=0$, $\mathsf{Ps}^{m}_{n}\left(x,0\right)$ reduces to the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$: …
##### 10: 30.17 Tables
###### §30.17 Tables
• Flammer (1957) includes 18 tables of eigenvalues, expansion coefficients, spheroidal wave functions, and other related quantities. Precision varies between 4S and 10S.

• Zhang and Jin (1996) includes 24 tables of eigenvalues, spheroidal wave functions and their derivatives. Precision varies between 6S and 8S.