§30.3 Eigenvalues

With $\mu =m=0,1,2,\mathrm{\dots }$, the spheroidal wave functions ${\mathsf{Ps}}_n^m(x,{\gamma }^2)$ 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 ${\lambda }_n^m\left({\gamma }^2\right)$, $n=m,m+1,m+2,\mathrm{\dots }$, of the parameter $\lambda$.

The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy

30.3.1
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.3.E1 Encodings: TeX, pMML, png See also: Annotations for §30.3(ii), §30.3 and Ch.30

30.3.2		$${\lambda }_n^m\left({\gamma }^2\right)=n(n+1)-\frac{1}{2}{\gamma }^2+O\left(n^{-2}\right),$$
$n\to \mathrm{\infty }$,
ⓘ Symbols: $O\left(x\right)$: order not exceeding, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.3.E2 Encodings: TeX, pMML, png See also: Annotations for §30.3(ii), §30.3 and Ch.30

30.3.3		$${\lambda }_n^m\left(0\right)=n(n+1),$$
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer and $n\ge m$: integer degree Permalink: http://dlmf.nist.gov/30.3.E3 Encodings: TeX, pMML, png See also: Annotations for §30.3(ii), §30.3 and Ch.30

30.3.4
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter Permalink: http://dlmf.nist.gov/30.3.E4 Encodings: TeX, pMML, png See also: Annotations for §30.3(ii), §30.3 and Ch.30

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

30.3.5		$${\beta }_p-\lambda -\frac{{\alpha }_{p-2}{\gamma }_p}{{\beta }_{p-2}-\lambda -}\frac{{\alpha }_{p-4}{\gamma }_{p-2}}{{\beta }_{p-4}-\lambda -}\mathrm{\cdots }=\frac{{\alpha }_p{\gamma }_{p+2}}{{\beta }_{p+2}-\lambda -}\frac{{\alpha }_{p+2}{\gamma }_{p+4}}{{\beta }_{p+4}-\lambda -}\mathrm{\cdots },$$
ⓘ 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 See also: Annotations for §30.3(iii), §30.3 and Ch.30

where ${\alpha }_k$, ${\beta }_k$, ${\gamma }_k$ are defined by

30.3.6		${\alpha }_k$	$=-(k+1)(k+2),$
		${\beta }_k$	$=(m+k)(m+k+1)-{\gamma }^2,$
		${\gamma }_k$	$={\gamma }^2,$
ⓘ 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 See also: Annotations for §30.3(iii), §30.3 and Ch.30

has the solutions $\lambda ={\lambda }_{m+2j}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. If $p$ is an odd positive integer, then Equation (30.3.5) has the solutions $\lambda ={\lambda }_{m+2j+1}^m\left({\gamma }^2\right)$, $j=0,1,2,\mathrm{\dots }$. 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		${\alpha }_k$	$={\gamma }^2{\displaystyle \frac{(k+2m+1)(k+2m+2)}{(2k+2m+3)(2k+2m+5)}},$
		${\beta }_k$	$=(k+m)(k+m+1)-2{\gamma }^2{\displaystyle \frac{(k+m)(k+m+1)-1+m^2}{(2k+2m-1)(2k+2m+3)}},$
		${\gamma }_k$	$={\gamma }^2{\displaystyle \frac{(k-1)k}{(2k+2m-3)(2k+2m-1)}}.$
ⓘ Symbols: $m$: nonnegative integer, ${\gamma }^2$: real parameter, ${\alpha }_k$: coefficent, ${\beta }_k$: coefficent and ${\gamma }_k$: coefficent A&S Ref: 21.7.4 Permalink: http://dlmf.nist.gov/30.3.E7 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §30.3(iii), §30.3 and Ch.30

30.3.8		$${\lambda }_n^m\left({\gamma }^2\right)=\sum _{k=0}^{\mathrm{\infty }}{\mathrm{\ell }}_{2k}{\gamma }^{2k},$$
.
ⓘ Symbols: ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter and ${\mathrm{\ell }}_j$: coefficients Referenced by: §30.16(i), item Permalink: http://dlmf.nist.gov/30.3.E8 Encodings: TeX, pMML, png See also: Annotations for §30.3(iv), §30.3 and Ch.30

For values of $r_n^m$ see Meixner et al. (1980, p. 109).

30.3.9		${\mathrm{\ell }}_0$	$=n(n+1),$
		$2{\mathrm{\ell }}_2$	$=-1-{\displaystyle \frac{(2m-1)(2m+1)}{(2n-1)(2n+3)}},$
		$2{\mathrm{\ell }}_4$	$={\displaystyle \frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-3){(2n-1)}^3(2n+1)}}-{\displaystyle \frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n+1){(2n+3)}^3(2n+5)}}.$
ⓘ Symbols: $m$: nonnegative integer, $n\ge m$: integer degree and ${\mathrm{\ell }}_j$: 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 See also: Annotations for §30.3(iv), §30.3 and Ch.30

30.3.10		$${\mathrm{\ell }}_6=(4m^2-1)(\begin{array}{l}\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)(2n+1){(2n+3)}^5(2n+5)(2n+7)}\\ -\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)(2n-3){(2n-1)}^5(2n+1)(2n+3)}),\end{array}$$
ⓘ Symbols: $m$: nonnegative integer, $n\ge m$: integer degree and ${\mathrm{\ell }}_j$: coefficients Permalink: http://dlmf.nist.gov/30.3.E10 Encodings: TeX, pMML, png See also: Annotations for §30.3(iv), §30.3 and Ch.30

30.3.11		$${\mathrm{\ell }}_8=2{(4m^2-1)}^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,$$
ⓘ Symbols: $m$: nonnegative integer, ${\mathrm{\ell }}_j$: coefficients, $A$, $B$, $C$ and $D$ Permalink: http://dlmf.nist.gov/30.3.E11 Encodings: TeX, pMML, png See also: Annotations for §30.3(iv), §30.3 and Ch.30

30.3.12		$A$	$=\begin{array}{l}{\displaystyle \frac{(n-m-1)(n-m)(n+m-1)(n+m)}{{(2n-5)}^2(2n-3){(2n-1)}^7(2n+1){(2n+3)}^2}}\\ -{\displaystyle \frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{{(2n-1)}^2(2n+1){(2n+3)}^7(2n+5){(2n+7)}^2}},\end{array}$
		$B$	$=\begin{array}{l}{\displaystyle \frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-7){(2n-5)}^2{(2n-3)}^3{(2n-1)}^4(2n+1)}}\\ -{\displaystyle \frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n+m+3)(n+m+4)}{(2n+1){(2n+3)}^4{(2n+5)}^3{(2n+7)}^2(2n+9)}},\end{array}$
		$C$	$={\displaystyle \frac{{(n-m+1)}^2{(n-m+2)}^2{(n+m+1)}^2{(n+m+2)}^2}{{(2n+1)}^2{(2n+3)}^7{(2n+5)}^2}}-{\displaystyle \frac{{(n-m-1)}^2{(n-m)}^2{(n+m-1)}^2{(n+m)}^2}{{(2n-3)}^2{(2n-1)}^7{(2n+1)}^2}},$
		$D$	$={\displaystyle \frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-3){(2n-1)}^4{(2n+1)}^2{(2n+3)}^4(2n+5)}}.$
ⓘ Defines: $A$ (locally), $B$ (locally), $C$ (locally) and $D$ (locally) Symbols: $m$: nonnegative integer and $n\ge m$: integer degree Permalink: http://dlmf.nist.gov/30.3.E12 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §30.3(iv), §30.3 and Ch.30

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