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##### 2: 28.20 Definitions and Basic Properties
###### §28.20(iv) Radial Mathieu Functions ${\operatorname{Mc}^{(j)}_{n}}$, ${\operatorname{Ms}^{(j)}_{n}}$
28.20.15 ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)={\operatorname{M}^{(j)}_{n}}% \left(z,h\right),$ $n=0,1,\dots$,
28.20.16 ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)=(-1)^{n}{\operatorname{M}^{(j)}_% {-n}}\left(z,h\right),$ $n=1,2,\dots$.
###### §28.20(vii) Shift of Variable
And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).
##### 3: 30.17 Tables
###### §30.17 Tables
• Hanish et al. (1970) gives $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2$, and their first derivatives, for $0\leq m\leq 2$, $m\leq n\leq m+49$, $-1600\leq\gamma^{2}\leq 1600$. The range of $z$ is given by $1\leq z\leq 10$ if $\gamma^{2}>0$, or $z=-\mathrm{i}\xi$, $0\leq\xi\leq 2$ if $\gamma^{2}<0$. Precision is 18S.

• EraŠevskaja et al. (1973, 1976) gives $S^{m(j)}\left(iy,-ic\right)$, $S^{m(j)}\left(z,\gamma\right)$ and their first derivatives for $j=1,2$, $0.5\leq c\leq 8$, $y=0,0.5,1,1.5$, $0.5\leq\gamma\leq 8$, $z=1.01,1.1,1.4,1.8$. Precision is 15S.

##### 6: 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 $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2,3,4$. … 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 …
##### 7: 30.18 Software
• SWF4: $S^{m(j)}_{n}\left(z,\gamma\right)$, $j=1,2$.

• ##### 8: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
33.5.6 $C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
33.5.9 $C_{\ell}\left(\eta\right)\sim\dfrac{e^{-\pi\eta/2}}{(2\ell+1)!!}\sim e^{-\pi% \eta/2}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{\ell+1}}.$
##### 9: 33.23 Methods of Computation
Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … Inside the turning points, that is, when $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$, there can be a loss of precision by a factor of approximately $|G_{\ell}|^{2}$. … Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. … WKBJ approximations (§2.7(iii)) for $\rho>\rho_{\operatorname{tp}}\left(\eta,\ell\right)$ are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for $F_{0}$ and $G_{0}$ in the region inside the turning point: $\rho<\rho_{\operatorname{tp}}\left(\eta,\ell\right)$.
##### 10: 30.14 Wave Equation in Oblate Spheroidal Coordinates
30.14.8 $w_{1}(\xi)=a_{1}S^{m(1)}_{n}\left(i\xi,\gamma\right)+b_{1}S^{m(2)}_{n}\left(i% \xi,\gamma\right).$
###### §30.14(v) The Interior Dirichlet Problem for Oblate Ellipsoids
30.14.9 $S^{m(1)}_{n}\left(\mathrm{i}\xi_{0},\gamma\right)=0.$