# §30.3 Eigenvalues

## §30.3(i) Definition

With $\mu=m=0,1,2,\dots$, the spheroidal wave functions $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$ 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 $z$. These solutions exist only for eigenvalues $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$, $n=m,m+1,m+2,\dots$, of the parameter $\lambda$.

## §30.3(ii) Properties

The eigenvalues $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$ are analytic functions of the real variable $\gamma^{2}$ and satisfy

 30.3.1 $\mathop{\lambda^{m}_{m}\/}\nolimits\!\left(\gamma^{2}\right)<\mathop{\lambda^{% m}_{m+1}\/}\nolimits\!\left(\gamma^{2}\right)<\mathop{\lambda^{m}_{m+2}\/}% \nolimits\!\left(\gamma^{2}\right)<\cdots,$
 30.3.2 $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)=n(n+1)-\tfrac{1}{% 2}\gamma^{2}+\mathop{O\/}\nolimits\!\left(n^{-2}\right),$ $n\to\infty$,
 30.3.3 $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(0\right)=n(n+1),$
 30.3.4 $-1<\frac{d\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)}{d(% \gamma^{2})}<0.$

## §30.3(iii) Transcendental Equation

If $p$ is an even nonnegative integer, then the continued-fraction equation

 30.3.5 $\beta_{p}-\lambda-\cfrac{\alpha_{p-2}\gamma_{p}}{\beta_{p-2}-\lambda-\cfrac{% \alpha_{p-4}\gamma_{p-2}}{\beta_{p-4}-\lambda-\cdots}}=\cfrac{\alpha_{p}\gamma% _{p+2}}{\beta_{p+2}-\lambda-\cfrac{\alpha_{p+2}\gamma_{p+4}}{\beta_{p+4}-% \lambda-\cdots}},$ Symbols: $\alpha_{k}$: coefficent, $\beta_{k}$: coefficent and $\gamma_{k}$: coefficent A&S Ref: 21.7.4 Referenced by: §30.3(iii), §30.3(iii) Permalink: http://dlmf.nist.gov/30.3.E5 Encodings: TeX, pMML, png

where $\alpha_{k}$, $\beta_{k}$, $\gamma_{k}$ are defined by

 30.3.6 $\displaystyle\alpha_{k}$ $\displaystyle=-(k+1)(k+2),$ $\displaystyle\beta_{k}$ $\displaystyle=(m+k)(m+k+1)-\gamma^{2},$ $\displaystyle\gamma_{k}$ $\displaystyle=\gamma^{2},$ Defines: $\alpha_{k}$: coefficent (locally), $\beta_{k}$: coefficent (locally) and $\gamma_{k}$: coefficent (locally) Symbols: $m$: nonnegative integer and $\gamma^{2}$: real parameter Referenced by: §30.4(iii) Permalink: http://dlmf.nist.gov/30.3.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

has the solutions $\lambda=\mathop{\lambda^{m}_{m+2j}\/}\nolimits\!\left(\gamma^{2}\right)$, $j=0,1,2,\dots$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda=\mathop{\lambda^{m}_{m+2j+1}\/}\nolimits\!\left(\gamma^{2}\right)$, $j=0,1,2,\dots$. If $p=0$ or $p=1$, the finite continued-fraction on the left-hand side of (30.3.5) equals 0; if $p>1$ its last denominator is $\beta_{0}-\lambda$ or $\beta_{1}-\lambda$.

In equation (30.3.5) we can also use

 30.3.7 $\displaystyle\alpha_{k}$ $\displaystyle=\gamma^{2}\frac{(k+2m+1)(k+2m+2)}{(2k+2m+3)(2k+2m+5)},$ $\displaystyle\beta_{k}$ $\displaystyle=(k+m)(k+m+1)-2\gamma^{2}\frac{(k+m)(k+m+1)-1+m^{2}}{(2k+2m-1)(2k% +2m+3)},$ $\displaystyle\gamma_{k}$ $\displaystyle=\gamma^{2}\frac{(k-1)k}{(2k+2m-3)(2k+2m-1)}.$

## §30.3(iv) Power-Series Expansion

 30.3.8 $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)=\sum_{k=0}^{% \infty}\ell_{2k}\gamma^{2k},$ $|\gamma^{2}|.

For values of $r_{n}^{m}$ see Meixner et al. (1980, p. 109).

 30.3.9 $\displaystyle\ell_{0}$ $\displaystyle=n(n+1),$ $\displaystyle 2\ell_{2}$ $\displaystyle=-1-\frac{(2m-1)(2m+1)}{(2n-1)(2n+3)},$ $\displaystyle 2\ell_{4}$ $\displaystyle=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-3)(2n-1)^{3}(2n+1)}-\frac{(n% -m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n+1)(2n+3)^{3}(2n+5)}.$ Symbols: $m$: nonnegative integer, $n\geq m$: integer degree and $\ell_{j}$: coefficients A&S Ref: 21.7.5 Permalink: http://dlmf.nist.gov/30.3.E9 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
 30.3.10 $\ell_{6}=(4m^{2}-1)\left(\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)(2n+1)(2n+3% )^{5}(2n+5)(2n+7)}-\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)(2n-3)(2n-1)^{5}(2n+1% )(2n+3)}\right),$
 30.3.11 $\ell_{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,$
 30.3.12 $\displaystyle A$ $\displaystyle=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)^{2}(2n-3)(2n-1)^{7}(2n+1)% (2n+3)^{2}}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)^{2}(2n+1)(2n+3)^{7}(2n+% 5)(2n+7)^{2}},$ $\displaystyle B$ $\displaystyle=\frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-% 7)(2n-5)^{2}(2n-3)^{3}(2n-1)^{4}(2n+1)}-\frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m% +1)(n+m+2)(n+m+3)(n+m+4)}{(2n+1)(2n+3)^{4}(2n+5)^{3}(2n+7)^{2}(2n+9)},$ $\displaystyle C$ $\displaystyle=\frac{(n-m+1)^{2}(n-m+2)^{2}(n+m+1)^{2}(n+m+2)^{2}}{(2n+1)^{2}(2% n+3)^{7}(2n+5)^{2}}-\frac{(n-m-1)^{2}(n-m)^{2}(n+m-1)^{2}(n+m)^{2}}{(2n-3)^{2}% (2n-1)^{7}(2n+1)^{2}},$ $\displaystyle D$ $\displaystyle=\frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-% 3)(2n-1)^{4}(2n+1)^{2}(2n+3)^{4}(2n+5)}.$

Further coefficients can be found with the Maple program SWF9; see §30.18(i).