# §30.16(i) Eigenvalues

For small $|\gamma^{2}|$ we can use the power-series expansion (30.3.8). Schäfke and Groh (1962) gives corresponding error bounds. If $|\gamma^{2}|$ is large we can use the asymptotic expansions in §30.9. Approximations to eigenvalues can be improved by using the continued-fraction equations from §30.3(iii) and §30.8; see Bouwkamp (1947) and Meixner and Schäfke (1954, §3.93).

Another method is as follows. Let $n-m$ be even. For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $\mathbf{A}=[A_{j,k}]$ with nonzero elements

 30.16.1 $\displaystyle A_{j,j}$ $\displaystyle=(m+2j-2)(m+2j-1)-2\gamma^{2}\frac{(m+2j-2)(m+2j-1)-1+m^{2}}{(2m+% 4j-5)(2m+4j-1)},$ $\displaystyle A_{j,j+1}$ $\displaystyle=-\gamma^{2}\frac{(2m+2j-1)(2m+2j)}{(2m+4j-1)(2m+4j+1)},$ $\displaystyle A_{j,j-1}$ $\displaystyle=-\gamma^{2}\frac{(2j-3)(2j-2)}{(2m+4j-7)(2m+4j-5)},$ Symbols: $m$: nonnegative integer, $A_{j,k}$: tridiagonal matrix elements and $\gamma^{2}$: real parameter Referenced by: ¶ ‣ §30.16(i), §30.16(ii) Permalink: http://dlmf.nist.gov/30.16.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

and real eigenvalues $\alpha_{1,d}$, $\alpha_{2,d}$, $\dots$, $\alpha_{d,d}$, arranged in ascending order of magnitude. Then

 30.16.2 $\alpha_{j,d+1}\leq\alpha_{j,d},$ Symbols: $d$: dimension and $\alpha_{j,k}$: real eigenvalues Permalink: http://dlmf.nist.gov/30.16.E2 Encodings: TeX, pMML, png

and

 30.16.3 $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)=\lim_{d\to\infty}% \alpha_{p,d},$ $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$.

The eigenvalues of $\mathbf{A}$ can be computed by methods indicated in §§3.2(vi), 3.2(vii). The error satisfies

 30.16.4 $\alpha_{p,d}-\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)=% \mathop{O\/}\nolimits\!\left(\frac{\gamma^{4d}}{4^{2d+1}((m+2d-1)!(m+2d+1)!)^{% 2}}\right),$ $d\to\infty$.

# ¶ Example

For $m=2$, $n=4$, $\gamma^{2}=10$,

 30.16.5 $\displaystyle\alpha_{2,2}$ $\displaystyle=14.18833\;246,$ $\displaystyle\alpha_{2,3}$ $\displaystyle=13.98002\;013,$ $\displaystyle\alpha_{2,4}$ $\displaystyle=13.97907\;459,$ $\displaystyle\alpha_{2,5}$ $\displaystyle=13.97907\;345,$ $\displaystyle\alpha_{2,6}$ $\displaystyle=13.97907\;345,$ Symbols: $\alpha_{j,k}$: real eigenvalues Permalink: http://dlmf.nist.gov/30.16.E5 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

which yields $\mathop{\lambda^{2}_{4}\/}\nolimits\!\left(10\right)=13.97907\;345$. If $n-m$ is odd, then (30.16.1) is replaced by

 30.16.6 $\displaystyle A_{j,j}$ $\displaystyle=(m+2j-1)(m+2j)-2\gamma^{2}\*\frac{(m+2j-1)(m+2j)-1+m^{2}}{(2m+4j% -3)(2m+4j+1)},$ $\displaystyle A_{j,j+1}$ $\displaystyle=-\gamma^{2}\frac{(2m+2j)(2m+2j+1)}{(2m+4j+1)(2m+4j+3)},$ $\displaystyle A_{j,j-1}$ $\displaystyle=-\gamma^{2}\frac{(2j-2)(2j-1)}{(2m+4j-5)(2m+4j-3)}.$ Symbols: $m$: nonnegative integer, $A_{j,k}$: tridiagonal matrix elements and $\gamma^{2}$: real parameter Referenced by: §30.16(ii) Permalink: http://dlmf.nist.gov/30.16.E6 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

# §30.16(ii) Spheroidal Wave Functions of the First Kind

If $|\gamma^{2}|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$.

If $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$ is known, then we can compute $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$ (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.

If $\mathop{\lambda^{m}_{n}\/}\nolimits\!\left(\gamma^{2}\right)$ is known, then $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$ can be found by summing (30.8.1). The coefficients $a^{m}_{n,r}(\gamma^{2})$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5).

A fourth method, based on the expansion (30.8.1), is as follows. Let $\mathbf{A}$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. Form the eigenvector $[e_{1,d},e_{2,d},\dots,e_{d,d}]^{\mathrm{T}}$ of $\mathbf{A}$ associated with the eigenvalue $\alpha_{p,d}$, $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$, normalized according to

 30.16.7 $\sum_{j=1}^{d}e_{j,d}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}% =\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$

Then

 30.16.8 $a_{n,k}^{m}(\gamma^{2})=\lim_{d\to\infty}e_{k+p,d},$
 30.16.9 $\mathop{\mathsf{Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)=\lim_{d\to% \infty}\sum_{j=1}^{d}(-1)^{j-p}e_{j,d}\mathop{\mathsf{P}^{m}_{n+2(j-p)}\/}% \nolimits\!\left(x\right).$

For error estimates see Volkmer (2004a).

# §30.16(iii) Radial Spheroidal Wave Functions

The coefficients $a_{n,k}^{m}(\gamma^{2})$ calculated in §30.16(ii) can be used to compute $\mathop{S^{m(j)}_{n}\/}\nolimits\!\left(z,\gamma\right)$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_{n}^{m}(\gamma)$ from (30.11.10) and (30.11.11).

For other methods see Van Buren and Boisvert (2002, 2004).