# spheroidal differential equation

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##### 1: 30.2 Differential Equations
###### §30.2(i) SpheroidalDifferentialEquation
The Liouville normal form of equation (30.2.1) is …
##### 2: 30.3 Eigenvalues
###### §30.3 Eigenvalues
With $\mu=m=0,1,2,\dots$, the spheroidal wave functions $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ are solutions of Equation (30.2.1) which are bounded on $(-1,1)$, or equivalently, which are of the form $(1-x^{2})^{\frac{1}{2}m}g(x)$ where $g(z)$ is an entire function of $z$. …
###### §30.3(iv) Power-Series Expansion
Further coefficients can be found with the Maple program SWF9; see §30.18(i).
##### 5: 30.16 Methods of Computation
###### §30.16(i) Eigenvalues
30.16.3 $\lambda^{m}_{n}\left(\gamma^{2}\right)=\lim_{d\to\infty}\alpha_{p,d},$ $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$.
30.16.4 $\alpha_{p,d}-\lambda^{m}_{n}\left(\gamma^{2}\right)=O\left(\frac{\gamma^{4d}}{% 4^{2d+1}((m+2d-1)!(m+2d+1)!)^{2}}\right),$ $d\to\infty$.
If $\lambda^{m}_{n}\left(\gamma^{2}\right)$ is known, then we can compute $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ (not normalized) by solving the differential equation (30.2.1) numerically with initial conditions $w(0)=1$, $w^{\prime}(0)=0$ if $n-m$ is even, or $w(0)=0$, $w^{\prime}(0)=1$ if $n-m$ is odd. …
##### 6: 30.7 Graphics
###### §30.7(i) Eigenvalues Figure 30.7.4: Eigenvalues λ n 10 ⁡ ( γ 2 ) , n = 10 , 11 , 12 , 13 , - 50 ≤ γ 2 ≤ 150 . Magnify
##### 8: 30.8 Expansions in Series of Ferrers Functions
30.8.4 $A_{k}f_{k-1}+\left(B_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right)\right)f_{k}+C_% {k}f_{k+1}=0,$
30.8.8 $\frac{\lambda^{m}_{n}\left(\gamma^{2}\right)-B_{k}}{A_{k}}\frac{a^{m}_{n,k}(% \gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=1+O\left(\frac{1}{k^{4}}\right).$
30.8.10 $A_{-N-1}{a^{\prime}}^{m}_{n,-N-2}(\gamma^{2})+{\left(B_{-N-1}-\lambda^{m}_{n}% \left(\gamma^{2}\right)\right){a^{\prime}}^{m}_{n,-N-1}(\gamma^{2})}+C^{\prime% }a^{m}_{n,-N}(\gamma^{2})=0,$
##### 9: 30.1 Special Notation
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 …
##### 10: 30.4 Functions of the First Kind
30.4.5 $\alpha_{k}g_{k+2}+(\beta_{k}-\lambda^{m}_{n}\left(\gamma^{2}\right))g_{k}+% \gamma_{k}g_{k-2}=0$