29 Lamé Functions Lamé Functions 29.2 Differential Equations 29.4 Graphics

§29.3 Definitions and Basic Properties

Permalink:: http://dlmf.nist.gov/29.3

§29.3(i) Eigenvalues
§29.3(ii) Distribution
§29.3(iii) Continued Fractions
§29.3(iv) Lamé Functions
§29.3(v) Normalization
§29.3(vi) Orthogonality
§29.3(vii) Power Series

§29.3(i) Eigenvalues

Notes:: See Erdélyi et al. (1955, §15.5.1) and Magnus and Winkler (1966, §2.1).
Defines:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation and $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation
Keywords:: Lamé functions
Referenced by:: §28.34(ii), §29.3(iv)
Permalink:: http://dlmf.nist.gov/29.3.i

For each pair of values of $\nu$ and there are four infinite unbounded sets of real eigenvalues for which equation (29.2.1) has even or odd solutions with periods $2\!\mathop{K\/}\nolimits\!$ or $4\!\mathop{K\/}\nolimits\!$ . They are denoted by $\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , $\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ , where $m=0,1,2,\ldots$ ; see Table 29.3.1.

Table 29.3.1: Eigenvalues of Lamé’s equation.

eigenvalue	parity	period
$\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	even	$2\!\mathop{K\/}\nolimits\!$
$\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	odd	$4\!\mathop{K\/}\nolimits\!$
$\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	even	$4\!\mathop{K\/}\nolimits\!$
$\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$	odd	$2\!\mathop{K\/}\nolimits\!$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : nonnegative integer, : real parameter, : real parameter and $\nu$ : real parameter
Keywords:: Lamé functions
Referenced by:: §29.3(i)
Permalink:: http://dlmf.nist.gov/29.3.T1

§29.3(ii) Distribution

Notes:: See Erdélyi (1941b) and Erdélyi et al. (1955, §15.5.1). For (29.3.6) see Volkmer (2004b).
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.ii

The eigenvalues interlace according to

29.3.1 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{a^{{m+1}}_{{% \nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E1
Encodings:: TeX, pMML, png

29.3.2 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{b^{{m+1}}_{{% \nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E2
Encodings:: TeX, pMML, png

29.3.3 $\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{b^{{m+1}}_{{% \nu}}\/}\nolimits\!\left(k^{2}\right),$

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E3
Encodings:: TeX, pMML, png

29.3.4 $\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)<\mathop{a^{{m+1}}_{{% \nu}}\/}\nolimits\!\left(k^{2}\right).$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E4
Encodings:: TeX, pMML, png

The eigenvalues coalesce according to

29.3.5 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)=\mathop{b^{{m}}_{{\nu}% }\/}\nolimits\!\left(k^{2}\right),$ $\nu=0,1,\dots,m-1$ .

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E5
Encodings:: TeX, pMML, png

If $\nu$ is distinct from $0,1,\dots,m-1$ , then

29.3.6 $\left(\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\mathop{b^{{m}}_% {{\nu}}\/}\nolimits\!\left(k^{2}\right)\right)\nu(\nu-1)\cdots(\nu-m+1)>0.$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Referenced by:: §29.3(ii)
Permalink:: http://dlmf.nist.gov/29.3.E6
Encodings:: TeX, pMML, png

If $\nu$ is a nonnegative integer, then

29.3.7 $\mathop{a^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)+\mathop{a^{{\nu-m}}_{{% \nu}}\/}\nolimits\!\left(1-k^{2}\right)=\nu(\nu+1),$ $m=0,1,\dots,\nu$ ,

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E7
Encodings:: TeX, pMML, png

29.3.8 ${\mathop{b^{{m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)+\mathop{b^{{\nu-m+1}}% _{{\nu}}\/}\nolimits\!\left(1-k^{2}\right)=\nu(\nu+1)},$ $m=1,2,\dots,\nu$ .

Symbols:: $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter and $\nu$ : real parameter
Permalink:: http://dlmf.nist.gov/29.3.E8
Encodings:: TeX, pMML, png

For the special case $k=k^{{\prime}}=\ifrac{1}{\sqrt{2}}$ see Erdélyi et al. (1955, §15.5.2).

