§30.3 Eigenvalues

Keywords:: spheroidal differential equation, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.3

§30.3(i) Definition
§30.3(ii) Properties
§30.3(iii) Transcendental Equation
§30.3(iv) Power-Series Expansion

§30.3(i) Definition

Notes:: See Meixner and Schäfke (1954, §3.23).
Defines:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation
Permalink:: http://dlmf.nist.gov/30.3.i

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

Notes:: See Meixner and Schäfke (1954, §3.2).
Permalink:: http://dlmf.nist.gov/30.3.ii

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 $\mathop{\lambda^{{m}}_{{m}}\/}\nolimits\!\left(\gamma^{2}\right)<\mathop{\lambda^{{m}}_{{m+1}}\/}\nolimits\!\left(\gamma^{2}\right)<\mathop{\lambda^{{m}}_{{m+2}}\/}\nolimits\!\left(\gamma^{2}\right)<\cdots,$

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : 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

30.3.2 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)=n(n+1)-\tfrac{1}{2}\gamma^{2}+\mathop{O\/}\nolimits\!\left(n^{{-2}}\right),$ $n\to\infty$ ,

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.3.E2
Encodings:: TeX, pMML, png

30.3.3 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(0\right)=n(n+1),$

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer and $n\geq m$ : integer degree
Permalink:: http://dlmf.nist.gov/30.3.E3
Encodings:: TeX, pMML, png

30.3.4 $-1<\frac{d\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)}{d(\gamma^{2})}<0.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree and $\gamma^{2}$ : real parameter
Permalink:: http://dlmf.nist.gov/30.3.E4
Encodings:: TeX, pMML, png

§30.3(iii) Transcendental Equation

Notes:: See Meixner and Schäfke (1954, §3.24).
Keywords:: spheroidal differential equation
Referenced by:: §30.16(i)
Permalink:: http://dlmf.nist.gov/30.3.iii

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

$\alpha _{k}=-(k+1)(k+2),$

$\beta _{k}=(m+k)(m+k+1)-\gamma^{2},$

$\gamma _{k}=\gamma^{2},$

Symbols:: $m$ : nonnegative integer, $\gamma^{2}$ : real parameter, $\alpha _{k}$ : coefficent, $\beta _{k}$ : coefficent and $\gamma _{k}$ : coefficent
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

$\alpha _{k}=\gamma^{2}\frac{(k+2m+1)(k+2m+2)}{(2k+2m+3)(2k+2m+5)},$

$\beta _{k}=(k+m)(k+m+1)-2\gamma^{2}\frac{(k+m)(k+m+1)-1+m^{2}}{(2k+2m-1)(2k+2m+3)},$

$\gamma _{k}=\gamma^{2}\frac{(k-1)k}{(2k+2m-3)(2k+2m-1)}.$

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

§30.3(iv) Power-Series Expansion

Notes:: See Meixner and Schäfke (1954, §3.531).
Keywords:: spheroidal differential equation, spheroidal wave functions
Permalink:: http://dlmf.nist.gov/30.3.iv

30.3.8 $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)=\sum _{{k=0}}^{{\infty}}\ell _{{2k}}\gamma^{{2k}},$ $|\gamma^{2}|<r_{n}^{m}$ .

Symbols:: $\mathop{\lambda^{{m}}_{{n}}\/}\nolimits\!\left(\gamma^{2}\right)$ : eigenvalues of the spheroidal differential equation, $m$ : nonnegative integer, $n\geq m$ : integer degree, $\gamma^{2}$ : real parameter and $\ell _{j}$ : coefficients
Referenced by:: §30.16(i), §30.18(i)
Permalink:: http://dlmf.nist.gov/30.3.E8
Encodings:: TeX, pMML, png

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

30.3.9

$\ell _{0}=n(n+1),$

$2\ell _{2}=-1-\frac{(2m-1)(2m+1)}{(2n-1)(2n+3)},$

$2\ell _{4}=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-3)(2n-1)^{3}(2n+1)}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n+1)(2n+3)^{3}(2n+5)}.$

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 $\ell _{6}=(4m^{2}-1)\left(\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)(2n+1)(2n+3)^{5}(2n+5)(2n+7)}-\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)(2n-3)(2n-1)^{5}(2n+1)(2n+3)}\right),$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree and $\ell _{j}$ : coefficients
Permalink:: http://dlmf.nist.gov/30.3.E10
Encodings:: TeX, pMML, png

30.3.11 $\ell _{8}=2(4m^{2}-1)^{2}A+\frac{1}{16}B+\frac{1}{8}C+\frac{1}{2}D,$

Symbols:: $m$ : nonnegative integer, $\ell _{j}$ : coefficients, $A$ , $B$ , $C$ and $D$
Permalink:: http://dlmf.nist.gov/30.3.E11
Encodings:: TeX, pMML, png

30.3.12

$A=\frac{(n-m-1)(n-m)(n+m-1)(n+m)}{(2n-5)^{2}(2n-3)(2n-1)^{7}(2n+1)(2n+3)^{2}}-\frac{(n-m+1)(n-m+2)(n+m+1)(n+m+2)}{(2n-1)^{2}(2n+1)(2n+3)^{7}(2n+5)(2n+7)^{2}},$

$B=\frac{(n-m-3)(n-m-2)(n-m-1)(n-m)(n+m-3)(n+m-2)(n+m-1)(n+m)}{(2n-7)(2n-5)^{2}(2n-3)^{3}(2n-1)^{4}(2n+1)}-\frac{(n-m+1)(n-m+2)(n-m+3)(n-m+4)(n+m+1)(n+m+2)(n+m+3)(n+m+4)}{(2n+1)(2n+3)^{4}(2n+5)^{3}(2n+7)^{2}(2n+9)},$

$C=\frac{(n-m+1)^{2}(n-m+2)^{2}(n+m+1)^{2}(n+m+2)^{2}}{(2n+1)^{2}(2n+3)^{7}(2n+5)^{2}}-\frac{(n-m-1)^{2}(n-m)^{2}(n+m-1)^{2}(n+m)^{2}}{(2n-3)^{2}(2n-1)^{7}(2n+1)^{2}},$

$D=\frac{(n-m-1)(n-m)(n-m+1)(n-m+2)(n+m-1)(n+m)(n+m+1)(n+m+2)}{(2n-3)(2n-1)^{4}(2n+1)^{2}(2n+3)^{4}(2n+5)}.$

Symbols:: $m$ : nonnegative integer, $n\geq m$ : integer degree, $A$ , $B$ , $C$ and $D$
Permalink:: http://dlmf.nist.gov/30.3.E12
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

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

§30.3 Eigenvalues

Contents

§30.3(i) Definition

§30.3(ii) Properties

§30.3(iii) Transcendental Equation

§30.3(iv) Power-Series Expansion