§29.3(iii) Continued Fractions

Notes:: See Ince (1940b).
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.iii

The quantity

29.3.9 $H=2\!\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$

Symbols:: $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, : nonnegative integer, : real parameter, $\nu$ : real parameter and
Referenced by:: §29.3(iii)
Permalink:: http://dlmf.nist.gov/29.3.E9
Encodings:: TeX, pMML, png

satisfies the continued-fraction equation

29.3.10 $\beta_{p}-H-\cfrac{\alpha_{{p-1}}\gamma_{p}}{\beta_{{p-1}}-H-\cfrac{\alpha_{{p% -2}}\gamma_{{p-1}}}{\beta_{{p-2}}-H-}}\dots=\cfrac{\alpha_{p}\gamma_{{p+1}}}{% \beta_{{p+1}}-H-\cfrac{\alpha_{{p+1}}\gamma_{{p+2}}}{\beta_{{p+2}}-H-}}\cdots,$

Symbols:: : nonnegative integer, , $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: §29.20(i), §29.3(iii), §29.3(iii), §29.3(iii), §29.3(iii)
Permalink:: http://dlmf.nist.gov/29.3.E10
Encodings:: TeX, pMML, png

where is any nonnegative integer, and

29.3.11 $\alpha_{p}=\begin{cases}(\nu-1)(\nu+2)k^{2},&p=0,\\ \tfrac{1}{2}(\nu-2p-1)(\nu+2p+2)k^{2},&p\geq 1,\end{cases}$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter and $\alpha_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.3.E11
Encodings:: TeX, pMML, png

29.3.12

$\beta_{p}=4p^{2}(2-k^{2}),$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p+2)(\nu+2p-1)k^{2}.$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\beta_{p}$ and $\gamma_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(i)
Permalink:: http://dlmf.nist.gov/29.3.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

The continued fraction following the second negative sign on the left-hand side of (29.3.10) is finite: it equals 0 if , and if , then the last denominator is $\beta_{0}-H$ . If $\nu$ is a nonnegative integer and $2p\leq\nu$ , then the continued fraction on the right-hand side of (29.3.10) terminates, and (29.3.10) has only the solutions (29.3.9) with $2m\leq\nu$ . If $\nu$ is a nonnegative integer and $2p>\nu$ , then (29.3.10) has only the solutions (29.3.9) with $2m>\nu$ .

The quantity $H=2\!\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.13 $\beta_{p}=\begin{cases}2-k^{2}+\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter and $\beta_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(ii)
Permalink:: http://dlmf.nist.gov/29.3.E13
Encodings:: TeX, pMML, png

29.3.14

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ and $\gamma_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(ii)
Permalink:: http://dlmf.nist.gov/29.3.E14
Encodings:: TeX, TeX, pMML, pMML, png, png

The quantity $H=2\!\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.15 $\beta_{p}=\begin{cases}2-k^{2}-\tfrac{1}{2}\nu(\nu+1)k^{2},&p=0,\\ (2p+1)^{2}(2-k^{2}),&p\geq 1,\end{cases}$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter and $\beta_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iii)
Permalink:: http://dlmf.nist.gov/29.3.E15
Encodings:: TeX, pMML, png

29.3.16

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-2)(\nu+2p+3)k^{2},$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p+1)(\nu+2p)k^{2}.$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ and $\gamma_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iii)
Permalink:: http://dlmf.nist.gov/29.3.E16
Encodings:: TeX, TeX, pMML, pMML, png, png

The quantity $H=2\!\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)-\nu(\nu+1)k^{2}$ satisfies equation (29.3.10) with

29.3.17

$\alpha_{p}=\tfrac{1}{2}(\nu-2p-3)(\nu+2p+4)k^{2},$

$\beta_{p}=(2p+2)^{2}(2-k^{2}),$

$\gamma_{p}=\tfrac{1}{2}(\nu-2p)(\nu+2p+1)k^{2}.$

Symbols:: : nonnegative integer, : real parameter, $\nu$ : real parameter, $\alpha_{p}$ , $\beta_{p}$ and $\gamma_{p}$
Referenced by:: ¶ ‣ §29.15(i), §29.6(iv)
Permalink:: http://dlmf.nist.gov/29.3.E17
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

§29.3(iv) Lamé Functions

Notes:: See Erdélyi et al. (1955, §15.5.1).
Defines:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function and $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.iv

The eigenfunctions corresponding to the eigenvalues of §29.3(i) are denoted by $\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ , $\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ . They are called Lamé functions with real periods and of order $\nu$ , or more simply, Lamé functions. See Table 29.3.2. In this table the nonnegative integer corresponds to the number of zeros of each Lamé function in $(0,\!\mathop{K\/}\nolimits\!)$ , whereas the superscripts , , or correspond to the number of zeros in $[0,2\!\mathop{K\/}\nolimits\!)$ .

Table 29.3.2: Lamé functions.

boundary conditions

eigenvalue

eigenfunction

parity of

$w(z-\!\mathop{K\/}\nolimits\!)$

period of

$\left.\ifrac{dw}{dz}\right|_{{z=0}}=\left.\ifrac{dw}{dz}\right|_{{z=\!\mathop{% K\/}\nolimits\!}}=0$

$\mathop{a^{{2m}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

even

$2\!\mathop{K\/}\nolimits\!$

$w(0)=\left.\ifrac{dw}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}=0$

$\mathop{a^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

odd

even

$4\!\mathop{K\/}\nolimits\!$

$\left.\ifrac{dw}{dz}\right|_{{z=0}}=w(\!\mathop{K\/}\nolimits\!)=0$

$\mathop{b^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

even

odd

$4\!\mathop{K\/}\nolimits\!$

$w(0)=w(\!\mathop{K\/}\nolimits\!)=0$

$\mathop{b^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$

$\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$

odd

$2\!\mathop{K\/}\nolimits\!$

Symbols:: $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}\right)$ : Lamé function, $\mathop{a^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{b^{{n}}_{{\nu}}\/}\nolimits\!\left(k^{2}\right)$ : eigenvalues of Lamé’s equation, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : complex variable, : real parameter, : real parameter, $\nu$ : real parameter and : eigenfunction
Keywords:: Lamé functions, Lamé’s equation
Referenced by:: §29.3(iv)
Permalink:: http://dlmf.nist.gov/29.3.T2

§29.3(v) Normalization

Notes:: See Jansen (1977, §3.1).
Keywords:: Lamé functions
Referenced by:: §29.20(i)
Permalink:: http://dlmf.nist.gov/29.3.v

29.3.18

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k% \right)\left(\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k% \right)\left(\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k% \right)\left(\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx=\frac{1}{4}\pi,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathrm{dn}\/}\nolimits\left(x,k% \right)\left(\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}\nolimits\!\left(x,k^{2}% \right)\right)^{2}dx=\frac{1}{4}\pi.$

For $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ see §22.2.

To complete the definitions, $\mathop{\mathit{Ec}^{{m}}_{{\nu}}\/}\nolimits\!\left(\!\mathop{K\/}\nolimits\!% ,k^{2}\right)$ is positive and $\left.\ifrac{d\mathop{\mathit{Es}^{{m}}_{{\nu}}\/}\nolimits\!\left(z,k^{2}% \right)}{dz}\right|_{{z=\!\mathop{K\/}\nolimits\!}}$ is negative.

§29.3(vi) Orthogonality

Notes:: Combine Hochstadt (1964, p. 148) and Table 29.3.2.
Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.vi

For $m\neq p$ ,

29.3.19

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Ec}^{{2m}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{{2p}}_{{\nu}}\/}\nolimits% \!\left(x,k^{2}\right)dx=0,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Ec}^{{2m+1}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Ec}^{{2p+1}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)dx=0,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Es}^{{2m+1}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{{2p+1}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)dx=0,$

$\int_{0}^{{\!\mathop{K\/}\nolimits\!}}\mathop{\mathit{Es}^{{2m+2}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)\mathop{\mathit{Es}^{{2p+2}}_{{\nu}}\/}% \nolimits\!\left(x,k^{2}\right)dx=0.$

For the values of these integrals when see §29.6.

§29.3(vii) Power Series

Keywords:: Lamé functions
Permalink:: http://dlmf.nist.gov/29.3.vii

For power-series expansions of the eigenvalues see Volkmer (2004b).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 29.2 Differential Equations 29.4 Graphics

§29.3 Definitions and Basic Properties

Contents

§29.3(i) Eigenvalues

§29.3(ii) Distribution

§29.3(iii) Continued Fractions

§29.3(iv) Lamé Functions

§29.3(v) Normalization

§29.3(vi) Orthogonality

§29.3(vii) Power Series